Cayley Diagrams of Small Groups

This page gives the Cayley diagrams, also known as Cayley graphs, of all groups of order less than 32. Their presentations are also given. The letters in the presentations correspond to the colours in the Cayley diagrams: black red green blue mauve grey.

Notation

N ⋊ H indicates a semidirect product of N by H. N is the normal subgroup.

QN, DN and DicN denote groups of order N (the quaternion, dihedral and dicyclic groups respectively).

Cyclic groups are denoted by C.

1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | 25 | 26 | 27 | 28 | 29 | 30 | 31

Order		Name	Presentation generators as permutations	Orders of elements Centre Derived subgroup Automorphisms GAP no., name (Sylow subgroup) Schur mulitplier
1	Abelian	1	<>	1 1 1 1/1=1 1, 1 1
2	Abelian	C2	< k \| k² > k=(ab)	1.2 C2 1 1 / 1 = 1 1, C2 1
3	Abelian	C3 ≅ A3	< k \| k³ > k=(abc)	1.3² C3 1 C2 / 1 = C2 1, C3 1
4	Abelian	C4	< k \| k⁴ > k=(abcd)	1.2.4² C4 1 C2 / 1 = C2 1, C4 1
4	Abelian	C2 × C2	< r, g \| r², g², rgrg > r=(ab)(cd) g=(ac)(bd) b=(ad)(bc)	1.2³ C2×C2 1 D6 / 1 = D6 2, C2 x C2 2
5	Abelian	C5	< k \| k⁵ > k=(abcde)	1.5⁴ C5 1 C4 / 1 = C4 1, C5 1
6	Abelian	C6 C3 × C2	< k \| k⁶ > k=(abcdef) < k,r \| k³, r², krk^-1r > k=(abc) r=(de)	1.2.3².6² C6 1 C2 / 1 = C2 2, C6 1
6	Other	D6 ≅S3 ≅ C3 ⋊ C2	< k,r \| k³, r², krkr > k=(abc) r=(bc)	1.2³.3² 1 C3 D6 / D6 = 1 1, S3 1
7	Abelian	C7	< k \| k⁷ > k=(abcdefg)	1.7⁶ C7 1 C6 / 1 = 1 1, C7 1
8	Abelian	C8	< k \| k⁸ > k=(abcdefgh)	1.2.4².8⁴ C8 1 C₂² / 1 = C₂² 1, C8 1
		C4 × C2	< k,r \| k⁴, r², krk^-1r > k=(abcd) r=(ef)	1.2¹⁺².4⁴ C4×C2 1 D8 / 1 = D8 2, C4 x C2 2
		C2 × C2 × C2	< r,g,b \| r², g², b², rg², gb², rb² > r=(ab) g=(cd) b=(ef)	1.2⁷ C2×C2×C2 1 PSL(3,2) / 1 = PSL(3,2) 5, C2 x C2 x C2 2x2x2
	Other	D8 = C4 ⋊ C2	< k,r \| r⁴, r², krkr > k=(abcd) r=(ac) < r,g,b \| b², g², r², bgbg, rbrg > b=(ab)(cd) g=(ac)(bd) r=(bc)	1.2¹⁺⁴.4² C2 C2 D8 / C₂² = C2 3, D8 2
	Other	Q8 a.k.a. Dic8	< r,b \| r⁴, b⁴, rbrrrb, rrbb > b=(abcd)(ehgf) r=(afch)(bgde) < r,g \| r⁴, g⁴, rgrrrg,rrgg > r=(abde)(fhcg) g=(acdf)(egbh) b=(bcef)(agdh)	1.2.4⁶ C2 C2 S4 / C₂² = D6 4, Q8 1
9	Abelian	C9	< k \| k⁹ > k=(abcdefghi)	1.3².6⁶ C9 1 C6 / 1 = C6 1, C9 1
9	Abelian	C3 × C3	< k, r \| k³, r³, krkkrr > k=(abc) r=(def)	1.3⁸ C3×C3 1 GL(2,3) / 1 = GL(2,3) 2, C3 x C3 3
10	Abelian	C10 ≅ C5 × C2	< k \| k¹⁰ > k=(abcdefghij) < k,r \| k⁵, r², krk^-1r > k=(abcde) r=(fg)	1.2.5⁴.10⁴ C10 1 C4 / 1 = C4 2, C10 1
10	Other	D10 = C5⋊C2	< k,r \| k⁵, r², krkr > k=(abcde) r=(be)(cd)	1.2⁵.5⁴ 1 C5 C5⋊C4 / D10 = C2 1, D10 1
11	Abelian	C11	< k \| k¹¹ > k=(abcdefghijk)	1.11¹⁰ C11 1 C10 / 1 = C10 1, C11 1
12	Abelian	C12 ≅ C4 × C3	< k \| k¹² > k=(abcdefghijkl) < k, r \| k³, r⁴, krkkrrr > k=(abc) r=(defg)	1.2.3².4².6².12⁴ C12 1 C₂² / 1 = C₂² 2, C12 C4 1
	Abelian	C6 × C2 ≅ C3 × C2 × C2	< k,r \| k⁶, r², krkkkkkrr > k=(abcdef) r=(gh)	1.2³.3².6⁶ C6×C2 1 D12 / 1 = D12 5, C6 x C2 C₂² 2
	Other direct products	D12 = D6 ⋊ C2 ≅ D6 × C2	< k,r \| k⁶, r², krkr > k=(abcdef) r=(bf)(ce) < k,r \| k³, r², g², krkr, kgkkg, rgrg > k=(abc) r=(bc) g=(de)	1.2¹⁺⁶.3².6² C2 C3 D12 / D6 = C2 4, D12 C₂² 2
	Other	Dic12 ≅ C3 ⋊ C4	< b,r \| b⁶, r⁴, brbrrr > b=(abc)(pr)(qs) r=(bc)(pqrs)	1.2.3².4⁶.6² C2 C3 D12 / D6 = C2 1, C3 : C4 C4 1
	Other	A4 = (C2×C2) ⋊ C3	< k,r \| k³, r², (kr)³ > k=(abc) r=(ab)(cd) The first diagram resembles a truncated tetrahedron, whose rotational symmetry group is A4. < r,b,e \| r², g², e³, rgrg, geree, rgegee > r=(ab)(cd) g=(ac)(bd) e=(bcd) The second diagram shows more clearly how the C3 (grey) acts on the C2×C2 (colour).	1.2³.3⁸ 1 C₂² S4 / A4 = C2 3, A4 C₂² 2
13	Abelian	C13	< k \| k¹³ > k=(abcdefghijklm)	1.13¹² C13 1 C12 / 1 = C12 1, C13 1
14	Abelian	C14 ≅ C7 × C2	< k \| k¹⁴ > k=(abcdefghijklmn) < k,r \| k⁷, r², krk⁶r² > k=(abcdefg) r=(pq)	1.2.7⁶.14⁶ C14 1 C6 / 1 = C6 2, C14 1
14	Other	D14 = C7 ⋊ C2	Here are three ways of drawing a Cayley diagram for D14. The first one I regard as usefully reflecting the structure of the group. The other two are given to show that it is possible to draw them like this, and omitted for other dihedral groups. < k,r \| k⁷, r², krkr > k=(abcdefg) r=(bg)(cf)(de) < r,g \| r², g², (rg)⁷ > r=(bg)(cf)(de) g=(af)(be)(cd) < k,r,g \| k⁷, r², g², kgr > k=(abcdefg) r=(bg)(cf)(de) g=(ag)(bf)(ce)	1.2⁷.7⁶ 1 C7 C7⋊C6 / D14 = C3 1, D14 1
15	Abelian	C15 ≅ C5 × C3	< k \| k¹⁵ > k=(abcdefghijklmno) < k,r \| k³, r⁵, krkkrrr > k=(abcde) r=(mno)	1.3².5⁴.15⁸ C15 1 C4×C2 / 1 = C4×C2 1, C15 1
16	Abelian	C16	< k \| k¹⁶ > k=(abcdefghijklmnop)	1.2.4².8⁴.16⁸ C16 1 C4×C2 / 1 = C4×C2 1, C16 1
		C4 × C4	< k,r \| k⁴, r⁴, krk^-1r^-1 > k=(abcd) r=(efgh)	1.2³.4¹² C4×C4 1 (C₂²×A4)⋊C2 / 1 = (C₂²×A4)⋊C2 2, C4 x C4 2
		C4 × C2 × C2	< r,g,e \| r², g², e⁴, rgrg, rere^-1, gege^-1 > r=(ab) g=(cd) e=(efgh)	1.2¹⁺⁶.4⁸ C4×C2×C2 1 "(((D8×C2)⋊C2)⋊C3)⋊C2" / 1 = ? 10, C4 x C2 x C2 2x2x2
		C2 × C2 × C2 × C2	< r,g,b,m \| r², g², b², m², rgrg, rbrb, rmrm, gbgb, gmgm, bmbm > r=(ab) g=(cd) b=(ef) m=(gh)	1.2¹⁵ C2×C2×C2×C2 1 A8 / 1 = A8 14, C2 x C2 x C2 x C2 2x2x2x2x2x2
		C8 × C2	< k,r \| k⁸, r², krk^-1r > k=(abcdefgh) r=(ij) Compare this diagram with those for D16, "Modular", and "Quasidihedral" below. These four diagrams show C2 acting in all four possible ways on C8, whose automorphism group is C2×C2.	1.2³.4⁴.8⁸ C8×C2 1 D8×C2 / 1 = D8×C2 5, C8 x C2 2
	Other direct products	D8 × C2	< k,r,g \| k⁴, r², g², krkr, kgkkkg, rgrg > k=(abcd) r=(ac) g=(pq)	1.2¹⁺²⁺⁸.4⁴ C2×C2 C2 "(((C4×C2)⋊C2)⋊C2)⋊C2" / C₂² = ? 11, C2 x D8 2x2x2
	Other direct products	Q8 × C2	< r,b,g \| r⁴, b⁴, g^r, rrbb, rgr^-1g, bgb^-1g > r=(abcd)(efgh) b=(aecg)(fbhd) g=(pq)	1.2¹⁺².4¹² C2×C2 C2 "(((C₂⁴)⋊C3)⋊C2)⋊C2" / C₂² = ? 12, C2 x Q8 2 2x2
	Other	D16 = C8 ⋊ C2	< k,r \| k⁸, r², krkr > k=(abcdefgh) r=(bh)(cg)(df)	1.2¹⁺⁸.4².8⁴ C2 C4 "(D8×C2)⋊C2" / D8 = C₂² 7, D16 2
		Modular = C8 ⋊ C2	< k,r \| k⁸, r², krkkkr > k=(abcdefgh) r=(bf)(dh) < k,r \| k⁸, r⁸, krkr >	1.2¹⁺².4⁴.8⁸ C4 C2 D8×C2 / C₂² = C₂² 6, (C4 x C2) : C2 1
		Quasidihedral, a.k.a. semidihedral = C8 ⋊ C2	< k,r \| k⁸, r², krkkkkkr > k=(abcdefgh) r=(bd)(cg)(fh)	1.2¹⁺⁴.4⁶.8⁴ C2 C4 D8×C2 / D8 = C2 8, QD16 1
		Dic16 a.k.a. Q16	< b,r \| b⁸, r⁴, (br)⁴, rbr³b > b=(abcdefgh)(pqrstuvw) r=(apet)(bwfs)(cvgr)(duhq)	1.2.4¹⁰.8⁴ C2 C4 (D8×C2)⋊C2 / D8 = C₂² 9, Q16 1
		C4 ⋊ C4	< k,r \| k⁴, r⁴, krkr³ > k=(abcd) r=(bd)(efgh)	1.2³.4¹² C2×C2 C2 C₂⁴⋊C2 / C₂² = ? 4, C4 : C4 2
		(C2 × C2) ⋊ C4	< r,g,e \| r², g², e⁴, rgrg, ereeeg, egeeer > r=(ab)(cd) g=(ac)(bd) e=(bc)(pqrs)	1.2³⁺⁴.4⁸ C2×C2 C2 C₂⁴⋊C2 / C₂² = ? 3, (C4 x C2) : C2 2x2
		Pauli = D8 ⋊ C2 = Q8 ⋊ C2 = (C4×C2) ⋊ C2	< k,r,b \| k⁴, r², b², krkr, kbkkkb, kkrbrb > k=(abcd)(efgh) r=(bd)(eg) b=(ae)(bf)(cg)(dh) The C2 acts on the D8 by the permutation (b,aab),(ab,aaab). The first diagram may make it clearer what happens: there are two black-and-red D8s, with the inner square of one rotated through π relative to the other. The second diagram is prettier, having some of the symmetry of a cube.	1.2⁷.4⁸ C4 C2 S4×C2 / C₂² = ? 13, C4 : C4 2x2
17	Abelian	C17	< k \| k¹⁷ > k=(abcdefghijklmnopq)	1.17¹⁶ C17 1 C16 / 1 = C16 1, C17 1
18	Abelian	C18 = C9 × C2	< k \| k¹⁸ > k=(abcdefghijklmnopqr) < k,r \| k⁹, r², krk^-1r^-1 > k=(abcdefghi) r=(mn)	1.2.3².6².9⁶.18⁶ C18 1 C6 / 1 = C6 2, C18 C9 1
	Abelian	C6 × C3 = C3 × C3 × C2	< k,r \| k⁶, r³, krk^-1r^-1 > k=(abcdef) r=(jkl)	1.2.3⁸.6⁸ C6×C3 1 GL(2,3) / 1 = GL(2,3) 5, C6 x C3 C₃² 3
	Other direct products	D6 × C3 ≅ (C3 × C3) ⋊ C2 with the C2 interchanging the generators of the two C3s	< k,r,g \| k³, r², g³, krkr, kgkkgg, rgrgg > k=(abc) r=(bc) g=(pqr) < k,r,g \| k³, r³, g², krkkrr, grgkk, gkgrr > k=(abc) r=(def) g=(ad)(be)(cf)	1.2³.3⁸.6⁶ C3 C3 D12 / D6 = C2 3, C3 x S3 C₃² 3
	Other	D18	< k,r \| k⁹, r², krkr > k=(abcdefghi) r=(bi)(ch)(dg)(ef)	1.2⁹.3².9⁶ 1 C9 "(C9⋊)⋊C2" / D18 = C3 1, D18 C9 1
	Other	(C3 × C3) ⋊ C2 with the C2 acting separately on the two C3s	< k,r,g \| k³, r³, c², krkkrr, kgkg, rgrgr > k=(abc) r=(def) g=(bc)(ef)	1.2⁹.3⁸ 1 C3×C3 "((C₃²⋊Q8)⋊C3)⋊C2" / C₃²⋊C2 = ? 4, (C3 x C3) : C2 C₃² 3
19	Abelian	C19	< k \| k¹⁹ > k=(abcdefghijklmnopqrs)	1.19¹⁸ C19 1 C18 / 1 = C18 1, C19
20	Abelian	C20 = C5 × C4	< k \| k²⁰ > k=(abcdefghijklmnopqrst) < k,r \| k⁵, r⁴, krk^-1r^-1 > k=(abcde) r=(mnop)	1.2.4².5⁴.10⁴.20⁸ C20 1 C4×C2 / 1 = C4×C2 2, C20 C4 1
	Abelian	C10 × C2 = C5 × C2 × C2	< k,r \| k¹⁰, r², krk^-1r^-1 > k=(abcdefghij) r=(mn) There is little interest in Cayley diagrams of direct products. This one is like a man who walks round in circles (black), and takes his hat on and off (red). The hat does not affect the walking, and the walking does not affect the hat.	1.2³.5⁴.10¹² C10×C2 1 D6×C4 / 1 = D6×C4 5, C10 x C2 C₂² 2
	Other direct products	D20 = C10 ⋊ C2 ≅ D10 × C2	< k,r \| k¹⁰, r², krkr > k=(abcdefghij) r=(bj)(ci)(dh)(eg) < k,b,g \| k⁵, r², g², krkr, kgk^-1g^-1, rgr^-1g^-1 > k=(abcde) r=(be)(cd) g=(pq)	1.2¹⁺¹⁰.5⁴.10⁴ C2 C5 "C2 x (C5 : C4)" / D10 = ? 4, D20 C₂² 2
	Other	Dic20 ≅ C5 ⋊_C2 C4	< b,r \| b⁵, r⁴, brbrrr > b=(abcde)(pr)(qs) r=(be)(cd)(pqrs) < k,r \| k⁵, r⁴, krkrrr >	1.2.4¹⁰.5⁴.10⁴ C2 C5 C2×(C5⋊_C2C4) / D10 = C2 1, C5 : C4 C4 1
	Other	Frob20 ≅ C5 ⋊_C4 C4	< k,r \| k⁴, r², (k²r)⁵ > k=(abcde) r=(bd)(ce) < k,g \| k⁵, g⁴, (kg)⁴ > k=(abcde) g=(bced)	1.2⁵.4¹⁰.5⁴ 1 C5 C5⋊_C4C4 / C5⋊_C4C4 = 1 3, C5 : C4 C4 1
21	Abelian	C21 = C7 × C3	< k \| k²¹ > k=(abcdefghijklmnopqrstu) < k,r \| k⁷, r³, krk^-1r^-1 > k=(abcdefg) r=(pqr)	1.3².7⁶.21¹² C21 1 C6×C2 / 1 = C6×C2 2, C21 1
21	Other	Frob21 ≅ C7 ⋊ C3	< k,r \| k⁷, r³, krk⁵r² > k=(abcdefg) r=(bce)(dgf)	1.3¹⁴.7⁶ 1 C7 Frob42 / Frob21 / C2 1
22	Abelian	C22 = C11 × C2	< k \| k²² > k=(abcdefghijklmnopqrstuv) < k,r \| k¹¹, r², krk^-1r^-1 > k=(abcdefghijk) r=(pq)	1.2.11¹⁰.22¹⁰ C22 1 C10 / 1 = C10 2, C22 1
22	Other	D22 = C11 ⋊ C2	< k,r \| k¹¹, r², krkr > k=(abcdefghijk) r=(bk)(cj()di)(eh)(fg)	1.2¹¹.11¹⁰ 1 C11 "(C11 : C5) : C2" / D22 = C5 1, D22 1
23	Abelian	C23	< k \| k²³ > k=(abcdefghijklmnopqrstuvw)	1.23²² C23 1 C22 / 1 = C22 1, C23 1
24	Abelian	C24 = C8 × C3	< k \| k²⁴ > k=(abcdefghijklmnopqrstuvwv) < k,r \| k⁸, r³, krk^-1r^-1 > k=(abcdefgh) r=(mn)	1.2.3².4².6².8⁴.12⁶.24⁶ C24 1 C₂³ / 1 = C₂³ 2, C24 C8 1
		C12 × C2 = C6 × C4 = C4 × C3 × C2	< k,r \| k¹², r², krk^-1r^-1 > k=(abcdefghijkl) r=(pq) < k,r \| k⁶, r⁴, krk^-1r^-1 > k=(abcdef) r=(pqrs)	1.2³.3².4⁴.6⁶.12⁸ C12×C2 1 D8×C2 / 1 = D8×C2 9, C12 x C2 C4×C2 2
		C3 × C2 × C2 × C2 = C6 × C2 × C2 = C2 × C2 × C2 × C3	< r,g,e \| r², g², e⁶, rgrg rere^-1, gege^-1 > r=(ab) g=(cd) e=(pqrstu) < r,g,b,e \| r², g², b², e³, rgrg, gbgb, brbr, rere^-1, gege^-1, bebe^-1, > r=(ab) g=(cd) b=(ef) e=(ghi)	1.2⁷.3².6¹⁴ C3×C2×C2×C2 1 PSL(3,2)×C2 / 1 = PSL(3,2)×C2 15, C6 x C2 x C2 C₂³ 2x2x2
	Other direct products	D12 × C2 = D6 × C2 × C2	< k,r,g \| k⁶, r², g², krkr, kgk^-1g^-1, rgr^-1g^-1 > k=(abcdef) r=(bf)(ce) g=(pq)	1.2¹⁵.3².6⁶ C2×C2 C3 S4×D6 / D6 = S4 14, C2 x C2 x S3 C₂³ 2x2x2
		D8 × C3	< k,r,g \| k⁴, r², g³, krkr, kgk^-1g^-1, rgr^-1g^-1 > k=(abcd) r=(bd) g=(pqr)	1.2⁵.3².4².6¹⁰.12⁴ C3×C2 C2 D8×C2 / C₂² = ? 10, C3 x D8 D8 2
		D6 × C4	< k,r,g \| k³, r², g⁴, krkr, kgk^-1g^-1, rgr^-1g^-1 > k=(abc) r=(bc) g=(pqrs)	1.2⁷.3².4⁸.6².12⁴ C4 C3 S3×C₂² / S3 = ? 5, C4 x S3 C4×C2 2
		Dic12 × C2 ≅ C6 ⋊_C2 C4	< b,r,g \| b⁶, r⁴, g², brbbbr, gbgb^-1, grgr^-1 > b=(abc)(mo)(np) r=(bc)(mnop) g=(st) < k,r \| k⁶, r⁴, krkr³ > k=(abcdef) r=(bf)(ce)(pqrs)	1.2¹⁺².3².4¹².6⁶ C₂² C3 D8×D6 / D6 = ? 7, C2 x (C3 : C4) C4×C2 2
		Q8 × C3	< r,b,g \| r⁴, b⁴, g³, rrbb, rgr^-1g^-1, bgb^-1g^-1 > r=(abcd)(efgh) b=(aecg)(fbhd) g=(pqr) These Cayley diagrams of direct products are tedious and uninformative.	1.2.3².4⁶.6².12¹² C3×C2 C2 S4×C2 / C₂² = ? 11, C3 x Q8 Q8 1
		A4 × C2 ≅ (C2×C2×C2) ⋊ C3	< k,r \| k³, r², (kr)³ central, (kr)⁶ > k=(abc) r=(ab)(cd)(pq) < k,r,g \| k³, r², g², (kr)³, kgk^-1g^-1, rgr^-1g^-1 > k=(abc) r=(ab)(cd) g=(pq) < r,g,b,e \| r², g², b², e³, rgrg, gbgb, brbr, gere^-1, bege^-1, rebe^-1 > r=(ab) g=(cd) b=(rf) e=(ace)(bdf)	1.2⁷.3⁸.6⁸ C2 C2×C2 S4 / A4 = C2 13, C2 x A4 C₂³ 2
	Other	D24 = C12 ⋊ C2	< k,r \| k¹², r², krkr> k=(abcdefghijkl) r=(bl)(ck)(dj)(ei)(fh)	1.2¹⁺¹².3².4².6².12⁴ C2 C6 D8×C3 / D12 = C2 6, D24 D8 2
		Dic24	< b,r \| b¹², r⁴, brbrrr > b=(abcdefghijkl)(mnopqrstuvwx) r=(asgm)(brhx)(cqiw)(dpjv)(eoku)(fnlt)	1.2.3².4²⁺¹².6².12⁴ C2 C6 D8×C3 / D12 = C2 4, C3 : Q8 Q8 1
		C3 ⋊ C8	< k,r \| k³, r⁸, krkr⁷ > k=(abc) r=(bc)(defghijk)	1.2.3².4².6².8¹².12⁴ C4 C3 D6×C₂² / S3 = D6×C₂² 1, C3 : C8 C8 1
		SL(2,3) ≅ Q8 ⋊ C3	< k,r \| k⁶, r⁴, krkrkr > k=(abcdef)(gh) r=(gahd)(ecbf) < r,b,g,e \| r⁴, b⁴, g⁴, e³, rrbb, bbgg, ggrr, rbgrbg, rebege>	1.2.3⁸.4⁶.6⁸ C2 Q8 S4 / A4 = C2 3, SL(2,3) Q8 1
		C3 ⋊ D8	< k,r,g \| k³, g⁴, r², gkgggk, rgrg, rkr^-1k^-1 > k=(abc) g=(ghij)(bc) r=(hj)	1.2¹⁺²⁺⁶.3².4⁶.6⁶ C2 C6 D6×C₂² / D12 = C2 8, (C6 x C2) : C2 D8 2
		S4 ≅ (C2×C2) ⋊ D6	< k,r \| k⁴, r², (kr)³ > k=(abcd) r=(ab) < k,r \| k³, r², (kr)⁴ > k=(abc) r=(cd) < b,g,r,e \| b², g², r², e³, bgbg, rgrb, rbrg, ege²b, ebe²bg > b=(hi)(jk) g=(hj)(ik) r=(ij) e=(hij)	1.2³⁺⁶.3⁸.4⁶ 1 A4 S4 / S4 = 1 12, S4 D8 2
25	Abelian	C25	< k \| k²⁵ > k=(abcdefghijklmnopqrstuvwxy)	1.5⁴.25²⁰ C25 1 C20 / 1 = C20 1, C25 1
25	Abelian	C5 × C5	< k,r \| k⁵, r⁵, krk^-1r^-1> k=(abcde) r=(fghij)	1.5²⁴ C5×C5 1 GL(2,5) / 1 = GL(2,5) 2, C5 x C5 1 5
26	Abelian	C26 = C13 × C2	< k \| k²⁶ > k=(abcdefghijklmnopqrstuvwxyz) < k,r \| k¹³, r², krk^-1r^-1 > k=(abcdefghijklm) r=(pq)	1.2.13¹².26¹² C26 1 C12 / 1 = C12 2, C26 1
26	Other	D26 = C13 ⋊ C2	< k,r \| k¹¹, r², krkr > k=(abcdefghijklm) r=(bm)(cl)(dk)(ej)(fi)(gh)	1.2¹³.13¹² 1 C13 "(C13 : C4) : C3" / D26 = C6 1, D26 1
27	Abelian	C27	< k \| k²⁷ > k=(abcdefghijklmnopqrstuvwxyzæ)	1.3².9⁶.27¹⁸ C27 1 C18 / 1 = C18 1, C27 1
		C9 × C3	< k,r \| k⁹, r³, krk^-1r^-1 > k=(abcdefghi) r=(pqr)	1.3⁸.9¹⁸ C9×C3 1 "C2 x (((C3 x C3) : C3) : C2)" / 1 = ? 2, C9 x C3 3
		C3 × C3 × C3	< k,r,g \| k³, r³, g³, krk^-1r^-1, rgr^-1g^-1, gkg^-1g^-1 > k=(abc) r=(def) g=(ghi)	1.3²⁷ C₃³ 1 Gl(3,3) / 1 = GL(3,3) 5, C3 x C3 x C3 3x3x3
	Other	C9 ⋊ C3	< k,r \| k⁹, r³, krk⁵r^-1 > k=(abcdefghi) r=(beh)(cif)	1.3²⁺⁶.9¹⁸ C3 C3 "((C3 x C3) : C3) : C2" / C₃² = ? 4, C9 : C3 1
	Other	(C3 × C3) ⋊ C3	< k,r,g \| k³, r³, g³, krk^-1r^-1, rgkkgg, krgrrgg > k=(abc)(def)(ghi) r=(adg)(beh)(cfi) g=(bdi)(cge)	1.3²⁺²⁴ C3 C3 "(((C3 x C3) : Q8) : C3) : C2" / C₃² = ? 3, (C3 x C3) : C3 3x3
28	Abelian	C28 = C7 × C4	< k \| k²⁸ > k=(abcdefghijklmnopqrstuvwxyzæð) < k,r \| k⁷, r⁴, krk^-1r^-1 > k=(abcdefg) r=(mnop)	1.2.4².7⁶.14⁶.28¹² C28 1 C6×C2 / 1 = C6×C2 2, C28 C4 1
	Abelian	C14 × C2 = C7 × C2 × C2	< k,r \| k¹⁴, r², krk^-1r^-1 > k=(abcdefg) r=(pq)	1.2³.7⁶.14¹⁸ C14×C2 1 D6×C6 / 1 = D6×C6 4, C14 x C2 C₂² 2
	Other direct products	D28 = C14 ⋊ C2 ≅ D14 × C2	< k,r \| k¹⁴, r², krkr > k=(abcdefghijklmn) r=(bn)(cm)(dl)(ek)(fj)(gj) < k,r,g \| k⁷, r², g², krkr, kgk^-1g^-1, rgr^-1g^-1 > k=(abcdefg) r=(bg)(cf)(de) g=(pq)	1.2¹⁺²⁺¹².7⁶.14⁶ C2 C7 (C7⋊C6)×C2 / D14 = C6 3, D28 C₂² 2
	Other	Dic28 ≅ C7 ⋊_C2 C4	< b,r \| b⁷, r⁴, brbrrr > b=(abcdefg)(pr)(qs) r=(bg)(cf)(de)(pqrs)	1.2.4¹⁴.7⁶.14⁶ C2 C7 (C7⋊C6)×C2 / D14 = C6 1, C7 : C4 C4 1
29	Abelian	C29	< k \| k²⁹ > k=(abcdefghijklmnopqrstuvwxyzæðñ)	1.29²⁸ C29 1 C28 / 1 = C28 1, C29 1
30	Abelian	C30 = C15 × C2 = C10 × C3 = C6 × C5 = C5 × C3 × C2	< k \| k³⁰ > k=(abcdefghijklmnopqrstuvwxyzæðñç) < k,r \| k¹⁵, r², krk^-1r^-1 > k=(abcdefghijklmno) r=(pq) < k,r \| k¹⁰, r³, krk^-1r^-1 > k=(abcdefghij) r=(klm) < k,r \| k⁶, r⁵, krk^-1r^-1 > k=(abcdef) r=(ghijk)	1.2.3².5⁴.6².10⁴.15⁸.30⁸ C30 1 C4×C2 / 1 = C4×C2 4, C30 1
	Other direct products	D10 × C3	< k,r,g \| k⁵, r², g³, krkr, kgk^-1g^-1, rgr^-1g^-1 > k=(abcde) r=(be)(cd) g=(pq)	1.2⁵.3².5⁴.6¹⁰.15⁸ C3 C5 (C5⋊C4)×C2 / D10 = C₂² 2, C3 x D10 1
	Other direct products	D6 × C5	< k,r,g \| k³, r², g⁵, krkr, kgk^-1g^-1, rgr^-1g^-1 > k=(abc) r=(bc) g=(pqrst)	1.2³.3².5⁴.10¹².15⁸ C5 C3 D6×C4 / D6 = C4 1, C5 x S3 1
	Other	D30 C15 ⋊ C2	< k,r \| k¹⁵, r², abab > k=(abcdefghijklmno) r=(bo)(cn)(dm)(el)(fk)(gj)(hi)	1.2¹⁵.3².5⁴.15⁸ 1 C15 (C5⋊C4)×D6 / D6 = ? 3, D30 1
31	Abelian	C31	< k \| k³¹ > k=(abcdefghijklmnopqrstuvwxyzæðñçþ)	1.31³⁰ C31 1 C30 / 1 = C30 1, C31 1

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Some more Cayley diagrams
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Cayley Diagrams of Small Groups

Notation

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