Cayley Diagrams of Small Groups

This page gives Cayley diagrams for all groups of order less than 32 which are not direct products. 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

Some of the notation used here is non-standard. If you want a page that uses standard notation, and has no little orange triangles indicating central extensions, see Cayley Diagrams of Small Groups.

An integer denotes the cyclic group of that order — the letter "C" (or "Z") indicating a cyclic group is omitted. Composite integers are not used.

A × B denotes the direct product of A and B.
N ⋊ H indicates the semidirect product of N by H, N being the normal subgroup.
C ↑ D indicates a central extension, C being the centre.

The orange triangles in the Cayley diagrams of central extensions are explained in the page on toll-bean extensions.

Contents

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
Cayley diagram Orders of elements.
Centre.
Derived subgroup.
1 Abelian 1 <> 1

1

1

2 Abelian 2 < k | k2 >
k=(ab)
1.2

2

1

3 Abelian 3 < k | k3 >
k=(abc)
1.32

3

1

4 Abelian 2×2     1.23   2×2   1
2↑2
< k | k4 >
k=(abcd)
1.2.42

2↑2

1

5 Abelian 5 < k | k5 >
k=(abcde)
1.54

5

1

6 Abelian 2×3     1.2.32.62   6   1
Other 3⋊2 < k,r | k3, r2, krkr >
k=(abc) r=(bc)
1.23.32

1

3

7 Abelian 7 < k | k7 >
k=(abcdefg)
1.76

7

1

8 Abelian 2↑2↑2 < k | k8 >
k=(abcdefgh)
1.2.42.84

2↑2↑2

1

(2↑2)×2     1.21+2.44   (2↑2)×2   1
2×2×2     1.27   2×2×2   1
Other (2↑2)⋊2 < k,r | r4, r2, krkr >
k=(abcd) r=(ac)

< r,g,b | b2, g2, r2, bgbg, rbrg, rgrb >
b=(ab)(cd) g=(ac)(bd) r=(bc)

1.21+4.42

2

2

2↑(2×2) < k,r | k4, r4, krkkkr, kkrr >
k=(abcd)(efgh) r=(ahcf)(bgde)
1.2.46

2

2

9 Abelian 3↑3 < k | k9 >
k=(abcdefghi)
1.32.66

3↑3

1

3×3     1.38   3×3   1
10 Abelian 2×5     1.2.54.104   2×5   1
Other 5⋊2 < k,r | k5, r2, krkr >
k=(abcde) r=(be)(cd)
1.25.54

1

5

11 Abelian 11 < k | k11 >
k=(abcdefghijk)
1.1110

11

1

12 Abelian (2↑2)×3     1.2.32.42.62.124   (2↑2)×3   1
2×2×3     1.23.32.66   2×2×3   1
Other d.p. 2×(3⋊2)     1.21+6.32.62   2   3
Other 3⋊(2↑2) < b,r | b6, r4, rkrrrk >
b=(abc)(pr)(qs) r=(bc)(pqrs)
1.2.32.46.62

2

3

(2×2)⋊3

(a.k.a. A4)

< k,r | k3, r2, (kr)3 >
k=(abc) r=(ab)(cd)
1.23.38

1

2×2

13 Abelian 13 < k | k13 >
k=(abcdefghijklm)
1.1312

13

1

14 Abelian 2×7     1.2.76.146   2×7   1
Other 7⋊2 < k,r | k7, r2, krkr >
k=(abcdefg) r=(bg)(cf)(de)
1.27.76

1

7

15 Abelian 3×5     1.32.54.158   3×5   1
16 Abelian 2↑2↑2↑2 < k | k16 >
k=(abcdefghijklmnop)
1.2.42.84.168

2↑2↑2↑2

1

2×(2↑2↑2)     1.23.44.88   2×(2↑2↑2)   1
(2↑2)×(2↑2)   1.23.412   (2↑2)×(2↑2)   1
2×2×(2↑2)     1.21+6.48   2×2×(2↑2)   1
2×2×2×2     1.215   2×2×2×2   1
Other d.p.s 2×((2↑2)⋊2)     1.21+2+8.44   2×2   2
2×(2↑(2×2))     1.21+2.412   2×2   2
Other (2↑2↑2)⋊2

(dihedral)

< k,r | k8, r2, krkr >
k=(abcdefgh) r=(bh)(cg)(df)
1.21+8.42.84

2

2↑2

(2↑2↑2)⋊2
≅ (2↑2)↑(2×2)

(modular)

< k,r | k8, r2, krkkkr >
k=(abcdefgh) r=(bf)(dh)
1.21+2.44.88

2↑2

2

(2↑2↑2)⋊2

(quasidihedral, semidihedral)

< k,r | k8, r2, krkkkkkr >
k=(abcdefgh) r=(bd)(cg)(fh)
1.21+4.46.84

2

2↑2

2↑((2↑2)⋊2)

(Q16)

< k,r | k8, r4, (kr)4, rkr3>
k=(abcdefgh)(pqrstuvw) r=(apet)(bwfs)(cvgr)(duhq)
1.2.410.84

2

2↑2

(2↑2)⋊(2↑2)

≅ 2↑(2×2×2)

< k,r | k4, r4, krkr3 >
k=(abcd) r=(bd)(efgh)

< b,g,r | b4, g4, r4, (bgr)2 >

1.23.412

2×2

2

(2×2)⋊(2↑2) < b,g,r | b2, g2, r4, bgbg, rbrrrg, rgrrrb >
b=(ab)(cd) g=(ac)(bd) r=(bc)(pqrs)
1.23+4.48

2×2

2

((2↑2)⋊2)⋊2
≅ (2↑(2×2))⋊2
≅ ((2↑2)×2)⋊2
≅ ((2↑2)↑(2×2)
< r,g,b | r2, g4, b4, (rg)2, (gb)4, (br)4 >
r=(bd)(eg) g=(agce)(bfdh) b=(afch)(bedg)

r and g generate (2↑2)⋊2; g and b generate 2↑(2×2); b and r generate (2↑2)×2. Thus, if we regard the Cayley diagram as a cube, two opposite faces portray D8, two others Q8, and the other two C4×C2. (br) generates the centre.

1.21+2+2+2.42+6

2↑2

2

17 Abelian 17 < k | k17 >
k=(abcdefghijklmnopq)
1.1716

17

1

18 Abelian 2×(3↑3)     1.2.32.62.96.186   2×(3↑3)   1
2×3×3     1.2.38.68   2×3×3   1
Other d.p. 3×(3⋊2)
≅ (3×3)⋊2

with the 2 interchanging the generators of the 3s
1.23.38.66

C3

C3

Other (3↑3)⋊2 < k,r | k9, r2, krkr >
k=(abcdefghi) r=(bi)(ch)(dg)(ef)
1.29.32.96

1

3↑3

(3×3)⋊2
with the 2 acting separately on the two 3s.
< k,r,g | k3, r3, c2, krkkrr, kgkg, rgrgr >
k=(abc) r=(def) g=(bc)(ef)
1.29.38

1

3×3

19 Abelian 19 < k | k19 >
k=(abcdefghijklmnopqrs)
1.1918

19

1

20 Abelian (2↑2)×5     1.2.42.54.104.204   (2↑2)×5   1
2×2×5     1.2.42.54.1012   2×2×5   1
Other d.p. 2×(5⋊2)     1.21+10.54.104   2   5
Other 5⋊2(2↑2) < k,r | k5, r4, krkrrr >
b=(abcde)(pr)(qs) r=(be)(cd)(pqrs)
1.2.410.54.104

2

5

5⋊2↑2(2↑2) < k,g | k5, g4, kgkkggg >
k=(abcde) g=(bced)
1.25.410.54

1

5

21 Abelian 3×7     1.32.76.2112   3×7   1
Other 7⋊3 < k,r | k7, r3, krk5r2 >
k=(abcdefg) r=(bce)(dgf)
1.314.76

1

7

22 Abelian 2×11     1.2.1110.2210   2×11   1
Other 11⋊2 < k,r | k11, r2, krkr >
k=(abcdefghijk) r=(bk)(cj()di)(eh)(fg)
1.211.1110

1

11

23 Abelian 23 < k | k23 >
k=(abcdefghijklmnopqrstuvw)
1.2322

23

1

24 Abelian (2↑2↑2)×3     1.2.32.42.62.84.126.246   (2↑2↑2)×3   1
2×(2↑2)×3     1.23.32.44.66.128   2×(2↑2)×3   1
2×2×2×3     1.27.32.614   2×2×2×3   1
Other d.p.s 2×2×(3⋊2)     1.215.32.66   2×2   3
3×((2↑2)⋊2)     1.25.32.42.610.124   2×3   2
(2↑2)×(3⋊2)     1.27.32.48.62.124   2↑2   3
2×(3⋊(2↑2))     1.21+2.32.412.66   2×2   3
3×(2↑(2×2))     1.2.32.48.62.1212   2×3   2
2×((2×2)⋊3)     1.27.38.68   2   2×2
Other ((2↑2)×3)⋊2) < k,r | k12, r2, krkr>
k=(abcdefghijkl) r=(bl)(ck)(dj)(ei)(fh)
1.21+12.32.42.62.124

2

6

2↑(2×(3⋊2))

< k,r | k12, r4, krkrrr >
b=(abcdefghijkl)(mnopqrstuvwx) r=(asgm)(brhx)(cqiw)(dpjv)(eoku)(fnlt)

1.2.32.42+12.62.124

2

6

3⋊(2↑2↑2) < k,r | k3, r8, krkr7 >
k=(abc) r=(bc)(defghijk)
1.2.32.42.62.812.124

2↑2

3

3⋊((2↑2)⋊2) < 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

2

2×3

(2↑(2×2))⋊3
≅ 2↑((2×2))⋊3)

a.k.a. SL(2,3)

< k,r | k3, r4, (kr)3 >
k=(abc) r=(ab)(cd)(pq)

< r,b,g,e | r4, b4, g4, e3, rrbb, bbgg, ggrr, rbgrbg, rebege>

1.2.38.46.68

2

2↑(2×2)

(2×2)⋊(3⋊2)
≅ ((2×2)⋊3)⋊2

a.k.a. S4

< k,r | k4, r2, (kr)3 >
k=(abcd) r=(ab)

< k,r | k3, r2, (kr)4 >
k=(abc) r=(cd)

1.23+6.38.46

1

(2×2)⋊3

25 Abelian 5↑5 < k | k25 >
k=(abcdefghijklmnopqrstuvwxy)
1.54.2520

5↑5

1

5×5     1.524   5×5   1
26 Abelian 2×13     1.2.1312.2612   2×13   1
Other 13⋊2 < k,r | k11, r2, krkr >
k=(abcdefghijklm) r=(bm)(cl)(dk)(ej)(fi)(gh)
1.213.1312

1

13

27 Abelian 3↑3↑3 < k | k27 >
k=(abcdefghijklmnopqrstuvwxyzæ)

1.32.96.2718

3↑3↑3

1

3×(3↑3)     1.32+6.918   3×(3↑3)   1
3×3×3     1.327   3×3×3   1
Other (3↑3)⋊3 < k,r | k9, r3, rkr2k5 >
k=(abcdefghi) r=(bhe)(cfi)
1.32+6.918

3

3

(3×3)⋊3 < k,r | k3, r3, (k,r)3 >

1.32+24

3

3

28 Abelian (2↑2)×7     1.2.42.76.2118   (2↑2)×7   1
2×2×7     1.23.76.1418   2×2×7   1
Other d.p. 2×(7⋊2)     1.21+2+12.76.146   2   7
Other 7⋊(2↑2) < b,r | b7, r4, brbrrr >
b=(abcdefg)(pr)(qs) r=(bg)(cf)(de)(pqrs)
1.2.414.76.146

2

7

29 Abelian 29 < k | k29 >
k=(abcdefghijklmnopqrstuvwxyzæðñ)
1.2928

29

1

30 Abelian 2×3×5     1.2.32.54.62.104.158.308   2×3×5   1
Other d.p.s 3×(5⋊2)     1.25.32.54.610.158   3   5
5×(3⋊2)     1.23.32.54.1012.158   5   3
Other (3×5)⋊2 < k,r | k15, r2, abab >
k=(abcdefghijklmno) r=(bo)(cn)(dm)(el)(fk)(gj)(hi)
1.215.32.54.158

1

3×5

31 Abelian 31 < k | k31 >
k=(abcdefghijklmnopqrstuvwxyzæðñçþ)
1.3130

31

1

Other pages giving Cayley diagrams of groups with fewer than 32 elements:

Another page giving Cayley diagrams of groups with fewer than 32 elements, drawn on the torus, using conventional notation and no orange triangles.
Some more Cayley diagrams
Some more pages on groups

Copyright N.S.Wedd 2007, 2008