Groups of orders 12, 24 and 48 containing A4

This page gives information on groups of orders 12, 24 and 48 which "contain" A4, as a normal subgroup, as a quotient group, or as a combination of these.

The information is currently not quite complete. The green table cell indicates missing information, some of the other information may be wrong.

O	Name	components, GAP no.	Presentation	Orders of Elements	Generating Permutations (and matrices)	Cayley Diagram	Centre	Derived Group	Sylow-2-subgroup
12	A4 PSL(2,3) Tetrahedral group	C₂²⋊ C3	< a,b,s \| a²=b²=s³=1, ba=ab, sa=bs, sb=abs >	1.2³.3⁸	(bc)(ad) (bdc)		1	C₂²	C₂²
24	A4×C2	A4 × C2		1.2¹⁺⁶.3⁸.6⁸		( direct product )	C2	C₂²	C₂³
	S4 PGL(2,3) Octahedral group	A4 ⋊ C2	< a,b,s,t \| a²=b²=s³=t²=1, ba=ab, sa=bs, sb=abs, ta=bt, tb=at, ts=s²t >	1.2³⁺⁶.3⁸.4⁶	(abcd) (bdc)		1	A4	D8
	SL(2,3) Binary tetrahedral group	C2 ↑ A4	< a,b,c,s \| a²=b²=c, c²=1, s³=1, ba=abc, sa=bs, sb=abs >	1.2.3⁸.4⁶.6⁸	(bcgf)(adhe) (bef)(cgd)		C2	Q8	Q8
48	A4×C4	A4×C4, 31		1.2¹⁺⁶.3⁸.4²⁺⁶.6⁸.12¹⁶		( direct product )	C4	C₂²	C4×C₂²
	A4×C2×C2	A4×C2×C2, 49		1.2³⁺¹².3⁸.6²⁴		( direct product )	C₂²	C₂²	C₂⁴
	Full octahedral group	S4×C2, 48		1.2¹⁹.3⁸.4¹².6⁸		( direct product )	C2	A4	D8×C2
	SL(2,3)×C2	SL(2,3)×C2, 32		1.2³.3⁸.4¹².6²⁴		( direct product )	C₂²	Q8	Q8×C2
	A4⋊C4	A4⋊C4 30	< a,b,s,t \| a²=b²=s³=t⁴=1, ba=ab, sa=bs, sb=abs, ta=bt, tb=at, ts=s²t >	1.2¹⁺⁶.3⁸.4²⁴.6⁸	(abcd)(pqrs) (bdc)		C2	A4	C₂²⋊C4
	GL(2,3)	(C2↑A4)⋊C2 29	< a,b,c,s,t \| a²=b²=c, c²=1, ba=abc, s³=t²=(st)²=1, sa=bs, sb=abs, ta=bt, tb=at, ts=s²t >	1.2¹⁺¹².3⁸.4⁶.6⁸.8¹²	(abfdhgce) (bef)(cgd)		C2	SL(2,3)	quasidihedral
	Binary octahedral group	C2↑(A4⋊C2) 28	< s,t \| s⁶=t⁸=1, ts=s⁵t³ >	1.2.3⁸.4¹⁸.6⁸.8¹²	(abij)(cekm)(dnlf)(gpoh), (bcd)(efg)(jkl)(mno)		C2	SL(2,3)	Q16
	C2↑(A4×C2)	C2↑(A4×C2) C4↑A4 33	< a,b,c,d,s \| a²=b²=d²=c, c²=1, s³=1, d central, ba=abc, sa=bs, sb=abs >	1.2.3⁸.4¹⁴.6²⁴	?		C4	Q8	Pauli
	C₄²⋊C3	C₄²⋊C3 3	< a,b,s \| a⁴=b⁴=s³=1, ba=ab, ca=bc, cb=a³b³c >	1.2³.3³².4¹²	(aecg)(bfdh)(imko)(jnlp), (bdc)(eni)(fok)(gml)(hpj)		1	C₄²	C₄²
	(C₂²,C₂²)⋊C3 The pullback of A4×A4 by C3	(C₂²,C₂²)⋊C3 C₂²⋊A4 50	< a,b,p,q,s \| a²=b²=p²=q²=s³=1, a,b,p,q all commute, sa=bs, sb=abs, sp=qs, sq=pqs >	1.2¹⁵.3³²	(ab)(cd), (pq)(rs), (abc)(pqr)		1	C₂⁴	C₂⁴

The image to the left shows how how some of the groups listed above are related (direct products are omitted from this diagram). The names of the groups, as given above, are shown in black. Green lines join groups to their normal subgroups. Red lines join groups to their quotient groups. The blue above each group shows it broken down into components; the light blue arcs indicate what the central extensions by C2 act on.

We can expand the diagram above to include all the groups listed in the table. The resulting diagram is below. The meanings of the colours of the lines is:

red	enlarge the group by adding a centre
dashed red	enlarge the group by adding two "centres", such that neither is central
green	enlarge the group by building a semidirect product of it
blue	enlarge the group by forming a direct product
pink	enlarge the group using it to build a semidirect product of something else

Notes

Orange triangles and up-arrows ↑ are used on this page to indicate central extensions.

The letters used for the permutations relate to the grids used used in the page GL(2,3), with the letters in these positions.

More miscellaneous short pages on finite groups
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