S₄×C₅₃ #11929685

order	1272
isoclinism	isoclinic class of S₄
automorphism	?
homology	Abelianization (H₁) = C₂×C₅₃ · Schur multiplier (H₂) = C₂

Aliases

direct product	S₄×C₅₃
unambiguous semidirect product	A₄⋊(C₂×C₅₃)

Splittings

Fitting, socle	C₂²×C₅₃ ⋊ D₃
derived	A₄ ⋊ C₂×C₅₃

Subgroups

	subgroup	G/H
center, hypercenter	C₅₃	S₄
commutator, cosocle, hypocenter	A₄	C₂×C₅₃
Frattini	C₁	G
Fitting, socle	C₂²×C₅₃	D₃
LCS	G ▷ A₄
UCS	C₁ ◁ C₅₃

22 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 57
transitive degree	μ_t(G) = 212

A minimal faithful permutation representation — 2 generators on 57 points:

(1,2,3,4)(5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57);(1,2)

Conjugacy classes & character table

265 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1272, 34);

S4×C53 #11929685