S₄×#7333 #72343018

order	6144
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂³

Aliases

direct product

S₄×#7333

Splittings

Fitting

C₂²×#7333 ⋊ D₃

Subgroups

	subgroup	G/H
center	C₄	S₄×#322
commutator	C₈×A₄	C₂×C₄×C₈
Frattini	#346	C₂²×S₄
Fitting	C₂²×#7333	D₃
socle	C₂³	D₃×#1313
prime radical	C₂²×S₄	#347
cosocle	A₄×#346	C₂³
hypercenter	#7333	S₄
hypocenter	A₄	C₂×#7333
LCS	G ▷ C₈×A₄ ▷ C₂×A₄ ▷ A₄
UCS	C₁ ◁ C₄ ◁ #346 ◁ #7333

1611 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 96

A minimal faithful permutation representation — 3 generators on 36 points:

(1,4,2,3)(5,8,13,12)(6,14,18,17)(7,15,23,22)(9,24,29,28)(10,25,11,19)(16,32,31,20)(21,33,30,35)(26,36,27,34);(2,4)(5,7,9,16,11,21,27,17,12,22,28,20,25,35,36,18,13,23,29,31,10,30,26,14,8,15,24,32,19,33,34,6);(2,4)(5,6,9,15,10,16,26,33,12,17,28,7,19,20,34,21,13,18,29,22,11,31,27,35,8,14,24,23,25,32,36,30)

Conjugacy classes & character table

230 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	S₄×AGL(1,ℤ/32)
maximal normal subgroup of	S₄×AGL(1,ℤ/32)

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:

gap> G := Group((1,4,2,3)(5,8,13,12)(6,14,18,17)(7,15,23,22)(9,24,29,28)(10,25,11,19)(16,32,31,20)(21,33,30,35)(26,36,27,34), (2,4)(5,7,9,16,11,21,27,17,12,22,28,20,25,35,36,18,13,23,29,31,10,30,26,14,8,15,24,32,19,33,34,6), (2,4)(5,6,9,15,10,16,26,33,12,17,28,7,19,20,34,21,13,18,29,22,11,31,27,35,8,14,24,23,25,32,36,30));

S4×#7333 #72343018