S₄×#1369 #72342878

order	3072
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂³

Aliases

direct product

S₄×#1369

Splittings

Fitting

C₂²×#1369 ⋊ D₃

Subgroups

	subgroup	G/H
center	C₄	S₄×#106
commutator	C₈×A₄	C₂²×C₈
Frattini	C₄×C₈	C₂²×S₄
Fitting	C₂²×#1369	D₃
socle	C₂³	D₃×#334
prime radical	C₂²×S₄	#108
cosocle	C₄×C₈×A₄	C₂³
hypercenter	#1369	S₄
hypocenter	A₄	C₂×#1369
LCS	G ▷ C₈×A₄ ▷ C₄×A₄ ▷ C₂×A₄ ▷ A₄
UCS	C₁ ◁ C₄ ◁ C₂×C₈ ◁ C₄×C₈ ◁ #1369

1059 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:

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

Conjugacy classes & character table

160 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	S₄×#7457, S₄×#7390, S₄×#12068
maximal normal subgroup of	S₄×#7457, S₄×#7390, S₄×#12068

GAP

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

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

S4×#1369 #72342878