S₄×#4223 #72254114

order	3936
automorphism	?
homology	Abelianization (H₁) = C₂×C₄ · Schur multiplier (H₂) = C₂²

Aliases

direct product

S₄×#4223

Splittings

Fitting, socle	C₂²×C₄₁ ⋊ C₄×D₃
derived	A₄×C₄₁ ⋊ C₂×C₄
Sylow	C₄₁ ⋊ C₄×S₄

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, hypocenter	A₄×C₄₁	C₂×C₄
Fitting, socle	C₂²×C₄₁	C₄×D₃
prime radical	S₄×D₄₁	C₂
cosocle	A₄×D₄₁	C₂²
LCS	G ▷ A₄×C₄₁
UCS	C₁

112 conjugacy classes of subgroups; only the 18 normal subgroups are shown.

normal subgroups only

This group has 112 conjugacy classes of subgroups — too many to draw. Only its 18 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 45
transitive degree	μ_t(G) = 164

A minimal faithful permutation representation — 2 generators on 45 points:

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

Conjugacy classes & character table

70 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	S₄×#4223⋊C₅
maximal normal subgroup of	S₄×#4223⋊C₅

GAP

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

gap> G := Group((1,4,3)(6,7,8,10)(9,12,13,19)(11,16,17,15)(14,21,22,32)(18,25,26,28)(20,29,30,24)(23,34,35,36)(27,38,39,44)(31,42,43,41)(33,45,40,37), (2,4)(5,6,9,14,23,36,45,29,16,7,11,18,27,40,32,44,42,25,12,20,31,43,30,19,28,41,38,21,33,39,26,17,10,15,24,37,34,35,22,13,8));