S₄×#6911 #72259566

order	5832
automorphism	?
homology	Abelianization (H₁) = C₂×C₃³ · Schur multiplier (H₂) = C₂×C₃⁴

Aliases

direct product

S₄×#6911

Splittings

Fitting

C₂²×#6911 ⋊ D₃

Subgroups

	subgroup	G/H
center	C₃	C₃×S₄×H(3,𝔽₃)
commutator, cosocle	C₃²×A₄	C₂×C₃³
Frattini	C₃²	C₃³×S₄
Fitting	C₂²×#6911	D₃
socle	C₂²×C₃	C₃×D₃×H(3,𝔽₃)
hypercenter	#6911	S₄
hypocenter	A₄	C₂×#6911
LCS	G ▷ C₃²×A₄ ▷ C₃×A₄ ▷ A₄
UCS	C₁ ◁ C₃ ◁ C₃³ ◁ #6911

1372 conjugacy classes of subgroups; only the 132 normal subgroups are shown.

normal subgroups only

This group has 1372 conjugacy classes of subgroups — too many to draw. Only its 132 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) = 31
transitive degree	μ_t(G) = 108

A minimal faithful permutation representation — 3 generators on 31 points:

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

Conjugacy classes & character table

175 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	S₄×#10624749
maximal normal subgroup of	S₄×#10624749

GAP

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

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