S₄×#6910 #72259574

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

Aliases

direct product

S₄×#6910

Splittings

Fitting

C₂²×#6910 ⋊ D₃

Subgroups

	subgroup	G/H
center	C₃	C₃×S₄×H(3,𝔽₃)
commutator, cosocle	C₃²×A₄	C₂×C₃³
Frattini	C₃²	C₃³×S₄
Fitting	C₂²×#6910	D₃
socle	C₂²×C₃	C₃×D₃×H(3,𝔽₃)
prime radical	C₃×S₄×H(3,𝔽₃)	C₃
hypercenter	#6910	S₄
hypocenter	A₄	C₂×#6910
LCS	G ▷ C₃²×A₄ ▷ C₃×A₄ ▷ A₄
UCS	C₁ ◁ C₃ ◁ C₃³ ◁ #6910

1156 conjugacy classes of subgroups; only the 132 normal subgroups are shown.

normal subgroups only

This group has 1156 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)(6,11,12)(7,15,16)(9,19,23)(14,26,27)(18,20,24)(21,30,25);(2,3,4)(5,16,27,17,15,14,8,7,26)(6,13,30,11,22,21,12,10,25)(9,20,31,23,24,28,19,18,29);(5,11,19,17,12,9,8,6,23)(7,22,20,16,10,24,15,13,18)(14,25,31,26,30,28,27,21,29)

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