S₄×#6732 #72275836

order	5760
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂²

non-solvable

Aliases

direct product

S₄×#6732

Splittings

derived

A₄×SL(2,𝔽₅) ⋊ C₂²

Subgroups

	subgroup	G/H
center, hypercenter	C₄	S₄×A₅
commutator, hypocenter	A₄×SL(2,𝔽₅)	C₂²
Frattini	C₂	C₂×S₄×A₅
Fitting	C₂²×C₄	D₃×A₅
socle	C₂³	C₂×D₃×A₅
cosocle	C₂×A₄	C₂²×A₅
LCS	G ▷ A₄×SL(2,𝔽₅)
UCS	C₁ ◁ C₄

899 conjugacy classes of subgroups; only the 23 normal subgroups are shown.

normal subgroups only

This group has 899 conjugacy classes of subgroups — too many to draw. Only its 23 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) = 28
transitive degree	μ_t(G) = 72

A minimal faithful permutation representation — 2 generators on 28 points:

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

Conjugacy classes & character table

90 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	S₄×GL(2,𝔽₅)
maximal normal subgroup of	S₄×GL(2,𝔽₅)
maximal quotient of	GL(2,𝔽₃)×#6732

GAP

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

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