SL(2,𝔽₃)×#6732 #72276705

order	5760
automorphism	?
homology	Abelianization (H₁) = C₂×C₃ · Schur multiplier (H₂) = C₁

non-solvable

Aliases

direct product

SL(2,𝔽₃)×#6732

Splittings

derived

Q₈×SL(2,𝔽₅) ⋊ C₂×C₃

Subgroups

	subgroup	G/H
center, hypercenter	C₂×C₄	A₄×A₅
commutator, hypocenter	Q₈×SL(2,𝔽₅)	C₂×C₃
Frattini, socle	C₂²	C₂×A₄×A₅
Fitting	C₄×Q₈	C₃×A₅
cosocle	C₂×Q₈	C₂×C₃×A₅
LCS	G ▷ Q₈×SL(2,𝔽₅)
UCS	C₁ ◁ C₂×C₄

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

normal subgroups only

This group has 435 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) = 32
transitive degree	μ_t(G) = 192

A minimal faithful permutation representation — 2 generators on 32 points:

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

Conjugacy classes & character table

126 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	SL(2,𝔽₃)×GL(2,𝔽₅), #72276844, GL(2,𝔽₃)×#6732
maximal normal subgroup of	SL(2,𝔽₃)×GL(2,𝔽₅), #72276844, GL(2,𝔽₃)×#6732

GAP

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

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