C₈×GL(2,𝔽₅) #72276990

order	3840
automorphism	?
homology	Abelianization (H₁) = C₄×C₈ · Schur multiplier (H₂) = C₄

non-solvable

Aliases

direct product

C₈×GL(2,𝔽₅)

Splittings

derived

SL(2,𝔽₅) ⋊ C₄×C₈

Subgroups

	subgroup	G/H
center, Fitting, hypercenter	C₄×C₈	S₅
commutator, hypocenter	SL(2,𝔽₅)	C₄×C₈
Frattini	C₄²	C₂×S₅
socle	C₂²	C₄×A₅⋊C₄
prime radical	C₂×#6732	C₂×C₄
cosocle	C₄×#6732	C₂²
LCS	G ▷ SL(2,𝔽₅)
UCS	C₁ ◁ C₄×C₈

572 conjugacy classes of subgroups; only the 44 normal subgroups are shown.

normal subgroups only

This group has 572 conjugacy classes of subgroups — too many to draw. Only its 44 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:

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

Conjugacy classes & character table

192 conjugacy classes — view the full character table.

GAP

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

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