C₅×#73124 #14510848

order	1920
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₅ · Schur multiplier (H₂) = C₂²

solvable

Aliases

direct product

C₅×#73124

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	C₄⋊S₄
commutator	#41⋊C₃	C₂×C₄×C₅
Frattini	C₂×C₄	C₂×C₅×S₄
Fitting	C₂×C₅×#131	D₃
prime radical	C₂×C₅×#41⋊C₃	C₂²
cosocle	C₂×#41⋊C₃	C₂²×C₅
hypercenter	C₂×C₅×C₈	S₄
hypocenter	SL(2,𝔽₃)	C₅×C₂²⋊C₄
LCS	G ▷ #41⋊C₃ ▷ SL(2,𝔽₃)
UCS	C₁ ◁ C₂²×C₅ ◁ C₂×C₄×C₅ ◁ C₂×C₅×C₈

312 conjugacy classes of subgroups; only the 52 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 52 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) = 41
transitive degree	μ_t(G) = 640

A minimal faithful permutation representation — 2 generators on 41 points:

(1,2,3,4,5)(6,8,9,7)(10,32,39,16,13,31,20,27,26,11,36,41,15,17,38,29,25,34,12,33,24,19,21,40)(14,18,22,28,23,35,37,30);(6,7,9,8)(11,16)(12,20)(13,23)(14,25)(17,30)(18,33)(19,29)(21,22)(24,26)(27,35)(28,32)(31,41)(34,40)(36,37)

Conjugacy classes & character table

200 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1920, 236546);