C₅×#73126 #14510850

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

solvable

Aliases

direct product

C₅×#73126

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×C₄⋊C₄
LCS	G ▷ #41⋊C₃ ▷ SL(2,𝔽₃)
UCS	C₁ ◁ C₂²×C₅ ◁ C₂×C₄×C₅ ◁ C₂×C₅×C₈

250 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 250 conjugacy classes of subgroups — too many to draw. Only its 60 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)(10,34,38,29,13,33,12,35,26,11,21,40,15,18,39,16,25,36,17,27,24,20,30,41)(14,19,31,32,23,37,22,28);(6,7,8,9)(11,16)(12,17)(13,23)(14,25)(18,32)(19,35)(20,29)(21,22)(24,26)(27,37)(28,34)(30,31)(33,41)(36,40)

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, 236548);