C₅×#85404 #14511142

order	1920
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂³

solvable

Aliases

direct product

C₅×#85404

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	C₂²×S₄
commutator, cosocle	C₂×SL(2,𝔽₃)	C₂³×C₅
Frattini	C₂²	C₂²×C₅×S₄
Fitting	C₅×#550	D₃
prime radical	C₅×C₄⋊(#41⋊C₃)	C₂
hypercenter	C₅×C₄⋊C₄	S₄
hypocenter	SL(2,𝔽₃)	C₂×C₅×D₄
LCS	G ▷ C₂×SL(2,𝔽₃) ▷ SL(2,𝔽₃)
UCS	C₁ ◁ C₂²×C₅ ◁ C₅×C₄⋊C₄

514 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 514 conjugacy classes of subgroups — too many to draw. Only its 72 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 — 3 generators on 41 points:

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

Conjugacy classes & character table

170 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1920, 236840);