C₅×#85440 #14511178

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

Aliases

direct product

C₅×#85440

Splittings

Fitting

C₂³×C₅×C₈ ⋊ D₃

Subgroups

	subgroup	G/H
center	C₄×C₅	C₄⋊S₄
commutator, cosocle	C₄×A₄	C₂³×C₅
Frattini	C₄	C₂²×C₅×S₄
Fitting	C₂³×C₅×C₈	D₃
socle	C₂³×C₅	C₂×D₁₂
hypercenter	C₂×C₅×C₈	S₄
hypocenter	A₄	C₅×#135
LCS	G ▷ C₄×A₄ ▷ C₂×A₄ ▷ A₄
UCS	C₁ ◁ C₄×C₅ ◁ C₂×C₄×C₅ ◁ C₂×C₅×C₈

546 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 546 conjugacy classes of subgroups — too many to draw. Only its 76 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) = 25
transitive degree	μ_t(G) = 240

A minimal faithful permutation representation — 3 generators on 25 points:

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

Conjugacy classes & character table

220 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1920, 236876);