C₅×#85458 #14511196

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

Aliases

direct product

C₅×#85458

Splittings

Fitting

C₂²×C₅×D₈ ⋊ D₃

Subgroups

	subgroup	G/H
center	C₂×C₅	D₄×S₄
commutator, cosocle	C₄×A₄	C₂³×C₅
Frattini	C₄	C₂²×C₅×S₄
Fitting	C₂²×C₅×D₈	D₃
socle	C₂³×C₅	D₃×D₄
hypercenter	C₅×D₈	S₄
hypocenter	A₄	C₅×#135
LCS	G ▷ C₄×A₄ ▷ C₂×A₄ ▷ A₄
UCS	C₁ ◁ C₂×C₅ ◁ C₄×C₅ ◁ C₅×D₈

552 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 552 conjugacy classes of subgroups — too many to draw. Only its 68 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)(11,16)(12,19)(13,22)(15,23)(18,25)(20,21);(10,12)(11,15)(13,19)(14,21)(16,18)(17,25)(20,22)(23,24);(7,8)(10,11)(12,18)(13,17)(14,16)(15,21)(19,23)(20,25)(22,24)

Conjugacy classes & character table

175 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1920, 236894);