C₅×#85511 #14511249

order	1920
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₅×#85511

Subgroups

	subgroup	G/H
center	C₄×C₅	C₂²×S₄
commutator	C₂×A₄	C₂²×C₄×C₅
Frattini	C₄	C₂²×C₅×S₄
Fitting	C₂²×C₅×#41	D₃
socle	C₂³×C₅	C₂²×Dic₃
prime radical	C₅×A₄×#41	C₂
cosocle	C₄×A₄	C₂³×C₅
hypercenter	C₅×#41	S₄
hypocenter	A₄	C₅×#131
LCS	G ▷ C₂×A₄ ▷ A₄
UCS	C₁ ◁ C₄×C₅ ◁ C₅×#41

628 conjugacy classes of subgroups; only the 124 normal subgroups are shown.

normal subgroups only

This group has 628 conjugacy classes of subgroups — too many to draw. Only its 124 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)(12,18)(15,21)(19,24)(22,25);(10,12)(11,15)(13,19)(14,18)(16,22)(17,21)(20,24)(23,25);(7,8)(10,11,13,16,14,17,20,23)(12,15,19,22,18,21,24,25)

Conjugacy classes & character table

250 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1920, 236947);