C₅×#85512 #14511250

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

Aliases

direct product

C₅×#85512

Subgroups

	subgroup	G/H
center	C₂×C₅	C₂×GL(2,ℤ/4)
commutator, cosocle	C₄×A₄	C₂³×C₅
Frattini	C₄	C₂²×C₅×S₄
Fitting	C₂²×C₅×#41	D₃
socle	C₂³×C₅	C₂×#67
prime radical	C₅×A₄×#41	C₂
hypercenter	C₅×#41	S₄
hypocenter	A₄	C₅×#137
LCS	G ▷ C₄×A₄ ▷ C₂×A₄ ▷ A₄
UCS	C₁ ◁ C₂×C₅ ◁ C₂×C₄×C₅ ◁ C₅×#41

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

normal subgroups only

This group has 604 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)(11,16)(12,19)(13,21)(15,22)(17,18)(23,24);(10,12)(11,15)(13,20)(14,19)(16,24)(17,22)(18,23)(21,25);(7,8)(10,11,14,18)(12,15,19,23)(13,16,21,17)(20,24,25,22)

Conjugacy classes & character table

160 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1920, 236948);