C₅×#85304 #14511042

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

Aliases

direct product

C₅×#85304

Subgroups

	subgroup	G/H
center	C₂²×C₅	C₂²×S₄
commutator, cosocle	C₂²×A₄	C₂³×C₅
Frattini	C₂²	C₂²×C₅×S₄
Fitting	C₂²×C₅×C₄⋊C₄	D₃
socle	C₂⁴×C₅	C₂²×D₃
prime radical	C₂²×C₅×A₄	C₂³
hypercenter	C₅×C₄⋊C₄	S₄
hypocenter	A₄	C₅×#125
LCS	G ▷ C₂²×A₄ ▷ A₄
UCS	C₁ ◁ C₂²×C₅ ◁ C₅×C₄⋊C₄

510 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 510 conjugacy classes of subgroups — too many to draw. Only its 80 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) = 480

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

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

Conjugacy classes & character table

190 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1920, 236740);