C₅×#73003 #14510727

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

solvable

Aliases

direct product

C₅×#73003

Splittings

Fitting

C₅×#395 ⋊ D₃

Subgroups

	subgroup	G/H
center, hypercenter	C₂×C₅	#1020⋊C₂
commutator, cosocle, hypocenter	C₂²⋊SL(2,𝔽₃)	C₂²×C₅
Frattini	C₂³	C₂×C₅×S₄
Fitting	C₅×#395	D₃
socle	C₂³×C₅	C₂×S₄
LCS	G ▷ C₂²⋊SL(2,𝔽₃)
UCS	C₁ ◁ C₂×C₅

278 conjugacy classes of subgroups; only the 24 normal subgroups are shown.

normal subgroups only

This group has 278 conjugacy classes of subgroups — too many to draw. Only its 24 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) = 29
transitive degree	μ_t(G) = 120

A minimal faithful permutation representation — 2 generators on 29 points:

(1,2,3,4,5)(6,8,29,7,13,24,11,17,27,12,20,26)(9,16,15,10,19,21,18,28,25,14,23,22);(7,12)(8,15)(9,14)(10,18)(13,21)(16,26)(17,25)(19,29)(20,22)(23,27)(24,28)

Conjugacy classes & character table

105 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1920, 236425);