C₅×#73001 #14510725

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

solvable

Aliases

direct product

C₅×#73001

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
prime radical	C₅×C₂²⋊SL(2,𝔽₃)	C₂²
LCS	G ▷ C₂²⋊SL(2,𝔽₃)
UCS	C₁ ◁ C₂×C₅

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

normal subgroups only

This group has 226 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) = 45
transitive degree	μ_t(G) = 240

A minimal faithful permutation representation — 2 generators on 45 points:

(1,2,3,4,5)(6,10,8,11,9,7)(12,13)(14,41,29,27,30,42,19,25,32,16,17,40)(15,38,36,24,31,45,23,34,37,20,21,44)(18,26,33,43)(22,35,28,39);(8,13)(10,11)(14,15,19,23)(16,24,27,20)(17,28,30,22)(18,31,33,21)(25,35,41,39)(26,34,43,38)(29,37,32,36)(40,44,42,45)

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, 236423);