C₅×#85350 #14511088

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

solvable

Aliases

direct product

C₅×#85350

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	C₂²×S₄
commutator, hypocenter	SL(2,𝔽₃)	C₂²×C₄×C₅
Frattini	C₂²	C₂²×C₅×S₄
Fitting	C₅×#519	D₃
prime radical	C₂²×C₅×SL(2,𝔽₃)	C₂²
cosocle	C₂×SL(2,𝔽₃)	C₂³×C₅
hypercenter	C₅×C₂²⋊C₄	S₄
LCS	G ▷ SL(2,𝔽₃)
UCS	C₁ ◁ C₂²×C₅ ◁ C₅×C₂²⋊C₄

514 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 514 conjugacy classes of subgroups — too many to draw. Only its 92 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) = 41
transitive degree	μ_t(G) = 320

A minimal faithful permutation representation — 3 generators on 41 points:

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

Conjugacy classes & character table

230 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1920, 236786);