C₅×#85359 #14511097

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

solvable

Aliases

direct product

C₅×#85359

Splittings

Fitting

C₅×#519 ⋊ D₃

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	C₂²×S₄
commutator, cosocle	C₂×SL(2,𝔽₃)	C₂³×C₅
Frattini	C₂²	C₂²×C₅×S₄
Fitting	C₅×#519	D₃
prime radical	C₂×C₅×#983	C₂
hypercenter	C₅×C₂²⋊C₄	S₄
hypocenter	SL(2,𝔽₃)	C₂×C₅×D₄
LCS	G ▷ C₂×SL(2,𝔽₃) ▷ SL(2,𝔽₃)
UCS	C₁ ◁ C₂²×C₅ ◁ C₅×C₂²⋊C₄

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

normal subgroups only

This group has 518 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) = 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)(10,24,29)(11,18,33,17,28,32)(12,22,39,21,36,38)(13,14,27)(16,34,41)(19,26)(20,30,40)(23,37);(10,12)(11,16)(13,21)(14,22)(15,23)(17,20)(18,30)(19,31)(24,36)(25,37)(26,35)(27,38)(28,34)(29,39)(32,40)(33,41);(7,9)(10,11)(12,20)(13,17)(14,26)(15,28)(16,21)(18,25)(19,24)(22,31)(23,30)(27,33)(29,32)(34,37)(35,36)(38,40)(39,41)

Conjugacy classes & character table

170 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1920, 236795);