C₂×#13088295 #72277390

order	2880
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

non-solvable

Aliases

direct product

C₂×#13088295

Splittings

derived

C₃×SL(2,𝔽₅) ⋊ C₂³

Subgroups

	subgroup	G/H
center, hypercenter	C₂²	D₃×S₅
commutator, cosocle, hypocenter	C₃×SL(2,𝔽₅)	C₂³
Frattini	C₂	C₂×D₃×S₅
Fitting, socle	C₂²×C₃	C₂×S₅
LCS	G ▷ C₃×SL(2,𝔽₅)
UCS	C₁ ◁ C₂²

692 conjugacy classes of subgroups; only the 34 normal subgroups are shown.

normal subgroups only

This group has 692 conjugacy classes of subgroups — too many to draw. Only its 34 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 — 3 generators on 45 points:

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

Conjugacy classes & character table

60 conjugacy classes — view the full character table.

Relationships

maximal subgroup of

C₂×SO⁺(4,𝔽₅)

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:

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