C₂×#72220369 #72252730

order	10368
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂²×C₃²

solvable

Aliases

direct product

C₂×#72220369

Splittings

Fitting	C₂×C₃²×H(5,𝔽₂) ⋊ C₃⋊D₃
derived	#72220368 ⋊ C₂²

Subgroups

	subgroup	G/H
center, socle, hypercenter	C₂²×C₃	C₃⋊(A₄⋊S₄)
commutator, cosocle, hypocenter	#72220368	C₂²
Frattini	C₂×C₃	C₂×C₃⋊(A₄⋊S₄)
Fitting	C₂×C₃²×H(5,𝔽₂)	C₃⋊D₃
LCS	G ▷ #72220368
UCS	C₁ ◁ C₂²×C₃

2929 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 2929 conjugacy classes of subgroups — too many to draw. Only its 106 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) = 35
transitive degree	μ_t(G) = 144

A minimal faithful permutation representation — 4 generators on 35 points:

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

Conjugacy classes & character table

142 conjugacy classes — view the full character table.

GAP

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

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