C₂×#72220370 #72252722

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

solvable

Aliases

direct product

C₂×#72220370

Splittings

Fitting	C₂×C₃²×H(5,𝔽₂) ⋊ C₂×C₃²
derived	C₃²×H(5,𝔽₂) ⋊ C₂²×C₃²

Subgroups

	subgroup	G/H
center, hypercenter	C₂²	#72213266
commutator, cosocle, hypocenter	C₃²×H(5,𝔽₂)	C₂²×C₃²
Frattini	C₂×C₃	C₂×D₃×A₄²
Fitting	C₂×C₃²×H(5,𝔽₂)	C₂×C₃²
socle	C₂²×C₃	D₃×A₄²
LCS	G ▷ C₃²×H(5,𝔽₂)
UCS	C₁ ◁ C₂²

3686 conjugacy classes of subgroups; only the 104 normal subgroups are shown.

normal subgroups only

This group has 3686 conjugacy classes of subgroups — too many to draw. Only its 104 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 — 2 generators on 35 points:

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

Conjugacy classes & character table

172 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)(3,23,24,17,12,22,7,9,19,14,15,8)(4,20,16,11,26,6,13,25,5,10,21,18)(27,28,35)(29,32,30)(31,34,33), (5,14,6)(9,15,10)(11,23,12)(16,17,18)(19,21,20)(24,26,25)(28,30)(29,33)(31,34)(32,35));