C₂×#72221889 #72244365

order	8748
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃² · Schur multiplier (H₂) = C₂×C₃⁵

Aliases

direct product

C₂×#72221889

Splittings

Fitting	C₂×#14554264 ⋊ C₂
derived	C₃⁵ ⋊ C₂²×C₃²
Sylow	#14554264 ⋊ C₂²

Subgroups

	subgroup	G/H
center, hypercenter	C₂	#72221889
commutator, cosocle, hypocenter	C₃⁵	C₂²×C₃²
Frattini	C₃³	C₂×C₃²×C₃⋊D₃
Fitting	C₂×#14554264	C₂
socle	C₂×C₃²	C₃×C₃⋊((C₃⋊D₃)⋊C₃)
LCS	G ▷ C₃⁵
UCS	C₁ ◁ C₂

6062 conjugacy classes of subgroups; only the 184 normal subgroups are shown.

normal subgroups only

This group has 6062 conjugacy classes of subgroups — too many to draw. Only its 184 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) = 38
transitive degree	μ_t(G) = 162

A minimal faithful permutation representation — 3 generators on 38 points:

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

Conjugacy classes & character table

198 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72221898
maximal normal subgroup of	C₂×#72221898

GAP

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

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