C₂×#72221908 #72244349

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

Aliases

direct product

C₂×#72221908

Splittings

Fitting	C₂×#14554265 ⋊ C₂
derived	C₃³×C₉ ⋊ C₂²×C₃²
Sylow	#14554265 ⋊ C₂²

Subgroups

	subgroup	G/H
center, hypercenter	C₂	#72221908
commutator, cosocle, hypocenter	C₃³×C₉	C₂²×C₃²
Frattini	C₃³	C₂×C₃²×C₃⋊D₃
Fitting	C₂×#14554265	C₂
socle	C₂×C₃²	C₃×C₃⋊((C₃⋊D₃)⋊C₃)
LCS	G ▷ C₃³×C₉
UCS	C₁ ◁ C₂

4298 conjugacy classes of subgroups; only the 184 normal subgroups are shown.

normal subgroups only

This group has 4298 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)(11,22,26,14,16,34)(12,30,32,18,19,31)(13,20)(15,33,36,21,25,27)(17,35,23,24,29,28);(10,15,17,13,24,21,20,11,14)(12,25,27,19,35,34,32,16,23)(18,26,29,31,36,33,30,28,22)(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×#72221920
maximal normal subgroup of	C₂×#72221920

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