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C₂×Q₈×C₉² #13047501

order	1296
nilpotency class	2
isoclinism	isoclinic class of D₄
automorphism	?
homology	Abelianization (H₁) = C₂³×C₉² · Schur multiplier (H₂) = C₂²×C₉

Aliases

C₂×Q₈×C₉²

Subgroups

	subgroup	G/H
center	C₂²×C₉²	C₂²
commutator	C₂	C₂³×C₉²
Frattini, cosocle	C₂×C₃²	C₂³×C₃²
socle, prime radical	C₂²×C₃²	C₂²×C₃²
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₉² ◁ G

437 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 1296

A minimal faithful permutation representation — 3 generators on 28 points:

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

Conjugacy classes & character table

810 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	C₂×Q₈×C₉⋊D₉, C₂×#13046820, C₂×#13047456
maximal subgroup of	C₂×Q₈×C₉⋊D₉, C₂×C₉²×SD₁₆, C₂×#13046820, …
maximal normal subgroup of	C₂×Q₈×C₉⋊D₉, C₂×C₉²×SD₁₆, C₂×#13046820, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1296, 1219);