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C₈×C₉⋊C₉ #10619515

order	648
nilpotency class	2
isoclinism	isoclinic class of H(3,𝔽₃)
automorphism	?
homology	Abelianization (H₁) = C₃×C₈×C₉ · Schur multiplier (H₂) = C₃

Aliases

direct product	C₈×C₉⋊C₉
unambiguous semidirect product	(C₈×C₉)⋊C₉

Subgroups

	subgroup	G/H
center	C₃²×C₈	C₃²
commutator	C₃	C₃×C₈×C₉
Frattini, cosocle	C₃²×C₄	C₂×C₃²
socle, prime radical	C₂×C₃²	C₃²×C₄
LCS	G ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₃²×C₈ ◁ G

68 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 26
transitive degree	μ_t(G) = 648

A minimal faithful permutation representation — 2 generators on 26 points:

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

Conjugacy classes & character table

264 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	(C₉⋊C₉)⋊C₁₆, Dic₃₆⋊C₉, C₈×D₉⋊C₉, …
maximal subgroup of	(C₉⋊C₉)⋊C₁₆, C₁₆×C₉⋊C₉, Dic₃₆⋊C₉, …
maximal normal subgroup of	(C₉⋊C₉)⋊C₁₆, C₁₆×C₉⋊C₉, Dic₃₆⋊C₉, …
maximal quotient of	C₁₆×C₉⋊C₉, C₂×C₈×C₉⋊C₉, C₈×#6855, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(648, 34);