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C₈₁⋊C₈ #10619482

order	648
isoclinism	isoclinic class of D₈₁
automorphism	?
homology	Abelianization (H₁) = C₈ · Schur multiplier (H₂) = C₁

Aliases

unambiguous semidirect product

C₈₁⋊C₈

Splittings

C₈₁ ⋊ C₈

Subgroups

	subgroup	G/H
center, hypercenter	C₄	D₈₁
commutator, hypocenter	C₈₁	C₈
Frattini	C₄×C₂₇	D₃
Fitting, cosocle	C₄×C₈₁	C₂
socle, prime radical	C₂×C₃	Dic₂₇
LCS	G ▷ C₈₁
UCS	C₁ ◁ C₄

20 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 89
transitive degree	μ_t(G) = 648

Polycyclic presentation:

2^3,3^4|a^2=b,b^2=c,d^3=e,e^3=f,f^3=g,d^a=d^2e^2f^2g^2,e^a=e^2f^2g^2,f^a=f^2g^2,g^a=g^2

Conjugacy classes & character table

168 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₈×D₈₁, #13046288, C₂×C₈₁⋊C₈, …
maximal normal subgroup of	C₈×D₈₁, #13046288, C₂×C₈₁⋊C₈, …
maximal quotient of	C₈₁⋊C₁₆, C₂×C₈₁⋊C₈, C₂₄₃⋊C₈, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(648, 1);