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(C₉⋊C₈)⋊A₄ #11720748

order	864
isoclinism	isoclinic class of D₉⋊A₄
automorphism	?
homology	Abelianization (H₁) = C₃×C₈ · Schur multiplier (H₂) = C₂

Aliases

unambiguous semidirect product	(C₉⋊C₈)⋊A₄
unambiguous semidirect product	(C₉⋊A₄)⋊C₈

Splittings

C₂²×C₉ ⋊ C₃×C₈

Subgroups

	subgroup	G/H
center, hypercenter	C₄	D₉⋊A₄
commutator, hypocenter	C₂²×C₉	C₃×C₈
Frattini	C₃×C₄	D₃×A₄
Fitting, cosocle	C₂²×C₄×C₉	C₂×C₃
socle	C₂³×C₃	C₃×Dic₃
prime radical	C₂×C₃×A₄	Dic₃
LCS	G ▷ C₂²×C₉
UCS	C₁ ◁ C₄

84 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 21
transitive degree	μ_t(G) = 216

Polycyclic presentation:

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

Conjugacy classes & character table

64 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₈×D₉⋊A₄, #3663⋊A₄, C₂×(C₉⋊C₈)⋊A₄, …
maximal normal subgroup of	C₈×D₉⋊A₄, #3663⋊A₄, C₂×(C₉⋊C₈)⋊A₄, …
maximal quotient of	(C₉⋊C₈)⋊SL(2,𝔽₃), (C₉⋊A₄)⋊C₁₆, #13136821, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(864, 715);