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C₉×A₄ #1091

order	108
isoclinism	isoclinic class of A₄
automorphism	Aut(G) = C₃×D₃×S₄ (order 432) · Inn(G) = A₄ · Out(G) = C₂×C₃×D₃
homology	Abelianization (H₁) = C₃×C₉ · Schur multiplier (H₂) = C₂×C₃

Aliases

C₉×A₄

Splittings

Fitting	C₂²×C₉ ⋊ C₃
derived, Sylow	C₂² ⋊ C₃×C₉

Subgroups

	subgroup	G/H
center, hypercenter	C₉	A₄
commutator, hypocenter	C₂²	C₃×C₉
Frattini	C₃	C₃×A₄
Fitting	C₂²×C₉	C₃
socle, cosocle	C₂²×C₃	C₃²
prime radical	C₃×A₄	C₃
LCS	G ▷ C₂²
UCS	C₁ ◁ C₉

23 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 13
transitive degree	μ_t(G) = 36

A minimal faithful permutation representation — 2 generators on 13 points:

(2,3,4);(1,4)(2,3)(5,6,8,7,9,11,10,12,13)

Conjugacy classes & character table

36 conjugacy classes — view the full character table.

Relationships

derived subgroup of	C₉⋊S₄, C₉⋊SL(2,ℤ/4), D₉×S₄, …
maximal subgroup of	C₉×S₄, C₉⋊S₄, A₄×D₉, …
maximal normal subgroup of	C₉×S₄, C₉⋊S₄, A₄×D₉, …
maximal quotient of	C₉×SL(2,𝔽₃), C₂×C₉×A₄, A₄×C₂₇, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(108, 18);