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C₂×C₉×A₄ #6266

order	216
isoclinism	isoclinic class of A₄
automorphism	Aut(G) = C₃×D₃×S₄ (order 432) · Inn(G) = A₄ · Out(G) = C₂×C₃×D₃
homology	Abelianization (H₁) = C₂×C₃×C₉ · Schur multiplier (H₂) = C₂×C₃

Aliases

C₂×C₉×A₄

Splittings

Fitting	C₂³×C₉ ⋊ C₃
derived	C₂² ⋊ C₂×C₃×C₉
Sylow	C₂³ ⋊ C₃×C₉

Subgroups

	subgroup	G/H
center, hypercenter	C₂×C₉	A₄
commutator, hypocenter	C₂²	C₂×C₃×C₉
Frattini	C₃	C₂×C₃×A₄
Fitting	C₂³×C₉	C₃
socle	C₂³×C₃	C₃²
prime radical	C₂×C₃×A₄	C₃
cosocle	C₂²×C₃	C₂×C₃²
LCS	G ▷ C₂²
UCS	C₁ ◁ C₂×C₉

52 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 15
transitive degree	μ_t(G) = 54

A minimal faithful permutation representation — 2 generators on 15 points:

(1,2)(4,5,6);(3,6)(4,5)(7,11,15,9,14,10,12,8,13)

Conjugacy classes & character table

72 conjugacy classes — view the full character table.

Relationships

derived subgroup of	#11722241, #11722242, #11722243, …
maximal subgroup of	C₉×SL(2,ℤ/4), C₉⋊SL(2,ℤ/4), A₄×Dic₉, …
maximal normal subgroup of	C₉×SL(2,ℤ/4), C₉⋊SL(2,ℤ/4), A₄×Dic₉, …
maximal quotient of	C₄×C₉×A₄, C₂×C₉×SL(2,𝔽₃), C₉×#41⋊C₃, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(216, 103);