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C₉×#157 #66417

order	324
isoclinism	isoclinic class of #157
automorphism	Aut(G) = C₂×C₃×AΓL(1,𝔽₉) (order 864) · Inn(G) = #157 · Out(G) = C₂³×C₃
homology	Abelianization (H₁) = C₄×C₉ · Schur multiplier (H₂) = C₃

Aliases

Splittings

Fitting, Sylow	C₃²×C₉ ⋊ C₄
derived	C₃² ⋊ C₄×C₉

Subgroups

	subgroup	G/H
center, hypercenter	C₉	#157
commutator, hypocenter	C₃²	C₄×C₉
Frattini	C₃	C₃×#157
Fitting	C₃²×C₉	C₄
socle	C₃³	C₃×C₄
prime radical, cosocle	C₃×C₃⋊D₃	C₂×C₃
LCS	G ▷ C₃²
UCS	C₁ ◁ C₉

38 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 15
transitive degree	μ_t(G) = 36

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

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

Conjugacy classes & character table

54 conjugacy classes — view the full character table.

Relationships

derived subgroup of	#13049183, #13049185, AGL(1,𝔽₉)⋊D₉
maximal subgroup of	C₉×AGL(1,𝔽₉), C₉⋊AGL(1,𝔽₉), D₉×#157, …
maximal normal subgroup of	C₉×AGL(1,𝔽₉), C₉⋊AGL(1,𝔽₉), D₉×#157, …
maximal quotient of	C₉×#622, C₂×C₉×#157, C₉×#1088, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(324, 109);