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C₃×AGL(1,𝔽₉) #6317

order	216
isoclinism	isoclinic class of AGL(1,𝔽₉)
automorphism	Aut(G) = C₂×AΓL(1,𝔽₉) (order 288) · Inn(G) = AGL(1,𝔽₉) · Out(G) = C₂²
homology	Abelianization (H₁) = C₃×C₈ · Schur multiplier (H₂) = C₁

Aliases

C₃×AGL(1,𝔽₉)

Splittings

Fitting, socle, Sylow	C₃³ ⋊ C₈
derived	C₃² ⋊ C₃×C₈

Subgroups

	subgroup	G/H
center, hypercenter	C₃	AGL(1,𝔽₉)
commutator, hypocenter	C₃²	C₃×C₈
Frattini	C₁	G
Fitting, socle	C₃³	C₈
prime radical	C₃×C₃⋊D₃	C₄
cosocle	#157	C₂×C₃
LCS	G ▷ C₃²
UCS	C₁ ◁ C₃

22 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 12
transitive degree	μ_t(G) = 24

A minimal faithful permutation representation — 2 generators on 12 points:

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

Conjugacy classes & character table

27 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×AGL(1,𝔽₉), C₃×AΓL(1,𝔽₉), AGL(1,𝔽₉)⋊D₃, …
maximal normal subgroup of	D₃×AGL(1,𝔽₉), C₃×AΓL(1,𝔽₉), AGL(1,𝔽₉)⋊D₃, …
maximal quotient of	C₃×#3771, C₂×C₃×AGL(1,𝔽₉), C₉×AGL(1,𝔽₉), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(216, 154);