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C₂×D₃×C₉ #1097

order	108
isoclinism	isoclinic class of D₃
automorphism	Aut(G) = C₂²×C₃×D₃ (order 72) · Inn(G) = D₃ · Out(G) = C₂²×C₃
holomorph	G ⋊ Aut(G) = AGL(1,ℤ/9)×#3811
homology	Abelianization (H₁) = C₂²×C₉ · Schur multiplier (H₂) = C₂

Aliases

direct product	C₂×D₃×C₉
unambiguous semidirect product	C₃⋊(C₂²×C₉)
unambiguous semidirect product	(C₂×C₃)⋊(C₂×C₉)

Splittings

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

Subgroups

	subgroup	G/H
center, hypercenter	C₂×C₉	D₃
commutator, hypocenter	C₃	C₂²×C₉
Frattini	C₃	C₂×C₃×D₃
Fitting	C₂×C₃×C₉	C₂
socle	C₂×C₃²	C₂×C₃
prime radical	C₂×C₃×D₃	C₃
cosocle	C₃²	C₂²×C₃
LCS	G ▷ C₃
UCS	C₁ ◁ C₂×C₉

34 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 14
transitive degree	μ_t(G) = 36

A minimal faithful permutation representation — 2 generators on 14 points:

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

Conjugacy classes & character table

54 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×Dic₉, #6194, #6195, …
maximal normal subgroup of	D₃×Dic₉, #6194, #6195, …
maximal quotient of	C₉×Dic₆, C₄×D₃×C₉, C₉×D₁₂, …
automorphism group of	C₂²×C₁₉, C₂²×C₂₇, D₃×C₁₉, …
outer automorphism group of	C₂²×C₁₉, C₂²×C₂₇, Q₈×C₁₉, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(108, 24);