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C₃×C₂²⋊C₉ #1093

order	108
isoclinism	isoclinic class of A₄
automorphism	Aut(G) = S₄×#272 (order 1296) · Inn(G) = A₄ · Out(G) = C₂×#272
homology	Abelianization (H₁) = C₃×C₉ · Schur multiplier (H₂) = C₂×C₃

Aliases

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

Splittings

C₂² ⋊ C₃×C₉

Subgroups

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

26 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 16
transitive degree	μ_t(G) = 54

A minimal faithful permutation representation — 2 generators on 16 points:

(2,3,4)(8,9,11,10,12,14,13,15,16);(1,4)(2,3)(5,6,7)

Conjugacy classes & character table

36 conjugacy classes — view the full character table.

Relationships

derived subgroup of	C₃⋊((C₂²⋊C₉)⋊C₂), C₃⋊((C₂²⋊C₉)⋊C₄), D₃×(C₂²⋊C₉)⋊C₂, …
maximal subgroup of	C₃×(C₂²⋊C₉)⋊C₂, C₃⋊((C₂²⋊C₉)⋊C₂), D₃×C₂²⋊C₉, …
maximal normal subgroup of	C₃×(C₂²⋊C₉)⋊C₂, C₃⋊((C₂²⋊C₉)⋊C₂), D₃×C₂²⋊C₉, …
maximal quotient of	C₃×Q₈⋊C₉, C₂×C₃×C₂²⋊C₉, C₃×C₂²⋊C₂₇, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(108, 20);