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C₂×D₃×Dic₃ #3803

order	144
isoclinism	isoclinic class of D₃²
automorphism	Aut(G) = D₃²×#120 (order 1152) · Inn(G) = D₃² · Out(G) = #120
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

C₂×D₃×Dic₃

Splittings

C₃² ⋊ C₂²×C₄

Subgroups

	subgroup	G/H
center, hypercenter	C₂²	D₃²
commutator, hypocenter	C₃²	C₂²×C₄
Frattini	C₂	C₂×D₃²
Fitting, socle	C₂²×C₃²	C₂²
prime radical	C₂²×C₃×D₃	C₂
cosocle	C₂×C₃²	C₂³
LCS	G ▷ C₃²
UCS	C₁ ◁ C₂²

116 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 12
transitive degree	μ_t(G) = 48

A minimal faithful permutation representation — 3 generators on 12 points:

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

Conjugacy classes & character table

36 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₄×D₃×Dic₃, D₃×Dic₃⋊C₄, #63877, …
maximal normal subgroup of	C₄×D₃×Dic₃, D₃×Dic₃⋊C₄, #63877, …
maximal quotient of	C₂×D₃×C₃⋊C₈, D₃×#208, #63814, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(144, 146);