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C₂²×C₃²×Dic₃ #89061

order	432
isoclinism	isoclinic class of D₃
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃²×C₄ · Schur multiplier (H₂) = C₂³×C₃

Aliases

C₂²×C₃²×Dic₃

Splittings

derived	C₃ ⋊ C₂²×C₃²×C₄
Sylow	C₃³ ⋊ C₂²×C₄

Subgroups

	subgroup	G/H
center, hypercenter	C₂³×C₃²	D₃
commutator, hypocenter	C₃	C₂²×C₃²×C₄
Frattini	C₂	C₂²×C₃²×D₃
Fitting, socle, prime radical	C₂³×C₃³	C₂
cosocle	C₂×C₃	C₂³×C₃²
LCS	G ▷ C₃
UCS	C₁ ◁ C₂³×C₃²

452 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 17
transitive degree	μ_t(G) = 144

A minimal faithful permutation representation — 3 generators on 17 points:

(6,7)(8,9,10)(14,15,16,17);(3,4)(5,6,7)(11,12,13);(1,2)

Conjugacy classes & character table

216 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₃×#63579, #11722061, C₃²×C₃⋊#95, …
maximal normal subgroup of	C₃×#63579, #11722061, C₃²×C₃⋊#95, …
maximal quotient of	C₂²×C₃²×C₃⋊C₈, C₂×C₃²×#208, C₂×C₃²×C₄×Dic₃, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(432, 708);