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C₂³×C₃×Dic₃ #64353

order	288
isoclinism	isoclinic class of D₃
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

C₂³×C₃×Dic₃

Splittings

derived	C₃ ⋊ C₂³×C₃×C₄
Sylow	C₃² ⋊ C₂³×C₄

Subgroups

	subgroup	G/H
center, hypercenter	C₂⁴×C₃	D₃
commutator, hypocenter	C₃	C₂³×C₃×C₄
Frattini	C₂	C₂³×C₃×D₃
Fitting, socle, prime radical	C₂⁴×C₃²	C₂
cosocle	C₂×C₃	C₂⁴×C₃
LCS	G ▷ C₃
UCS	C₁ ◁ C₂⁴×C₃

539 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 16
transitive degree	μ_t(G) = 96

A minimal faithful permutation representation — 4 generators on 16 points:

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

Conjugacy classes & character table

144 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10591693, C₂×#63579, #10591696, …
maximal normal subgroup of	#10591693, C₂×#63579, #10591696, …
maximal quotient of	C₂³×C₃×C₃⋊C₈, C₂²×C₃×#208, C₂²×C₃×C₄×Dic₃, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(288, 1001);