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C₂³×C₃⋊Dic₃ #64368

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

Aliases

C₂³×C₃⋊Dic₃

Splittings

C₃² ⋊ C₂³×C₄

Subgroups

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

708 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 16
transitive degree	μ_t(G) = 288

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

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

Conjugacy classes & character table

96 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#63579, #10591697, #10591705, …
maximal normal subgroup of	C₂×#63579, #10591697, #10591705, …
maximal quotient of	C₂³×C₃⋊(C₃⋊C₈), C₂²×#3748, C₂²×C₄×C₃⋊Dic₃, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(288, 1016);