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C₂²×C₃⋊(C₃⋊C₈) #64129

order	288
isoclinism	isoclinic class of C₃⋊D₃
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂³×C₃

Aliases

C₂²×C₃⋊(C₃⋊C₈)

Splittings

C₃² ⋊ C₂²×C₈

Subgroups

	subgroup	G/H
center, hypercenter	C₂²×C₄	C₃⋊D₃
commutator, hypocenter	C₃²	C₂²×C₈
Frattini	C₄	C₂²×C₃⋊D₃
Fitting	C₂²×C₃²×C₄	C₂
socle, prime radical	C₂³×C₃²	C₄
cosocle	C₃²×C₄	C₂³
LCS	G ▷ C₃²
UCS	C₁ ◁ C₂²×C₄

228 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 18
transitive degree	μ_t(G) = 288

A minimal faithful permutation representation — 3 generators on 18 points:

(1,2)(3,4)(6,7)(9,10)(11,12,13,15,14,16,17,18);(3,4)(5,6,7);(1,2)(8,9,10)

Conjugacy classes & character table

96 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10589155, #10589163, #10589165, …
maximal normal subgroup of	#10589155, #10589163, #10589165, …
maximal quotient of	C₂×C₄×C₃⋊(C₃⋊C₈), C₂×C₄⋊(C₃⋊(C₃⋊C₈)), #10592624, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(288, 777);