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C₄×#624 #63882

order	288
isoclinism	isoclinic class of D₃²
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂²×C₄

Aliases

Splittings

C₃² ⋊ C₂×C₄²

Subgroups

	subgroup	G/H
center, hypercenter	C₂×C₄	D₃²
commutator, hypocenter	C₃²	C₂×C₄²
Frattini	C₂²	C₂×D₃²
Fitting	C₂×C₃²×C₄	C₂²
socle, cosocle	C₂²×C₃²	C₂³
prime radical	C₂²×C₃⋊D₃	C₂²
LCS	G ▷ C₃²
UCS	C₁ ◁ C₂×C₄

243 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 14
transitive degree	μ_t(G) = 48

A minimal faithful permutation representation — 3 generators on 14 points:

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

Conjugacy classes & character table

72 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10590043, #10590048, #10590643, …
maximal normal subgroup of	#10590043, #10590048, #10590643, …
maximal quotient of	#10590247, C₄×#3710, #10590321, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(288, 530);