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C₂×#3773 #64234

order	288
isoclinism	isoclinic class of GO⁺(4,𝔽₂)
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

Splittings

C₃² ⋊ C₂×C₄⋊C₄

Subgroups

	subgroup	G/H
center, hypercenter	C₂²	GO⁺(4,𝔽₂)
commutator	C₃⋊D₃	C₂²×C₄
Frattini	C₂	C₂×GO⁺(4,𝔽₂)
Fitting, socle	C₂²×C₃²	D₄
prime radical	C₂²×C₃⋊D₃	C₂²
cosocle	C₂×C₃⋊D₃	C₂³
hypocenter	C₃²	C₂×C₄⋊C₄
LCS	G ▷ C₃⋊D₃ ▷ C₃²
UCS	C₁ ◁ C₂²

146 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 12
transitive degree	μ_t(G) = 48

A minimal faithful permutation representation — 3 generators on 12 points:

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

Conjugacy classes & character table

36 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10593166, #10593338, #10593339, …
maximal normal subgroup of	#10593166, #10593338, #10593339, …
maximal quotient of	C₂×#63732, #10593327, #10593328, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(288, 882);