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C₂²×#638 #64369

order	288
isoclinism	isoclinic class of #634
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶×C₃

Aliases

C₂²×#638

Splittings

Fitting	C₂⁴×C₃² ⋊ C₂
Sylow	C₃² ⋊ C₂²×D₄

Subgroups

	subgroup	G/H
center	C₂³	C₂×C₃⋊D₃
commutator, cosocle	C₂×C₃²	C₂⁴
Frattini	C₂	C₂³×C₃⋊D₃
Fitting	C₂⁴×C₃²	C₂
socle	C₂³×C₃²	C₂²
hypercenter	C₂⁴	C₃⋊D₃
hypocenter	C₃²	C₂²×D₄
LCS	G ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂³ ◁ C₂⁴

708 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 14
transitive degree	μ_t(G) = 144

A minimal faithful permutation representation — 4 generators on 14 points:

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

Conjugacy classes & character table

84 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#63581, #10591704, #10591711, …
maximal normal subgroup of	C₂×#63581, #10591704, #10591711, …
maximal quotient of	C₂²×#3750, C₂×#64133, C₂²×#3752, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(288, 1017);