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C₂²×#625 #64325

order	288
isoclinism	isoclinic class of #625
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

C₂²×#625

Splittings

C₃² ⋊ C₂²×D₄

Subgroups

	subgroup	G/H
center, hypercenter	C₂³	D₃²
commutator, cosocle	C₂×C₃²	C₂⁴
Frattini	C₂	C₂²×D₃²
Fitting, socle	C₂³×C₃²	C₂²
hypocenter	C₃²	C₂²×D₄
LCS	G ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂³

499 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 14
transitive degree	μ_t(G) = 96

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

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

Conjugacy classes & character table

60 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10591363, #10591381, #10591385, …
maximal normal subgroup of	#10591363, #10591381, #10591385, …
maximal quotient of	C₂²×#3713, C₂×#63822, C₂×#63823, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(288, 973);