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C₂×#3776 #64240

order	288
isoclinism	isoclinic class of #3774
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

Splittings

C₃² ⋊ C₂×Q₁₆

Subgroups

	subgroup	G/H
center, hypercenter	C₂²	GO⁺(4,𝔽₂)
commutator, cosocle	C₃⋊Dic₃	C₂³
Frattini	C₂	C₂×GO⁺(4,𝔽₂)
Fitting, socle, prime radical	C₂²×C₃²	D₄
hypocenter	C₃²	C₂×Q₁₆
LCS	G ▷ C₃⋊Dic₃ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂²

98 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 96

A minimal faithful permutation representation — 3 generators on 24 points:

(3,6,5,8,7,4)(9,11,13,19)(10,14,16,23)(12,21,22,18)(15,24,20,17);(6,8)(9,10,13,16)(11,17,19,24)(12,20,22,15)(14,21,23,18);(1,2)

Conjugacy classes & character table

30 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10593169, #10593352, #10593369, …
maximal normal subgroup of	#10593169, #10593352, #10593369, …
maximal quotient of	#10593379, #10593393, #10593395, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(288, 888);