C₂×#3688 #63687

order	288
isoclinism	isoclinic class of #3688
automorphism	Aut(G) = D₄×((C₂²⋊C₉)⋊C₂)⋊C₃ (order 1728) · Inn(G) = (C₂²⋊C₉)⋊C₂ · Out(G) = C₃×D₄
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

	subgroup	G/H
center, hypercenter	C₂²	(C₂²⋊C₉)⋊C₂
commutator, cosocle, hypocenter	Q₈⋊C₉	C₂²
Frattini	C₂×C₃	C₂×S₄
Fitting	C₂×C₃×Q₈	D₃
socle, prime radical	C₂²×C₃	S₄
LCS	G ▷ Q₈⋊C₉
UCS	C₁ ◁ C₂²

65 conjugacy classes of subgroups.

minimal degree	μ(G) = 27
transitive degree	μ_t(G) = 288

A minimal faithful permutation representation — 2 generators on 27 points:

(4,6)(5,9)(7,8)(10,11)(12,17,21,24,16,26,23,18)(13,22,25,15,19,14,27,20);(1,2)(3,11,6,9,8,7,5,4,10)(14,20,23)(15,21,22)(17,24,27)(18,25,26)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(288, 335);