C₂×#3689 #63688

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

Q₈⋊C₉ ⋊ C₂²

	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	C₂²×C₃	S₄
LCS	G ▷ Q₈⋊C₉
UCS	C₁ ◁ C₂²

81 conjugacy classes of subgroups.

minimal degree	μ(G) = 19
transitive degree	μ_t(G) = 144

A minimal faithful permutation representation — 2 generators on 19 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(288, 336);