#1007⋊C₃ #64275

order	288
automorphism	Aut(G) = C₂²×D₃×S₄ (order 576) · Inn(G) = C₂×D₃×A₄ · Out(G) = C₂²
homology	Abelianization (H₁) = C₂²×C₃ · Schur multiplier (H₂) = C₂

Aliases

	subgroup	G/H
center, Frattini	C₂	C₂×D₃×A₄
commutator, cosocle, hypocenter	C₃×Q₈	C₂²×C₃
Fitting	C₂×C₃×Q₈	C₂×C₃
socle	C₂×C₃	C₂²×A₄
hypercenter	C₂²	D₃×A₄
LCS	G ▷ C₃×Q₈
UCS	C₁ ◁ C₂ ◁ C₂²

99 conjugacy classes of subgroups.

minimal degree	μ(G) = 19
transitive degree	μ_t(G) = 48

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

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

Polycyclic presentation:

2^2,3,2^2,3,2|d^2=g,e^2=g,b^a=bg,f^a=f^2,d^c=de,e^c=d,e^d=eg

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(288, 923);