2₋¹⁺⁴⋊C₉ #63701

order	288
automorphism	Aut(G) = C₃×D₄×S₄ (order 576) · Inn(G) = C₂²×A₄ · Out(G) = C₂²×C₃
homology	Abelianization (H₁) = C₂²×C₉ · Schur multiplier (H₂) = C₂

Aliases

2₋¹⁺⁴⋊C₉

2₋¹⁺⁴ ⋊ C₉

79 conjugacy classes of subgroups.

minimal degree	μ(G) = 25
transitive degree	μ_t(G) = 144

A minimal faithful permutation representation — 2 generators on 25 points:

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

Polycyclic presentation:

2^2,3^2,2^3|c^3=d,e^2=g,f^2=g,b^a=bg,e^c=f,f^c=ef,f^e=fg

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(288, 349);