#127⋊C₉ #63427

order	288
isoclinism	isoclinic class of #127⋊C₃
automorphism	Aut(G) = C₃×#4626⋊C₂ (order 1152) · Inn(G) = #127⋊C₃ · Out(G) = C₂²×C₃
homology	Abelianization (H₁) = C₂×C₉ · Schur multiplier (H₂) = C₄

Aliases

derived	C₄² ⋊ C₂×C₉
Sylow	#127 ⋊ C₉

50 conjugacy classes of subgroups.

minimal degree	μ(G) = 21
transitive degree	μ_t(G) = 36

A minimal faithful permutation representation — 2 generators on 21 points:

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

Polycyclic presentation:

2,3^2,2^4|b^3=c,d^2=fg,e^2=f,d^a=dfg,e^a=ef,d^b=e,e^b=deg,f^b=g,g^b=fg

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(288, 75);