#126⋊C₉ #63426

order	288
isoclinism	isoclinic class of #126⋊C₃
automorphism	Aut(G) = C₃×#560⋊C₃ (order 576) · Inn(G) = #126⋊C₃ · Out(G) = C₂×C₃
homology	Abelianization (H₁) = C₂×C₉ · Schur multiplier (H₂) = C₂

Aliases

derived	C₄² ⋊ C₂×C₉
Sylow	#126 ⋊ C₉

34 conjugacy classes of subgroups.

minimal degree	μ(G) = 25
transitive degree	μ_t(G) = 72

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

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

Polycyclic presentation:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(288, 74);