(C₂²⋊C₉)⋊C₈ #63420

order	288
isoclinism	isoclinic class of (C₂²⋊C₉)⋊C₂
automorphism	Aut(G) = C₂²×((C₂²⋊C₉)⋊C₂)⋊C₃ (order 864) · Inn(G) = (C₂²⋊C₉)⋊C₂ · Out(G) = C₂²×C₃
homology	Abelianization (H₁) = C₈ · Schur multiplier (H₂) = C₂

Aliases

(C₂²⋊C₉)⋊C₈

C₂²⋊C₉ ⋊ C₈

44 conjugacy classes of subgroups.

minimal degree	μ(G) = 21
transitive degree	μ_t(G) = 72

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

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

Polycyclic presentation:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(288, 68);