(C₂²⋊C₉)⋊C₄ #3690

order	144
isoclinism	isoclinic class of (C₂²⋊C₉)⋊C₂
automorphism	Aut(G) = C₂×((C₂²⋊C₉)⋊C₂)⋊C₃ (order 432) · Inn(G) = (C₂²⋊C₉)⋊C₂ · Out(G) = C₂×C₃
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₂

Aliases

(C₂²⋊C₉)⋊C₄

C₂²⋊C₉ ⋊ C₄

32 conjugacy classes of subgroups.

minimal degree	μ(G) = 17
transitive degree	μ_t(G) = 36

A minimal faithful permutation representation — 2 generators on 17 points:

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

Polycyclic presentation:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(144, 33);