#1094⋊C₄ #88602

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

Aliases

unambiguous semidirect product	#1094⋊C₄
unambiguous semidirect product	((C₂²⋊C₉)⋊C₄)⋊C₃

72 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)(10,14,17,12,13,16)(11,15);(6,7,8)(9,17,12,15,13,14,11,10,16)

Polycyclic presentation:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(432, 249);