(C₃⋊Dic₃)⋊C₉ #66315

order	324
isoclinism	isoclinic class of (C₃⋊D₃)⋊C₃
automorphism	Aut(G) = C₂×C₃×((C₃⋊D₃)⋊C₃)⋊C₂ (order 648) · Inn(G) = (C₃⋊D₃)⋊C₃ · Out(G) = C₂²×C₃
homology	Abelianization (H₁) = C₄×C₉ · Schur multiplier (H₂) = C₃

Aliases

derived	C₃² ⋊ C₄×C₉
Sylow	C₃²⋊C₉ ⋊ C₄

45 conjugacy classes of subgroups.

minimal degree	μ(G) = 22
transitive degree	μ_t(G) = 36

A minimal faithful permutation representation — 2 generators on 22 points:

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

Polycyclic presentation:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(324, 7);