C₃⋊(C₉⋊C₈) #6179

order	216
isoclinism	isoclinic class of C₃⋊D₉
automorphism	Aut(G) = C₂²×#72210336 (order 11664) · Inn(G) = C₃⋊D₉ · Out(G) = C₂²×#272
homology	Abelianization (H₁) = C₈ · Schur multiplier (H₂) = C₃

Aliases

unambiguous semidirect product	C₃⋊(C₉⋊C₈)
unambiguous semidirect product	C₉⋊(C₃⋊C₈)

C₃×C₉ ⋊ C₈

40 conjugacy classes of subgroups.

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 216

A minimal faithful permutation representation — 3 generators on 20 points:

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

Polycyclic presentation:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(216, 16);