C₄⋊(C₃⋊(C₃⋊D₃)) #6310

order	216
isoclinism	isoclinic class of #6308
automorphism	Aut(G) order 2426112
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂×C₃³

Aliases

Fitting	C₃³×C₄ ⋊ C₂
Sylow	C₃³ ⋊ D₄

224 conjugacy classes of subgroups.

minimal degree	μ(G) = 13
transitive degree	μ_t(G) = 108

A minimal faithful permutation representation — 4 generators on 13 points:

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

Polycyclic presentation:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(216, 147);