C₉⋊SL(2,𝔽₃) #6202

order	216
isoclinism	isoclinic class of C₉⋊SL(2,𝔽₃)
automorphism	Aut(G) = C₃×D₃×A₄ (order 216) · Inn(G) = C₃×A₄ · Out(G) = C₂×C₃
homology	Abelianization (H₁) = C₃² · Schur multiplier (H₂) = C₁

Aliases

Fitting	Q₈×C₉ ⋊ C₃
Sylow	Q₈ ⋊ 3₋¹⁺²

27 conjugacy classes of subgroups.

minimal degree	μ(G) = 17
transitive degree	μ_t(G) = 72

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

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

Polycyclic presentation:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(216, 39);