3₋¹⁺²⋊C₄ #1082

order	108
isoclinism	isoclinic class of AGL(1,ℤ/9)
automorphism	Aut(G) = C₂×AGL(1,ℤ/9) (order 108) · Inn(G) = AGL(1,ℤ/9) · Out(G) = C₂
holomorph	G ⋊ Aut(G) = #72228965
homology	Abelianization (H₁) = C₃×C₄ · Schur multiplier (H₂) = C₁

Aliases

unambiguous semidirect product	3₋¹⁺²⋊C₄
unambiguous semidirect product	Dic₉⋊C₃

derived	C₉ ⋊ C₃×C₄
Sylow	3₋¹⁺² ⋊ C₄

20 conjugacy classes of subgroups.

minimal degree	μ(G) = 13
transitive degree	μ_t(G) = 36

A minimal faithful permutation representation — 2 generators on 13 points:

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

Polycyclic presentation:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(108, 9);