(C₃⋊Dic₃)⋊C₃ #1081

order	108
isoclinism	isoclinic class of (C₃⋊D₃)⋊C₃
automorphism	Aut(G) = C₂×((C₃⋊D₃)⋊C₃)⋊C₂ (order 216) · Inn(G) = (C₃⋊D₃)⋊C₃ · Out(G) = C₂²
homology	Abelianization (H₁) = C₃×C₄ · Schur multiplier (H₂) = C₃

Aliases

derived	C₃² ⋊ C₃×C₄
Sylow	H(3,𝔽₃) ⋊ C₄

25 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:

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

Polycyclic presentation:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(108, 8);