#3697⋊C₃ #88708

order	432
isoclinism	isoclinic class of 3₋¹⁺²⋊Q₈
automorphism	Aut(G) = C₂×D₄×AGL(1,ℤ/9) (order 864) · Inn(G) = C₂×AGL(1,ℤ/9) · Out(G) = C₂³
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂²

Aliases

Fitting	C₂×C₄×3₋¹⁺² ⋊ C₂
Sylow	3₋¹⁺² ⋊ #41

128 conjugacy classes of subgroups.

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

A minimal faithful permutation representation — 3 generators on 17 points:

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

Polycyclic presentation:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(432, 355);