Dic₃₆ #3661

order	144
isoclinism	isoclinic class of Dic₃₆
automorphism	Aut(G) = AGL(1,ℤ/8)×AGL(1,ℤ/9) (order 1728) · Inn(G) = D₃₆ · Out(G) = C₂³×C₃
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

Dic₃₆

C₉ ⋊ Q₁₆

27 conjugacy classes of subgroups.

minimal degree	μ(G) = 25
transitive degree	μ_t(G) = 144

A minimal faithful permutation representation — 2 generators on 25 points:

(1,2,3,4,5,6,7,8)(9,16,15,14,13,12,11,10)(17,18,19,20,21,22,23,24,25);(1,9,5,13)(2,10,6,14)(3,11,7,15)(4,12,8,16)(18,25)(19,24)(20,23)(21,22)

Polycyclic presentation:

2^4,3^2|a^2=d,b^2=cd,c^2=d,e^3=f^2,b^a=bc,c^a=cd,e^a=e^2f,f^a=f^2

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(144, 4);