D₇₂ #3665

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) = C₂

Aliases

D₇₂

Fitting	C₈×C₉ ⋊ C₂
Sylow	C₉ ⋊ D₈

33 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:

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

Polycyclic presentation:

2^4,3^2|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, 8);