D₃₆ #609

order	72
isoclinism	isoclinic class of Dic₁₈
automorphism	Aut(G) = D₄×AGL(1,ℤ/9) (order 432) · Inn(G) = C₂×D₉ · Out(G) = C₂²×C₃
holomorph	G ⋊ Aut(G) = #72220092
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral	D₃₆
unambiguous semidirect product	C₄⋊D₉

Fitting	C₄×C₉ ⋊ C₂
Sylow	C₉ ⋊ D₄

24 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)(5,6,7,8,9,10,11,12,13);(2,4)(6,13)(7,12)(8,11)(9,10)

Polycyclic presentation:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(72, 6);