Dic₁₈ #607

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) = C₁

Aliases

dicyclic	Dic₁₈
unambiguous semidirect product	C₉⋊Q₈

C₉ ⋊ Q₈

	subgroup	G/H
center	C₂	C₂×D₉
commutator, cosocle	C₂×C₉	C₂²
Frattini, socle, prime radical	C₂×C₃	C₂×D₃
Fitting	C₄×C₉	C₂
hypercenter	C₄	D₉
hypocenter	C₉	Q₈
LCS	G ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₂ ◁ C₄

18 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)(5,8,7,6)(9,10,11,12,13,14,15,16,17);(1,5,3,7)(2,6,4,8)(10,17)(11,16)(12,15)(13,14)

Polycyclic presentation:

2^3,3^2|a^2=c,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, 4);