Dic₁₉ #660

order	76
isoclinism	isoclinic class of D₁₉
automorphism	Aut(G) = C₂×AGL(1,𝔽₁₉) (order 684) · Inn(G) = D₁₉ · Out(G) = C₂×C₉
holomorph	G ⋊ Aut(G) = #72227356
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic	Dic₁₉
unambiguous semidirect product	C₁₉⋊C₄

C₁₉ ⋊ C₄

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₁₉
commutator, hypocenter	C₁₉	C₄
Fitting, socle, prime radical, cosocle	C₂×C₁₉	C₂
LCS	G ▷ C₁₉
UCS	C₁ ◁ C₂

6 conjugacy classes of subgroups.

minimal degree	μ(G) = 23
transitive degree	μ_t(G) = 76

A minimal faithful permutation representation — 2 generators on 23 points:

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

Polycyclic presentation:

2^2,19|a^2=b,c^a=c^18

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(76, 1);