Dic₅₁ #6074

order	204
isoclinism	isoclinic class of D₅₁
automorphism	Aut(G) = C₂×D₃×AGL(1,𝔽₁₇) (order 3264) · Inn(G) = D₅₁ · Out(G) = C₂×C₁₆
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic	Dic₅₁
unambiguous semidirect product	C₃⋊Dic₁₇

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₅₁
commutator, hypocenter	C₃×C₁₇	C₄
Fitting, socle, prime radical, cosocle	C₂×C₃×C₁₇	C₂
LCS	G ▷ C₃×C₁₇
UCS	C₁ ◁ C₂

12 conjugacy classes of subgroups.

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 204

A minimal faithful permutation representation — 2 generators on 24 points:

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

Polycyclic presentation:

2^2,3,17|a^2=b,c^a=c^2,d^a=d^16

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(204, 3);