Dic₁₇ #593

order	68
isoclinism	isoclinic class of D₁₇
automorphism	Aut(G) = C₂×AGL(1,𝔽₁₇) (order 544) · Inn(G) = D₁₇ · Out(G) = C₂×C₈
holomorph	G ⋊ Aut(G) = #72218906
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

Dic₁₇

C₁₇ ⋊ C₄

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₁₇
commutator, hypocenter	C₁₇	C₄
Fitting, socle, prime radical, cosocle	C₂×C₁₇	C₂
LCS	G ▷ C₁₇
UCS	C₁ ◁ C₂

6 conjugacy classes of subgroups.

minimal degree	μ(G) = 21
transitive degree	μ_t(G) = 68

A minimal faithful permutation representation — 2 generators on 21 points:

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

Polycyclic presentation:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(68, 1);