D₆₈ #3627

order	136
isoclinism	isoclinic class of Dic₃₄
automorphism	Aut(G) = D₄×AGL(1,𝔽₁₇) (order 2176) · Inn(G) = C₂×D₁₇ · Out(G) = C₂×C₁₆
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral	D₆₈
unambiguous semidirect product	C₄⋊D₁₇

Fitting	C₄×C₁₇ ⋊ C₂
Sylow	C₁₇ ⋊ D₄

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

Polycyclic presentation:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(136, 6);