D₁₃₃ #63175

order	266
isoclinism	isoclinic class of D₁₃₃
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral	D₁₃₃
unambiguous semidirect product	C₇⋊D₁₉
unambiguous semidirect product	C₁₉⋊D₇

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₇×C₁₉	C₂
LCS	G ▷ C₇×C₁₉
UCS	C₁

8 conjugacy classes of subgroups.

minimal degree	μ(G) = 26
transitive degree	μ_t(G) = 133

A minimal faithful permutation representation — 2 generators on 26 points:

(1,2,3,4,5,6,7)(8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26);(2,7)(3,6)(4,5)(9,26)(10,25)(11,24)(12,23)(13,22)(14,21)(15,20)(16,19)(17,18)

Polycyclic presentation:

2,7,19|b^a=b^6,c^a=c^18

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(266, 3);