D₁₅₃ #64558

order	306
isoclinism	isoclinic class of D₁₅₃
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral	D₁₅₃
unambiguous semidirect product	C₉⋊D₁₇
unambiguous semidirect product	C₁₇⋊D₉

	subgroup	G/H
center, hypercenter	C₁	G
commutator, Fitting, cosocle, hypocenter	C₉×C₁₇	C₂
Frattini	C₃	D₅₁
socle	C₃×C₁₇	D₃
LCS	G ▷ C₉×C₁₇
UCS	C₁

12 conjugacy classes of subgroups.

minimal degree	μ(G) = 26
transitive degree	μ_t(G) = 153

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

Polycyclic presentation:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(306, 3);