C₃⋊D₅₁ #64564

order	306
isoclinism	isoclinic class of C₃⋊D₅₁
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₃

Aliases

unambiguous semidirect product	C₃⋊D₅₁
unambiguous semidirect product	C₁₇⋊(C₃⋊D₃)

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₃²×C₁₇	C₂
LCS	G ▷ C₃²×C₁₇
UCS	C₁

24 conjugacy classes of subgroups.

minimal degree	μ(G) = 23
transitive degree	μ_t(G) = 153

A minimal faithful permutation representation — 3 generators on 23 points:

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

Polycyclic presentation:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(306, 9);