C₃⋊D₃₉ #6626

order	234
isoclinism	isoclinic class of C₃⋊D₃₉
automorphism	Aut(G) order 67392
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₃

Aliases

unambiguous semidirect product	C₃⋊D₃₉
unambiguous semidirect product	C₁₃⋊(C₃⋊D₃)

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₃²×C₁₃	C₂
LCS	G ▷ C₃²×C₁₃
UCS	C₁

24 conjugacy classes of subgroups.

minimal degree	μ(G) = 19
transitive degree	μ_t(G) = 117

A minimal faithful permutation representation — 3 generators on 19 points:

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

Polycyclic presentation:

2,3^2,13|b^a=b^2,c^a=c^2,d^a=d^12

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(234, 15);