C₃⋊D₅₇ #66799

order	342
isoclinism	isoclinic class of C₃⋊D₅₇
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₃

Aliases

unambiguous semidirect product	C₃⋊D₅₇
unambiguous semidirect product	C₁₉⋊(C₃⋊D₃)

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₃²×C₁₉	C₂
LCS	G ▷ C₃²×C₁₉
UCS	C₁

24 conjugacy classes of subgroups.

minimal degree	μ(G) = 25
transitive degree	μ_t(G) = 171

A minimal faithful permutation representation — 3 generators on 25 points:

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

Polycyclic presentation:

2,3^2,19|b^a=b^2,c^a=c^2,d^a=d^18

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(342, 17);