C₃⋊D₆₃ #67389

order	378
isoclinism	isoclinic class of C₃⋊D₆₃
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₃

Aliases

unambiguous semidirect product	C₃⋊D₆₃
unambiguous semidirect product	C₇⋊(C₃⋊D₉)
unambiguous semidirect product	C₉⋊D₂₁

	subgroup	G/H
center, hypercenter	C₁	G
commutator, Fitting, cosocle, hypocenter	C₃×C₇×C₉	C₂
Frattini	C₃	C₃⋊D₂₁
socle	C₃²×C₇	D₃
LCS	G ▷ C₃×C₇×C₉
UCS	C₁

40 conjugacy classes of subgroups.

minimal degree	μ(G) = 19
transitive degree	μ_t(G) = 189

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

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

Polycyclic presentation:

2,3^2,7,3|b^3=e^2,b^a=b^2e,c^a=c^2,d^a=d^6,e^a=e^2

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(378, 42);