C₃⋊(D₇⋊C₉) #67368

order	378
isoclinism	isoclinic class of C₃⋊AGL(1,𝔽₇)
automorphism	Aut(G) = C₃×D₃×AGL(1,𝔽₇) (order 756) · Inn(G) = C₃⋊AGL(1,𝔽₇) · Out(G) = C₂×C₃
homology	Abelianization (H₁) = C₂×C₉ · Schur multiplier (H₂) = C₁

Aliases

unambiguous semidirect product	C₃⋊(D₇⋊C₉)
unambiguous semidirect product	D₂₁⋊C₉
unambiguous semidirect product	(C₇⋊C₉)⋊D₃

	subgroup	G/H
center, Frattini, hypercenter	C₃	C₃⋊AGL(1,𝔽₇)
commutator, hypocenter	C₃×C₇	C₂×C₉
Fitting, socle, cosocle	C₃²×C₇	C₂×C₃
prime radical	C₃×D₂₁	C₃
LCS	G ▷ C₃×C₇
UCS	C₁ ◁ C₃

28 conjugacy classes of subgroups.

minimal degree	μ(G) = 19
transitive degree	μ_t(G) = 126

A minimal faithful permutation representation — 2 generators on 19 points:

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

Polycyclic presentation:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(378, 21);