C₃⋊D₂₁ #1266

order	126
isoclinism	isoclinic class of C₃⋊D₂₁
automorphism	Aut(G) = AGL(1,𝔽₇)×AGL(2,𝔽₃) (order 18144) · Inn(G) = G · Out(G) = C₃×GL(2,𝔽₃)
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₃

Aliases

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

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

24 conjugacy classes of subgroups.

minimal degree	μ(G) = 13
transitive degree	μ_t(G) = 63

A minimal faithful permutation representation — 3 generators on 13 points:

(2,3)(4,5)(6,7)(9,10)(12,13);(1,2,4,6,7,5,3)(11,12,13);(8,9,10)

Polycyclic presentation:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(126, 15);