D₇ #26

order	14
isoclinism	isoclinic class of D₇
automorphism	Aut(G) = AGL(1,𝔽₇) (order 42) · Inn(G) = G · Out(G) = C₃
holomorph	G ⋊ Aut(G) = #64423⋊C₂
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral	D₇
unambiguous semidirect product	C₇⋊C₂

C₇ ⋊ C₂

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

4 conjugacy classes of subgroups.

minimal degree	μ(G) = 7
transitive degree	μ_t(G) = 7

A minimal faithful permutation representation — 2 generators on 7 points:

(1,2,3,4,5,6,7);(2,7)(3,6)(4,5)

Polycyclic presentation:

2,7|b^a=b^6

Here ζ is a primitive root of unity of order 7 (ζ⁷ = 1).

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(14, 1);