order	28
isoclinism	isoclinic class of D₇
automorphism	Aut(G) = C₂×AGL(1,𝔽₇) (order 84) · Inn(G) = D₇ · Out(G) = C₂×C₃
holomorph	G ⋊ Aut(G) = #72210376
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral	D₁₄
direct product	C₂×D₇
unambiguous semidirect product	C₇⋊C₂²
unambiguous semidirect product	(C₂×C₇)⋊C₂

Splittings

Fitting, socle	C₂×C₇ ⋊ C₂
derived, Sylow	C₇ ⋊ C₂²

	subgroup	G/H
center, hypercenter	C₂	D₇
commutator, cosocle, hypocenter	C₇	C₂²
Frattini	C₁	G
Fitting, socle	C₂×C₇	C₂
LCS	G ▷ C₇
UCS	C₁ ◁ C₂

10 conjugacy classes of subgroups.

minimal degree	μ(G) = 9
transitive degree	μ_t(G) = 14

A minimal faithful permutation representation — 2 generators on 9 points:

(1,2)(3,4,5,6,7,8,9);(4,9)(5,8)(6,7)

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(28, 3);