C₇×D₇ #1028

order	98
isoclinism	isoclinic class of D₇
automorphism	Aut(G) = C₂×C₃×AGL(1,𝔽₇) (order 252) · Inn(G) = D₇ · Out(G) = C₂×C₃²
holomorph	G ⋊ Aut(G) = AGL(1,𝔽₇)×#64423⋊C₂
homology	Abelianization (H₁) = C₂×C₇ · Schur multiplier (H₂) = C₁

Aliases

affine special semilinear	AΣL(1,𝔽₄₉)
direct product	C₇×D₇
unambiguous semidirect product	C₇⋊(C₂×C₇)

Fitting, socle, Sylow	C₇² ⋊ C₂
derived	C₇ ⋊ C₂×C₇

	subgroup	G/H
center, hypercenter	C₇	D₇
commutator, cosocle, hypocenter	C₇	C₂×C₇
Frattini	C₁	G
Fitting, socle	C₇²	C₂
LCS	G ▷ C₇
UCS	C₁ ◁ C₇

11 conjugacy classes of subgroups.

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

A minimal faithful permutation representation — 2 generators on 14 points:

(1,6,5,13,12,11,4,3,2,10,9,14,8,7);(1,4,8,12,9,5,2)(3,6,10,13,14,11,7)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(98, 3);