AGL(1,𝔽₈) #292

order	56
isoclinism	isoclinic class of AGL(1,𝔽₈)
automorphism	Aut(G) = AΓL(1,𝔽₈) (order 168) · Inn(G) = G · Out(G) = C₃
holomorph	G ⋊ Aut(G) = #72214179
homology	Abelianization (H₁) = C₇ · Schur multiplier (H₂) = C₁

Aliases

affine general linear	AGL(1,𝔽₈)
unambiguous semidirect product	C₂³⋊C₇

C₂³ ⋊ C₇

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₂³	C₇
LCS	G ▷ C₂³
UCS	C₁

6 conjugacy classes of subgroups.

minimal degree	μ(G) = 8
transitive degree	μ_t(G) = 8

A minimal faithful permutation representation — 2 generators on 8 points:

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

Polycyclic presentation:

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

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(56, 11);