ΓL(1,𝔽₈) #55

order	21
isoclinism	isoclinic class of ΓL(1,𝔽₈)
automorphism	Aut(G) = AGL(1,𝔽₇) (order 42) · Inn(G) = G · Out(G) = C₂
holomorph	G ⋊ Aut(G) = ΓL(1,𝔽₈)⋊AGL(1,𝔽₇)
homology	Abelianization (H₁) = C₃ · Schur multiplier (H₂) = C₁

Aliases

general semilinear	ΓL(1,𝔽₈)
unambiguous semidirect product	C₇⋊C₃

C₇ ⋊ C₃

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₇	C₃
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:

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

Polycyclic presentation:

3,7|b^a=b^2

Here i is the imaginary unit (i² = −1).

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(21, 1);