order	84
isoclinism	isoclinic class of AGL(1,𝔽₇)
automorphism	Aut(G) = C₂×AGL(1,𝔽₇) (order 84) · Inn(G) = AGL(1,𝔽₇) · Out(G) = C₂
holomorph	G ⋊ Aut(G) = #72222066
homology	Abelianization (H₁) = C₃×C₄ · Schur multiplier (H₂) = C₁

Aliases

unambiguous semidirect product	ΓL(1,𝔽₈)⋊C₄
unambiguous semidirect product	Dic₇⋊C₃

Splittings

	subgroup	G/H
center, Frattini, hypercenter	C₂	AGL(1,𝔽₇)
commutator, hypocenter	C₇	C₃×C₄
Fitting, socle, cosocle	C₂×C₇	C₂×C₃
prime radical	C₂×ΓL(1,𝔽₈)	C₂
LCS	G ▷ C₇
UCS	C₁ ◁ C₂

12 conjugacy classes of subgroups.

minimal degree	μ(G) = 11
transitive degree	μ_t(G) = 28

A minimal faithful permutation representation — 2 generators on 11 points:

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

Polycyclic presentation:

2,3,2,7|a^2=c,d^a=d^6,d^b=d^4

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(84, 1);