C₃₁×AGL(1,𝔽₈) #13181931

order	1736
isoclinism	isoclinic class of AGL(1,𝔽₈)
automorphism	Aut(G) = C₂×C₃×C₅×AΓL(1,𝔽₈) (order 5040) · Inn(G) = AGL(1,𝔽₈) · Out(G) = C₂×C₃²×C₅
homology	Abelianization (H₁) = C₇×C₃₁ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₃₁×AGL(1,𝔽₈)
unambiguous semidirect product	C₂³⋊(C₇×C₃₁)
unambiguous semidirect product	(C₂³×C₃₁)⋊C₇

Fitting, socle, Hall	C₂³×C₃₁ ⋊ C₇
derived, Sylow	C₂³ ⋊ C₇×C₃₁

	subgroup	G/H
center, hypercenter	C₃₁	AGL(1,𝔽₈)
commutator, cosocle, hypocenter	C₂³	C₇×C₃₁
Frattini	C₁	G
Fitting, socle	C₂³×C₃₁	C₇
LCS	G ▷ C₂³
UCS	C₁ ◁ C₃₁

12 conjugacy classes of subgroups.

minimal degree	μ(G) = 39
transitive degree	μ_t(G) = 248

A minimal faithful permutation representation — 2 generators on 39 points:

(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31)(33,34,35,37,39,36,38);(32,36)(33,34)(35,39)(37,38)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1736, 30);