C₅²×AGL(1,𝔽₈) #13063845

order	1400
isoclinism	isoclinic class of AGL(1,𝔽₈)
automorphism	?
homology	Abelianization (H₁) = C₅²×C₇ · Schur multiplier (H₂) = C₅

Aliases

direct product	C₅²×AGL(1,𝔽₈)
unambiguous semidirect product	C₂³⋊(C₅²×C₇)
unambiguous semidirect product	(C₂³×C₅)⋊(C₅×C₇)
unambiguous semidirect product	(C₂³×C₅²)⋊C₇

Fitting, socle, Hall	C₂³×C₅² ⋊ C₇
derived, Sylow	C₂³ ⋊ C₅²×C₇

	subgroup	G/H
center, hypercenter	C₅²	AGL(1,𝔽₈)
commutator, cosocle, hypocenter	C₂³	C₅²×C₇
Frattini	C₁	G
Fitting, socle	C₂³×C₅²	C₇
LCS	G ▷ C₂³
UCS	C₁ ◁ C₅²

48 conjugacy classes of subgroups.

minimal degree	μ(G) = 18
transitive degree	μ_t(G) = 200

A minimal faithful permutation representation — 2 generators on 18 points:

(1,2,3,4,5)(12,13,14,16,18,15,17);(6,7,8,9,10)(11,15)(12,13)(14,18)(16,17)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1400, 134);