C₂×C₁₃×ΓL(1,𝔽₈) #10587725

order	546
isoclinism	isoclinic class of ΓL(1,𝔽₈)
automorphism	?
homology	Abelianization (H₁) = C₂×C₃×C₁₃ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₂×C₁₃×ΓL(1,𝔽₈)
unambiguous semidirect product	(C₂×C₇)⋊(C₃×C₁₃)

	subgroup	G/H
center, hypercenter	C₂×C₁₃	ΓL(1,𝔽₈)
commutator, cosocle, hypocenter	C₇	C₂×C₃×C₁₃
Frattini	C₁	G
Fitting, socle	C₂×C₇×C₁₃	C₃
LCS	G ▷ C₇
UCS	C₁ ◁ C₂×C₁₃

16 conjugacy classes of subgroups.

minimal degree	μ(G) = 22
transitive degree	μ_t(G) = 182

A minimal faithful permutation representation — 2 generators on 22 points:

(1,2)(3,4,5,6,7,8,9,10,11,12,13,14,15)(17,18,19)(20,22,21);(16,17,18,20,19,21,22)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(546, 14);