ΓL(1,𝔽₈)×PSL(2,𝔽₁₁) #28357801

order	13860
automorphism	?
homology	Abelianization (H₁) = C₃ · Schur multiplier (H₂) = C₂

Aliases

direct product	ΓL(1,𝔽₈)×PSL(2,𝔽₁₁)
unambiguous semidirect product	C₇⋊(C₃×PSL(2,𝔽₁₁))
unambiguous semidirect product	(C₇×PSL(2,𝔽₁₁))⋊C₃

Fitting, Sylow	C₇ ⋊ C₃×PSL(2,𝔽₁₁)
socle, derived	C₇×PSL(2,𝔽₁₁) ⋊ C₃

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, socle, hypocenter	C₇×PSL(2,𝔽₁₁)	C₃
Fitting, cosocle	C₇	C₃×PSL(2,𝔽₁₁)
LCS	G ▷ C₇×PSL(2,𝔽₁₁)
UCS	C₁

72 conjugacy classes of subgroups.

minimal degree	μ(G) = 18
transitive degree	μ_t(G) = 77

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

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13860, 468);