C₂×ΓL(1,𝔽₈)×AGL(1,ℤ/8) #13060734

order	1344
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₃ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product	C₂×ΓL(1,𝔽₈)×AGL(1,ℤ/8)
unambiguous semidirect product	(C₂×C₇×AGL(1,ℤ/8))⋊C₃

Fitting, Hall	C₂×C₇×AGL(1,ℤ/8) ⋊ C₃
Sylow	C₇ ⋊ C₂×C₃×AGL(1,ℤ/8)

596 conjugacy classes of subgroups.

minimal degree	μ(G) = 17
transitive degree	μ_t(G) = 112

A minimal faithful permutation representation — 4 generators on 17 points:

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

maximal subgroup of	D₅×ΓL(1,𝔽₈)×AGL(1,ℤ/8), C₂×AGL(1,ℤ/8)×AGL(1,𝔽₇), C₄×ΓL(1,𝔽₈)×AGL(1,ℤ/8), …
maximal normal subgroup of	C₂×AGL(1,ℤ/8)×AGL(1,𝔽₇), C₄×ΓL(1,𝔽₈)×AGL(1,ℤ/8), AGL(1,ℤ/8)×ΓL(1,𝔽₈)⋊C₄, …
maximal quotient of	D₅×ΓL(1,𝔽₈)×AGL(1,ℤ/8), C₄×ΓL(1,𝔽₈)×AGL(1,ℤ/8), ΓL(1,𝔽₈)×#3151, …

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1344, 10023);