ΓL(1,𝔽₈)×#478 #13055009

order	1344
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂

Aliases

direct product	ΓL(1,𝔽₈)×#478
unambiguous semidirect product	(C₇×#478)⋊C₃

Fitting, Hall	C₇×#478 ⋊ C₃
Sylow	C₇ ⋊ C₃×#478

204 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 204 conjugacy classes of subgroups — too many to draw. Only its 81 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

minimal degree	μ(G) = 23
transitive degree	μ_t(G) = 224

A minimal faithful permutation representation — 3 generators on 23 points:

(2,3,4)(5,7,6)(8,9,11,13)(10,14,15,12)(16,17,18,20)(19,21,23,22);(1,2,3,5,4,6,7)(8,10,11,15)(9,12,13,14)(17,20)(21,22);(17,21)(19,23)(20,22)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1344, 4298);