ΓL(1,𝔽₈)×#469 #13055000

order	1344
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product	ΓL(1,𝔽₈)×#469
unambiguous semidirect product	(C₇×#469)⋊C₃

Fitting, Hall	C₇×#469 ⋊ C₃
Sylow	C₇ ⋊ C₃×#469

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

normal subgroups only

This group has 260 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) = 19
transitive degree	μ_t(G) = 224

A minimal faithful permutation representation — 3 generators on 19 points:

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

maximal subgroup of	ΓL(1,𝔽₈)×#3184, AGL(1,𝔽₇)×#469, #72259571
maximal normal subgroup of	ΓL(1,𝔽₈)×#3184, AGL(1,𝔽₇)×#469, #72259571

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1344, 4289);