Finite Group DatabasebetaFinite Group Databasebeta| order | 1344 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C25 |
| unambiguous semidirect product | #89992⋊C3 |
| Hall | #89992 ⋊ C3 |
| Hall | ΓL(1,𝔽8) ⋊ #386 |
| Sylow | C7 ⋊ C3×#386 |
572 conjugacy classes of subgroups; only the 144 normal subgroups are shown.
This group has 572 conjugacy classes of subgroups — too many to draw. Only its 144 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 224 |
Polycyclic presentation:
2^3,3,2^3,7|a^2=g,c^a=cf,h^a=h^6,c^b=ce,h^d=h^2104 conjugacy classes — view the full character table.
| maximal subgroup of | #72218419, #120×ΓL(1,𝔽8)⋊C4, C4×#6500⋊C3, … |
| maximal normal subgroup of | #120×ΓL(1,𝔽8)⋊C4, C4×#6500⋊C3, AGL(1,𝔽7)×#386, … |
| maximal quotient of | #72218419, #72230204, #72276714, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1344, 1408);