Finite Group DatabasebetaFinite Group Databasebeta| order | 1344 |
| isoclinism | isoclinic class of #548⋊C3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C7 · Schur multiplier (H2) = C2 |
| direct product | C7×C4⋊(#41⋊C3) |
| unambiguous semidirect product | C4⋊(C7×#41⋊C3) |
| unambiguous semidirect product | #550⋊(C3×C7) |
| Fitting, Hall | C7×#550 ⋊ C3 |
| Sylow | #550 ⋊ C3×C7 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C7 | C22×A4 |
| commutator, cosocle | C2×Q8 | C22×C3×C7 |
| Frattini | C22 | C22×C7×A4 |
| Fitting | C7×#550 | C3 |
| hypercenter | C7×C4⋊C4 | A4 |
| hypocenter | Q8 | C3×C7×D4 |
| LCS | G ▷ C2×Q8 ▷ Q8 | |
| UCS | C1 ◁ C22×C7 ◁ C7×C4⋊C4 | |
174 conjugacy classes of subgroups; only the 46 normal subgroups are shown.
This group has 174 conjugacy classes of subgroups — too many to draw. Only its 46 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 448 |
A minimal faithful permutation representation — 2 generators on 27 points:
(1,2,3,4,5,6,7)(9,11)(12,14,17,13)(15,21,24,18,16,27,23,19,22,20,25,26);(8,9)(10,11)(12,26)(13,24)(14,22)(15,21)(16,18)(17,27)(19,23)(20,25)245 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1344, 10042);