Finite Group DatabasebetaFinite Group Databasebeta| order | 1344 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C7 · Schur multiplier (H2) = C29 |
| direct product | C7×#5968 |
| Fitting | C3×C7×2−1+4 ⋊ C2 |
| Hall | C3×C7 ⋊ #585 |
| Sylow | C3 ⋊ C7×#585 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C7 | C24×D3 |
| commutator, cosocle | C2×C3 | C25×C7 |
| Frattini | C2 | C24×D3×C7 |
| Fitting | C3×C7×2−1+4 | C2 |
| socle | C2×C3×C7 | C25 |
| hypercenter | C7×2−1+4 | D3 |
| hypocenter | C3 | C7×#585 |
| LCS | G ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C2×C7 ◁ C7×2−1+4 | |
1620 conjugacy classes of subgroups; only the 886 normal subgroups are shown.
This group has 1620 conjugacy classes of subgroups — too many to draw. Only its 886 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 336 |
A minimal faithful permutation representation — 5 generators on 26 points:
(1,2,3,4,5,6,7)(8,9,10)(11,14,15,21)(12,18,17,25)(13,19,20,22)(16,24,23,26);(12,17)(16,23)(18,25)(24,26);(11,13,15,20)(12,16,17,23)(14,22,21,19)(18,26,25,24);(11,12)(13,16)(14,18)(15,17)(19,24)(20,23)(21,25)(22,26);(9,10)(14,21)(18,25)(19,22)(24,26)357 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1344, 11654);