Finite Group DatabasebetaFinite Group Databasebeta| order | 1344 |
| isoclinism | isoclinic class of #4656 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C7 · Schur multiplier (H2) = C25 |
| direct product | C7×#4671 |
| Fitting | C3×C7×#95 ⋊ C2 |
| Hall | C3×C7 ⋊ #392 |
| Sylow | C3 ⋊ C7×#392 |
| subgroup | G/H | |
|---|---|---|
| center | C23×C7 | C22×D3 |
| commutator, cosocle | C23×C3 | C23×C7 |
| Frattini | C23 | C22×D3×C7 |
| Fitting | C3×C7×#95 | C2 |
| socle | C23×C3×C7 | C23 |
| hypercenter | C7×#95 | D3 |
| hypocenter | C3 | C7×#392 |
| LCS | G ▷ C23×C3 ▷ C3 | |
| UCS | C1 ◁ C23×C7 ◁ C7×#95 | |
644 conjugacy classes of subgroups; only the 138 normal subgroups are shown.
This group has 644 conjugacy classes of subgroups — too many to draw. Only its 138 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 672 |
A minimal faithful permutation representation — 3 generators on 22 points:
(1,2,3,4,5,6,7)(8,9,10)(16,18)(19,20,21,22);(11,12,13,14)(15,16)(17,18);(9,10)(12,14)(20,22)294 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1344, 4997);