Finite Group DatabasebetaFinite Group Databasebeta| order | 1344 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(3,ℤ/4) |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C42×C7 · Schur multiplier (H2) = C2×C4 |
| direct product | C3×C7×H(3,ℤ/4) |
| subgroup | G/H | |
|---|---|---|
| center | C3×C4×C7 | C42 |
| commutator | C4 | C3×C42×C7 |
| Frattini, cosocle | C22×C4 | C22×C3×C7 |
| socle | C2×C3×C7 | #95 |
| prime radical | C23×C3×C7 | D4 |
| LCS | G ▷ C4 ▷ C1 | |
| UCS | C1 ◁ C3×C4×C7 ◁ G | |
188 conjugacy classes of subgroups; only the 108 normal subgroups are shown.
This group has 188 conjugacy classes of subgroups — too many to draw. Only its 108 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 — 2 generators on 26 points:
(1,2,3)(4,5,6,7,8,9,10)(11,12)(13,19,21,25)(14,17)(15,18)(16,20,23,26)(22,24);(11,13)(12,16,18,23)(14,20)(15,21)(17,19,24,25)(22,26)462 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1344, 777);