Finite Group DatabasebetaFinite Group Databasebeta| order | 1344 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of D16 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C4×C7 · Schur multiplier (H2) = C2 |
| direct product | C3×C7×#367 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C22×C3×C7 | D8 |
| commutator | C8 | C2×C3×C4×C7 |
| Frattini, cosocle | C2×C8 | C22×C3×C7 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3×C7 ◁ C2×C3×C4×C7 ◁ C2×C3×C7×C8 ◁ G | |
92 conjugacy classes of subgroups.
| minimal degree | μ(G) = 30 |
| transitive degree | μt(G) = 1344 |
A minimal faithful permutation representation — 2 generators on 30 points:
(1,2,3)(4,5,6,7,8,9,10)(11,12,13,14)(16,20)(17,24)(18,25)(21,26)(22,30)(23,29)(27,28);(15,16,21,30,18,22,17,20,19,23,24,28,25,27,26,29)462 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1344, 807);