Finite Group DatabasebetaFinite Group Databasebeta| order | 1344 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(5,𝔽2) |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3×C4×C7 · Schur multiplier (H2) = C25 |
| direct product | C3×C7×#519 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3×C7 | C24 |
| commutator | C2 | C23×C3×C4×C7 |
| Frattini, cosocle | C22 | C24×C3×C7 |
| prime radical | C23×C3×C7 | C23 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3×C7 ◁ G | |
532 conjugacy classes of subgroups.
| minimal degree | μ(G) = 30 |
| transitive degree | μt(G) = 672 |
A minimal faithful permutation representation — 4 generators on 30 points:
(1,2,3)(4,5,6,7,8,9,10)(11,12,13,14)(16,20)(18,26)(21,23)(24,30);(15,16,19,20)(17,23,25,21)(18,22,26,27)(24,29,30,28);(15,17,19,25)(16,21,20,23)(18,24,26,30)(22,28,27,29);(15,18)(16,22)(17,24)(19,26)(20,27)(21,28)(23,29)(25,30)714 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1344, 11228);