Finite Group DatabasebetaFinite Group Databasebeta| order | 1344 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(5,𝔽2) |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C3×C7 · Schur multiplier (H2) = C29 |
| direct product | C2×C3×C7×2−1+4 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3×C7 | C24 |
| commutator, Frattini, cosocle | C2 | C25×C3×C7 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3×C7 ◁ G | |
1588 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 672 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2)(19,20,21)(22,23,24,25,26,27,28);(4,8)(6,14)(9,11)(12,18);(3,4,7,8)(5,11,13,9)(6,10,14,15)(12,17,18,16);(3,5,7,13)(4,9,8,11)(6,12,14,18)(10,16,15,17);(3,6)(4,10)(5,12)(7,14)(8,15)(9,16)(11,17)(13,18)714 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1344, 11682);