Finite Group DatabasebetaFinite Group Databasebeta| order | 1312 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C41 · Schur multiplier (H2) = C2 |
| direct product | #125×C41 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C22×C41 | C23 |
| commutator, Frattini, cosocle | C22 | C23×C41 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C41 ◁ G | |
56 conjugacy classes of subgroups.
| minimal degree | μ(G) = 57 |
| transitive degree | μt(G) = 1312 |
A minimal faithful permutation representation — 3 generators on 57 points:
(3,7)(5,8)(9,10,12,14)(11,13,15,16)(17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57);(1,2,4,6)(3,5,7,8)(9,10)(11,13)(12,14)(15,16);(1,3)(2,5)(4,7)(6,8)(9,11,12,15)(10,13,14,16)574 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1312, 32);