Finite Group DatabasebetaFinite Group Databasebeta| order | 1248 |
| automorphism | Aut(G) order 119808 |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2 |
| dihedral | D624 |
| Fitting | C3×C13×C16 ⋊ C2 |
| Hall | C3×C13 ⋊ D16 |
| Sylow | C13 ⋊ D48 |
| Sylow | C3 ⋊ D208 |
56 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 624 |
A minimal faithful permutation representation — 2 generators on 32 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16)(17,18,19)(20,21,22,23,24,25,26,27,28,29,30,31,32);(2,16)(3,15)(4,14)(5,13)(6,12)(7,11)(8,10)(18,19)(21,32)(22,31)(23,30)(24,29)(25,28)(26,27)Polycyclic presentation:
2^5,3,13|b^2=cd,c^2=de,d^2=e,b^a=bc,c^a=cd,d^a=de,f^a=f^2,g^a=g^12315 conjugacy classes — view the full character table.
| maximal subgroup of | D1248, D3120, D1872, … |
| maximal normal subgroup of | D1248 |
| maximal quotient of | D1248, Dic624, D3120, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1248, 224);