Finite Group DatabasebetaFinite Group Databasebeta| order | 1248 |
| automorphism | Aut(G) order 119808 |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C1 |
| dicyclic | Dic312 |
| Hall | C3×C13 ⋊ Q32 |
| Sylow | C13 ⋊ Dic24 |
| Sylow | C3 ⋊ Dic104 |
48 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 1248 |
A minimal faithful permutation representation — 2 generators on 48 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16)(17,32,31,30,29,28,27,26,25,24,23,22,21,20,19,18)(33,34,35)(36,37,38,39,40,41,42,43,44,45,46,47,48);(1,17,9,25)(2,18,10,26)(3,19,11,27)(4,20,12,28)(5,21,13,29)(6,22,14,30)(7,23,15,31)(8,24,16,32)(34,35)(37,48)(38,47)(39,46)(40,45)(41,44)(42,43)Polycyclic presentation:
2^5,3,13|a^2=e,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 | Dic624, Dic936, Dic1560, … |
| maximal normal subgroup of | Dic624 |
| maximal quotient of | Dic936, Dic1560, Dic2184, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1248, 225);