Finite Group DatabasebetaFinite Group Databasebeta| order | 12900 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C4 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | (C43⋊C3)⋊Dic25 |
| unambiguous semidirect product | Dic1075⋊C3 |
| derived, Hall | C25×C43 ⋊ C3×C4 |
| Hall | Dic1075 ⋊ C3 |
| Hall | C25×C43⋊C3 ⋊ C4 |
| Hall | C43⋊C3 ⋊ Dic25 |
| Sylow | C43 ⋊ C3×Dic25 |
| Sylow | C25 ⋊ (C43⋊C3)⋊C4 |
36 conjugacy classes of subgroups.
| minimal degree | μ(G) = 72 |
| transitive degree | μt(G) = 4300 |
A minimal faithful permutation representation — 2 generators on 72 points:
(2,6,7,3,4,8)(5,14,15,9,10,24)(11,27,28,17,18,37)(12,26,22,20,16,13)(19,29,34,21,30,40)(23,25,39,31,32,35)(33,38,42,36,41,43)(45,47)(46,50)(48,53)(49,56)(51,59)(52,58)(54,55)(57,63)(60,64)(61,65)(62,67)(66,68)(69,70,71,72);(1,2,5,11,19,20,4,8,13,23,33,32,10,16,24,34,42,41,18,29,35,37,28,39,30,27,38,43,40,15,26,14,25,36,31,22,7,6,12,21,17,9,3)(44,58,68,61,47,50,64,62,53,56,59,57,55,54,63,51,49,48,67,60,46,45,65,66,52)Polycyclic presentation:
2,3,5,43,2,5|a^2=e,c^5=f^4,c^a=c^4f,d^a=d^42,f^a=f^4,d^b=d^6434 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(12900, 4);