Finite Group DatabasebetaFinite Group Databasebeta| order | 12900 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | D75⋊D43 |
| unambiguous semidirect product | D129⋊D25 |
| unambiguous semidirect product | D1075⋊D3 |
| Fitting, derived, Hall | C3×C25×C43 ⋊ C22 |
| Hall | C25×C43 ⋊ C2×D3 |
| Hall | C3×C43 ⋊ C2×D25 |
| Hall | C3×C25 ⋊ C2×D43 |
| Sylow | C43 ⋊ D3×D25 |
| Sylow | C25 ⋊ D3×D43 |
| Sylow | C3 ⋊ D25×D43 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, Fitting, cosocle, hypocenter | C3×C25×C43 | C22 |
| Frattini | C5 | D15⋊D43 |
| socle | C3×C5×C43 | C2×D5 |
| LCS | G ▷ C3×C25×C43 | |
| UCS | C1 | |
60 conjugacy classes of subgroups.
| minimal degree | μ(G) = 71 |
| transitive degree | μt(G) = 3225 |
A minimal faithful permutation representation — 2 generators on 71 points:
(1,2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40,42,43,41,39,37,35,33,31,29,27,25,23,21,19,17,15,13,11,9,7,5,3)(44,64)(45,59)(46,57)(47,68)(48,63)(49,51)(50,58)(52,62)(53,66)(54,67)(55,60)(56,65)(70,71);(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)(32,33)(34,35)(36,37)(38,39)(40,41)(42,43)(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)Polycyclic presentation:
2^2,3,5,43,5|d^5=f^4,d^a=d^4f,e^a=e^42,f^a=f^4,c^b=c^2,d^b=d^4f,f^b=f^4861 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(12900, 30);