Finite Group DatabasebetaFinite Group Databasebeta| order | 12900 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C4 · Schur multiplier (H2) = C5 |
| unambiguous semidirect product | (C43⋊C3)⋊#1039 |
| derived, Hall | C52×C43 ⋊ C3×C4 |
| Hall | C43⋊#1039 ⋊ C3 |
| Hall | C52×C43⋊C3 ⋊ C4 |
| Hall | C43⋊C3 ⋊ #1039 |
| Sylow | C43 ⋊ C3×#1039 |
| Sylow | C52 ⋊ (C43⋊C3)⋊C4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | C5⋊(C5⋊(D43⋊C3)) |
| commutator, hypocenter | C52×C43 | C3×C4 |
| Fitting, socle, cosocle | C2×C52×C43 | C2×C3 |
| prime radical | C2×C52×C43⋊C3 | C2 |
| LCS | G ▷ C52×C43 | |
| UCS | C1 ◁ C2 | |
96 conjugacy classes of subgroups.
| minimal degree | μ(G) = 57 |
| transitive degree | μt(G) = 4300 |
A minimal faithful permutation representation — 3 generators on 57 points:
(2,6,7,3,4,8)(5,14,15,9,10,24)(11,27,28,17,18,36)(12,26,22,20,16,13)(19,32,34,23,25,37)(21,38,39,30,29,33)(31,41,42,40,35,43)(44,45,46,47)(49,50)(51,52)(54,55)(56,57);(1,2,5,11,19,22,7,6,12,21,31,39,15,26,14,25,35,42,28,37,38,27,18,29,34,36,43,41,32,10,16,24,33,40,30,20,4,8,13,23,17,9,3)(53,54,56,57,55);(48,49,51,52,50)Polycyclic presentation:
2,3,5^2,43,2|a^2=f,c^a=c^4,d^a=d^4,e^a=e^42,e^b=e^36434 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(12900, 45);