Finite Group DatabasebetaFinite Group Databasebeta| order | 12900 |
| automorphism | ? |
| homology | Abelianization (H1) = C4 · Schur multiplier (H2) = C5 |
| unambiguous semidirect product | C3⋊(C43⋊#1039) |
| unambiguous semidirect product | C43⋊(C3⋊#1039) |
| derived, Hall | C3×C52×C43 ⋊ C4 |
| Hall | C52×C43 ⋊ Dic3 |
| Hall | C3×C43 ⋊ #1039 |
| Hall | C3×C52 ⋊ Dic43 |
| Sylow | C43 ⋊ C3⋊#1039 |
| Sylow | C52 ⋊ Dic129 |
| Sylow | C3 ⋊ C43⋊#1039 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | C3⋊(C5⋊D215) |
| commutator, hypocenter | C3×C52×C43 | C4 |
| Fitting, socle, prime radical, cosocle | C2×C3×C52×C43 | C2 |
| LCS | G ▷ C3×C52×C43 | |
| UCS | C1 ◁ C2 | |
96 conjugacy classes of subgroups.
| minimal degree | μ(G) = 60 |
| transitive degree | μt(G) = 12900 |
A minimal faithful permutation representation — 3 generators on 60 points:
(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)(45,46)(47,48,49,50)(52,53)(54,55)(57,58)(59,60);(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)(44,45,46)(56,57,59,60,58);(51,52,54,55,53)Polycyclic presentation:
2,3,5^2,43,2|a^2=f,b^a=b^2,c^a=c^4,d^a=d^4,e^a=e^423228 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(12900, 68);