Finite Group DatabasebetaFinite Group Databasebeta| order | 20100 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | (C5⋊(D67⋊C3))⋊D5 |
| unambiguous semidirect product | ((C5⋊D5)⋊D67)⋊C3 |
| Fitting, socle, derived, Hall | C52×C67 ⋊ C22×C3 |
| Hall | (C5⋊D5)⋊D67 ⋊ C3 |
| Hall | C52×C67⋊C3 ⋊ C22 |
| Hall | C67⋊C3 ⋊ D52 |
| Sylow | C67 ⋊ C3×D52 |
| Sylow | C52 ⋊ C2×D67⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C52×C67 | C22×C3 |
| LCS | G ▷ C52×C67 | |
| UCS | C1 | |
96 conjugacy classes of subgroups.
| minimal degree | μ(G) = 77 |
| transitive degree | μt(G) = 670 |
A minimal faithful permutation representation — 2 generators on 77 points:
(1,2,8)(3,12,24)(4,6,13)(5,16,7)(9,25,41)(10,26,23)(11,29,15)(14,38,20)(17,42,59)(18,43,37)(19,48,28)(21,39,22)(27,57,33)(30,53,64)(31,60,52)(32,56,47)(34,55,45)(35,40,58)(36,44,49)(46,67,50)(51,66,62)(54,61,63)(68,69)(70,71)(74,75)(76,77);(2,3)(4,6)(5,9)(7,8)(10,14)(11,17)(12,20)(13,22)(15,24)(16,26)(18,27)(19,30)(21,33)(23,35)(25,36)(28,41)(29,44)(31,46)(32,49)(34,50)(37,52)(38,45)(39,43)(40,54)(42,53)(47,59)(48,61)(51,63)(55,64)(56,65)(57,62)(58,60)(66,67)(69,70)(71,72)(73,74,76,77,75)Polycyclic presentation:
2^2,3,5^2,67|d^a=d^4,f^a=f^66,d^b=d^4,e^b=e^4,f^c=f^29202 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20100, 88);