Finite Group DatabasebetaFinite Group Databasebeta| order | 30150 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C5 |
| unambiguous semidirect product | C3⋊(D67⋊(C52⋊C3)) |
| unambiguous semidirect product | D201⋊(C52⋊C3) |
| Fitting, socle, derived | C3×C52×C67 ⋊ C2×C3 |
| Hall | C3×C52⋊(C67⋊C3) ⋊ C2 |
| Hall | C52×C67 ⋊ C3×D3 |
| Sylow | C67 ⋊ D3×C52⋊C3 |
| Sylow | C52 ⋊ C3⋊(D67⋊C3) |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C3×C52×C67 | C2×C3 |
| LCS | G ▷ C3×C52×C67 | |
| UCS | C1 | |
52 conjugacy classes of subgroups.
| minimal degree | μ(G) = 85 |
| transitive degree | μt(G) = 3015 |
A minimal faithful permutation representation — 2 generators on 85 points:
(2,3)(4,5,7)(6,9,11)(8,13,16)(10,15,12)(14,17,18)(20,24,25,21,22,26)(23,32,33,27,28,42)(29,45,46,35,36,59)(30,44,40,38,34,31)(37,64,65,48,49,77)(39,62,63,51,47,56)(41,43,61,53,54,57)(50,72,73,67,58,82)(52,79,80,68,66,75)(55,60,78,70,71,76)(69,83,84,81,74,85);(1,2,3)(4,6,10,16,18)(5,9,15,8,14)(7,13,11,17,12)(19,20,23,29,37,50,63,33,44,32,43,58,74,84,65,78,82,79,64,71,36,47,57,59,75,81,68,51,38,22,26,31,41,55,70,53,40,25,24,30,39,52,69,80,46,61,62,45,60,49,66,73,76,77,85,83,72,54,28,34,42,56,67,48,35,27,21)Polycyclic presentation:
2,3^2,5^2,67|c^a=c^2,f^a=f^66,d^b=de,e^b=d^2e^3,f^b=f^29858 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(30150, 82);