Finite Group DatabasebetaFinite Group Databasebeta| order | 33300 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C9 · Schur multiplier (H2) = C2×C5 |
| unambiguous semidirect product | #30659271⋊C2 |
| unambiguous semidirect product | ((C5⋊D5)⋊#66526)⋊C2 |
| unambiguous semidirect product | #30659275⋊C2 |
| Fitting, socle, derived, Hall | C52×C37 ⋊ C22×C9 |
| Hall | C52⋊#66526 ⋊ C22 |
| Hall | C5⋊D5×D37 ⋊ C9 |
| Sylow | C37 ⋊ C2×(C5⋊D5)⋊C9 |
| Sylow | C52 ⋊ C2×#66526⋊C2 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, hypocenter | C52×C37 | C22×C9 |
| prime radical | C5⋊D5×D37⋊C3 | C3 |
| cosocle | C52×C37⋊C3 | C22×C3 |
| LCS | G ▷ C52×C37 | |
| UCS | C1 | |
100 conjugacy classes of subgroups.
| minimal degree | μ(G) = 52 |
| transitive degree | μt(G) = 555 |
A minimal faithful permutation representation — 2 generators on 52 points:
(1,2,4)(3,13,9,7,6,10)(5,15,14,11,12,8)(16,17,27,26,51,39,31,40,43)(18,44,32,33,42,22,41,25,52)(19,24,23,20,38,47,28,34,49)(21,45,48,30,50,35,29,37,36);(1,3,8,14,7)(2,5,6,13,11)(4,9,15,12,10)(17,18)(19,22)(20,23)(21,29)(24,32)(25,33)(26,37)(27,28)(30,41)(31,34)(35,40)(36,43)(38,48)(39,45)(42,51)(44,52)(46,49)(47,50)Polycyclic presentation:
2^2,3^2,5^2,37|c^3=d,g^a=g^36,e^b=e^4,f^b=f^4,e^c=ef^3,f^c=e^4f^3,g^c=g^16,g^d=g^2688 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(33300, 418);