Finite Group DatabasebetaFinite Group Databasebeta| order | 33300 |
| automorphism | ? |
| homology | Abelianization (H1) = C32×C4 · Schur multiplier (H2) = C3 |
| unambiguous semidirect product | #26012816⋊C3 |
| unambiguous semidirect product | #26012838⋊C3 |
| unambiguous semidirect product | #26012842⋊C3 |
| Fitting, socle, derived, Hall | C52×C37 ⋊ C32×C4 |
| Hall | C52⋊C3×C37⋊C3 ⋊ C4 |
| Hall | #16814877 ⋊ C32 |
| Sylow | C37 ⋊ C3×#1043⋊C3 |
| Sylow | C52 ⋊ C3×#3866⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, hypocenter | C52×C37 | C32×C4 |
| prime radical | (C37⋊C3)⋊((C5⋊D5)⋊C3) | C2 |
| cosocle | C5⋊D185 | C2×C32 |
| LCS | G ▷ C52×C37 | |
| UCS | C1 | |
96 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)(3,12,15,13,7,10,8,5,9,6,14,11)(16,17,24,18)(19,34,23,21)(20,35,38,28)(22,39,31,43)(25,29,44,52)(26,49,42,40)(27,50,37,33)(30,51,48,46)(32,41,36,47);(1,3,6,13,8)(2,5,7,14,12)(4,9,10,11,15)(17,19,25)(18,22,32)(20,26,27)(21,29,43)(23,33,40)(24,36,50)(28,42,41)(30,44,52)(31,47,34)(35,48,38)(37,46,51)(39,45,49)Polycyclic presentation:
2,3^2,2,5^2,37|a^2=d,e^a=e^2,f^a=f^2,g^a=g^31,g^b=g^26,e^c=ef,f^c=e^2f^3,e^d=e^4,f^d=f^4,g^d=g^3675 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(33300, 427);