Finite Group DatabasebetaFinite Group Databasebeta| order | 36900 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C9 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | #16861518⋊C9 |
| derived, Hall | C52×C41 ⋊ C4×C9 |
| Hall | C41×C52⋊C9 ⋊ C4 |
| Hall | #16861518 ⋊ C9 |
| Hall | C52⋊C9 ⋊ #4223 |
| Sylow | C41 ⋊ #1043⋊C9 |
| Sylow | C52 ⋊ C9×#4223 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C3 | #16861518⋊C3 |
| commutator, hypocenter | C52×C41 | C4×C9 |
| Fitting, socle | C3×C52×C41 | C3×C4 |
| prime radical, cosocle | C3×C5⋊D205 | C2×C3 |
| LCS | G ▷ C52×C41 | |
| UCS | C1 ◁ C3 | |
60 conjugacy classes of subgroups.
| minimal degree | μ(G) = 65 |
| transitive degree | μt(G) = 1845 |
A minimal faithful permutation representation — 2 generators on 65 points:
(1,2,4)(3,5,15,9,7,13,8,14,10,6,12,11)(16,23,22,18,17,24,21,20,19)(26,27,28,30)(29,32,33,39)(31,36,37,35)(34,41,42,52)(38,45,46,48)(40,49,50,44)(43,54,55,56)(47,58,59,64)(51,62,63,61)(53,65,60,57);(1,3,9,6,8)(2,5,7,12,14)(4,10,11,13,15)(25,26,29,34,43,56,65,49,36,27,31,38,47,60,52,64,62,45,32,40,51,63,50,39,48,61,58,41,53,59,46,37,30,35,44,57,54,55,42,33,28)Polycyclic presentation:
2,3,2,5^2,41,3|a^2=c,b^3=g,d^a=d^3,e^a=e^3,f^a=f^9,d^b=de,e^b=d^2e^3,d^c=d^4,e^c=e^4,f^c=f^40372 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(36900, 203);