Finite Group DatabasebetaFinite Group Databasebeta| order | 36972 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | #20259721⋊D3 |
| Fitting, socle, derived | C3×C13×C79 ⋊ C22×C3 |
| Hall | C3×#15690493 ⋊ C22 |
| Hall | C13×C79 ⋊ C2×C3×D3 |
| Sylow | C79 ⋊ D3×D13⋊C3 |
| Sylow | C13 ⋊ D3×D79⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C3×C13×C79 | C22×C3 |
| LCS | G ▷ C3×C13×C79 | |
| UCS | C1 | |
88 conjugacy classes of subgroups.
| minimal degree | μ(G) = 95 |
| transitive degree | μt(G) = 3081 |
A minimal faithful permutation representation — 2 generators on 95 points:
(2,3)(4,5,12,15,13,10)(6,11,8,16,7,14)(17,18,24)(19,28,40)(20,22,29)(21,32,23)(25,41,57)(26,42,39)(27,45,31)(30,54,36)(33,58,79)(34,59,53)(35,64,44)(37,55,38)(43,76,49)(46,80,94)(47,81,72)(48,85,63)(50,75,61)(51,56,77)(52,60,73)(62,92,68)(65,74,82)(66,91,89)(67,69,84)(70,78,93)(71,90,88)(83,95,86);(1,2,3)(5,6)(7,9)(8,12)(10,11)(13,15)(14,16)(18,19)(20,22)(21,25)(23,24)(26,30)(27,33)(28,36)(29,38)(31,40)(32,42)(34,43)(35,46)(37,49)(39,51)(41,52)(44,57)(45,60)(47,62)(48,65)(50,68)(53,70)(54,61)(55,59)(56,74)(58,71)(63,79)(64,82)(66,83)(67,76)(69,86)(72,88)(73,89)(75,90)(77,81)(78,80)(84,94)(85,95)(87,92)(91,93)Polycyclic presentation:
2^2,3^2,13,79|e^a=e^12,f^a=f^78,d^b=d^2,e^b=e^12,e^c=e^3,f^c=f^23297 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(36972, 205);