Finite Group DatabasebetaFinite Group Databasebeta| order | 36972 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C4 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | C3⋊#27134499 |
| derived | C3×C13×C79 ⋊ C3×C4 |
| Hall | C3×#15690494 ⋊ C4 |
| Hall | C13×C79 ⋊ C3×Dic3 |
| Sylow | C79 ⋊ Dic3×C13⋊C3 |
| Sylow | C13 ⋊ C3⋊((C79⋊C3)⋊C4) |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | C3⋊#20259714 |
| commutator, hypocenter | C3×C13×C79 | C3×C4 |
| Fitting, socle, cosocle | C2×C3×C13×C79 | C2×C3 |
| prime radical | C2×C3×#15690494 | C2 |
| LCS | G ▷ C3×C13×C79 | |
| UCS | C1 ◁ C2 | |
56 conjugacy classes of subgroups.
| minimal degree | μ(G) = 99 |
| transitive degree | μt(G) = 12324 |
A minimal faithful permutation representation — 2 generators on 99 points:
(2,3)(4,5,6,7)(9,10,12)(11,14,17)(13,16,15)(18,20,19)(22,26,27,23,24,28)(25,34,35,29,30,44)(31,47,48,37,38,61)(32,46,42,40,36,33)(39,66,67,50,51,82)(41,64,65,53,49,58)(43,45,63,55,56,59)(52,87,88,69,70,98)(54,85,86,72,68,80)(57,62,84,74,71,81)(60,93,92,77,78,76)(73,94,95,79,89,96)(75,83,99,90,91,97);(1,2,3)(8,9,11,10,13,17,14,18,20,12,15,19,16)(21,22,25,31,39,52,71,38,49,59,61,80,95,67,84,93,85,66,83,90,74,55,42,27,26,32,41,54,73,77,56,30,36,44,58,76,88,99,70,89,94,87,97,98,92,65,35,46,34,45,60,79,72,53,40,24,28,33,43,57,75,91,51,68,78,81,82,96,86,48,63,64,47,62,69,50,37,29,23)Polycyclic presentation:
2,3^2,13,79,2|a^2=f,c^a=c^2,e^a=e^78,d^b=d^3,e^b=e^551056 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(36972, 129);