Finite Group DatabasebetaFinite Group Databasebeta| order | 36972 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C4 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | C3⋊#27134506 |
| derived | C3×C13×C79 ⋊ C3×C4 |
| Hall | C3×#15690493 ⋊ C4 |
| Hall | C13×C79 ⋊ C3×Dic3 |
| Sylow | C79 ⋊ C3⋊(Dic13⋊C3) |
| Sylow | C13 ⋊ C3⋊((C79⋊C3)⋊C4) |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | C3⋊#20259721 |
| commutator, hypocenter | C3×C13×C79 | C3×C4 |
| Fitting, socle, cosocle | C2×C3×C13×C79 | C2×C3 |
| prime radical | C2×C3×#15690493 | 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)(9,13,14,10,11,15)(12,19,18,16,17,20)(22,26,27,23,24,28)(25,34,35,29,30,44)(31,47,48,37,38,61)(32,46,42,40,36,33)(39,66,67,50,51,83)(41,64,65,53,49,58)(43,45,63,55,56,59)(52,87,88,69,70,98)(54,86,71,72,68,80)(57,62,85,74,75,81)(60,94,93,78,79,77)(73,99,91,90,89,96)(76,84,95,92,82,97);(1,2,3)(8,9,12,11,15,18,17,19,20,14,13,16,10)(21,22,25,31,39,52,71,48,63,64,47,62,82,51,68,79,81,83,96,90,72,53,40,24,28,33,43,57,76,93,65,35,46,34,45,60,70,89,88,95,97,98,99,87,78,56,30,36,44,58,77,92,74,55,42,27,26,32,41,54,73,91,67,85,94,86,66,84,75,38,49,59,61,80,69,50,37,29,23)Polycyclic presentation:
2,3^2,13,79,2|a^2=f,c^a=c^2,d^a=d^12,e^a=e^78,d^b=d^3,e^b=e^231044 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(36972, 122);