Finite Group DatabasebetaFinite Group Databasebeta| order | 36972 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C4 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | C3⋊#27134502 |
| derived | C3×C13×C79 ⋊ C3×C4 |
| Hall | C3×#15690493 ⋊ C4 |
| Hall | C13×C79 ⋊ C3×Dic3 |
| Sylow | C79 ⋊ C3⋊(Dic13⋊C3) |
| Sylow | C13 ⋊ Dic3×C79⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | C3⋊#20259717 |
| 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)(22,23,25)(24,27,32)(26,30,29)(28,34,42)(31,37,39)(33,40,36)(35,44,55)(38,47,52)(41,50,49)(43,53,46)(45,57,70)(48,60,68)(51,63,65)(54,66,62)(56,58,59)(61,73,83)(64,76,81)(67,79,78)(69,74,75)(71,84,72)(77,85,92)(80,88,90)(82,86,87)(89,93,97)(91,94,95)(96,98,99);(1,2,3)(8,9,12,11,15,18,17,19,20,14,13,16,10)(21,22,24,28,35,45,58,42,52,65,78,87,73,44,56,69,82,91,96,98,88,63,37,23,26,31,38,48,61,74,53,32,39,49,62,75,57,71,70,83,92,97,99,93,76,47,27,33,41,51,64,77,86,66,40,25,29,36,46,59,72,84,55,68,81,90,95,85,60,34,43,54,67,80,89,94,79,50,30)Polycyclic presentation:
2,3^2,13,79,2|a^2=f,c^a=c^2,d^a=d^12,d^b=d^3,e^b=e^231122 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(36972, 124);