Finite Group DatabasebetaFinite Group Databasebeta| order | 37386 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | C3⋊#27159577 |
| Fitting, socle, derived | C3×C31×C67 ⋊ C2×C3 |
| Hall | C3×#20262033 ⋊ C2 |
| Hall | C31×C67 ⋊ C3×D3 |
| Sylow | C67 ⋊ C3⋊(D31⋊C3) |
| Sylow | C31 ⋊ D3×C67⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C3×C31×C67 | C2×C3 |
| LCS | G ▷ C3×C31×C67 | |
| UCS | C1 | |
36 conjugacy classes of subgroups.
| minimal degree | μ(G) = 101 |
| transitive degree | μt(G) = 6231 |
A minimal faithful permutation representation — 2 generators on 101 points:
(2,3)(5,9,10,6,7,11)(8,17,18,12,13,25)(14,28,29,20,21,33)(15,27,23,22,19,16)(24,26,34,30,31,32)(36,37,39)(38,41,46)(40,44,43)(42,48,56)(45,51,53)(47,54,50)(49,58,69)(52,61,66)(55,64,63)(57,67,60)(59,71,83)(62,73,81)(65,76,78)(68,79,75)(70,82,72)(74,86,93)(77,87,92)(80,90,89)(84,94,85)(88,95,99)(91,96,98)(97,100,101);(1,2,3)(4,5,8,14,22,7,11,16,24,31,13,19,25,29,34,21,28,32,33,18,27,17,26,30,23,10,9,15,20,12,6)(35,36,38,42,49,59,67,46,53,63,75,71,84,94,69,81,92,98,95,73,48,57,68,80,91,97,96,76,51,37,40,45,52,62,74,79,54,39,43,50,60,72,85,82,56,66,78,89,86,58,70,83,93,99,101,100,87,61,41,47,55,65,77,88,90,64,44)Polycyclic presentation:
2,3^2,31,67|c^a=c^2,d^a=d^30,d^b=d^5,e^b=e^291080 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(37386, 47);