Finite Group DatabasebetaFinite Group Databasebeta| order | 37386 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | C3⋊#27159581 |
| Fitting, socle, derived | C3×C31×C67 ⋊ C2×C3 |
| Hall | C3×#20262033 ⋊ C2 |
| Hall | C31×C67 ⋊ C3×D3 |
| Sylow | C67 ⋊ C3⋊(D31⋊C3) |
| Sylow | C31 ⋊ C3⋊(D67⋊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)(36,40,41,37,38,42)(39,48,49,43,44,58)(45,61,62,51,52,75)(46,60,56,54,50,47)(53,80,81,64,65,93)(55,78,79,67,63,72)(57,59,77,69,70,73)(66,88,89,83,74,98)(68,95,96,84,82,91)(71,76,94,86,87,92)(85,99,100,97,90,101);(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)(35,36,39,45,53,66,79,49,60,48,59,74,90,100,81,94,98,95,80,87,52,63,73,75,91,97,84,67,54,38,42,47,57,71,86,69,56,41,40,46,55,68,85,96,62,77,78,61,76,65,82,89,92,93,101,99,88,70,44,50,58,72,83,64,51,43,37)Polycyclic presentation:
2,3^2,31,67|c^a=c^2,d^a=d^30,e^a=e^66,d^b=d^5,e^b=e^291047 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(37386, 45);