Finite Group DatabasebetaFinite Group Databasebeta| order | 37820 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C5 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | #34038658⋊C2 |
| unambiguous semidirect product | #34038664⋊C2 |
| unambiguous semidirect product | #34038670⋊C2 |
| Fitting, socle, derived, Hall | C31×C61 ⋊ C22×C5 |
| Hall | #22628160 ⋊ C22 |
| Hall | D31×D61 ⋊ C5 |
| Sylow | C61 ⋊ C2×D31⋊C5 |
| Sylow | C31 ⋊ C2×D61⋊C5 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C31×C61 | C22×C5 |
| LCS | G ▷ C31×C61 | |
| UCS | C1 | |
40 conjugacy classes of subgroups.
| minimal degree | μ(G) = 92 |
| transitive degree | μt(G) = 1891 |
A minimal faithful permutation representation — 2 generators on 92 points:
(2,6,7,18,5,3,4,11,12,9)(8,23,24,25,10,14,15,27,29,17)(13,30,22,28,16,20,21,31,19,26)(32,33,39,61,42)(34,44,74,92,75)(35,46,81,52,51)(36,37,53,89,50)(38,57,65,84,64)(40,48,72,83,45)(41,66,62,82,68)(43,69,87,47,76)(49,54,90,67,56)(55,58,70,77,91)(59,79,80,71,86)(60,88,73,63,78);(1,2,5,10,12,19,29,23,7,13,22,30,24,26,14,6,4,8,16,27,21,31,20,11,15,25,28,18,17,9,3)(33,34)(35,37)(36,40)(38,43)(39,41)(42,50)(44,54)(45,52)(46,58)(47,49)(48,62)(51,67)(53,70)(55,72)(56,65)(57,77)(59,79)(60,73)(61,75)(63,74)(64,86)(66,69)(68,84)(71,88)(76,78)(80,81)(82,89)(83,87)(85,91)(90,92)Polycyclic presentation:
2^2,5,31,61|e^a=e^60,d^b=d^30,d^c=d^16,e^c=e^20128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(37820, 42);