Finite Group DatabasebetaFinite Group Databasebeta| order | 36722 |
| automorphism | ? |
| homology | Abelianization (H1) = C2 · Schur multiplier (H2) = C1 |
| dihedral | D18361 |
| unambiguous semidirect product | C7⋊D2623 |
| unambiguous semidirect product | C43⋊D427 |
| unambiguous semidirect product | C61⋊D301 |
| Fitting, socle, derived, Hall | C7×C43×C61 ⋊ C2 |
| Hall | C43×C61 ⋊ D7 |
| Hall | C7×C61 ⋊ D43 |
| Hall | C7×C43 ⋊ D61 |
| Sylow | C61 ⋊ D301 |
| Sylow | C43 ⋊ D427 |
| Sylow | C7 ⋊ D2623 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C7×C43×C61 | C2 |
| LCS | G ▷ C7×C43×C61 | |
| UCS | C1 | |
16 conjugacy classes of subgroups.
| minimal degree | μ(G) = 111 |
| transitive degree | μt(G) = 18361 |
A minimal faithful permutation representation — 2 generators on 111 points:
(1,2,3,4,5,6,7)(8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50)(51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111);(2,7)(3,6)(4,5)(9,50)(10,49)(11,48)(12,47)(13,46)(14,45)(15,44)(16,43)(17,42)(18,41)(19,40)(20,39)(21,38)(22,37)(23,36)(24,35)(25,34)(26,33)(27,32)(28,31)(29,30)(52,111)(53,110)(54,109)(55,108)(56,107)(57,106)(58,105)(59,104)(60,103)(61,102)(62,101)(63,100)(64,99)(65,98)(66,97)(67,96)(68,95)(69,94)(70,93)(71,92)(72,91)(73,90)(74,89)(75,88)(76,87)(77,86)(78,85)(79,84)(80,83)(81,82)Polycyclic presentation:
2,7,43,61|b^a=b^6,c^a=c^42,d^a=d^609182 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(36722, 9);