Finite Group DatabasebetaFinite Group Databasebeta| order | 22010 |
| automorphism | ? |
| homology | Abelianization (H1) = C2 · Schur multiplier (H2) = C1 |
| dihedral | D11005 |
| unambiguous semidirect product | C5⋊D2201 |
| unambiguous semidirect product | C31⋊D355 |
| unambiguous semidirect product | C71⋊D155 |
| Fitting, socle, derived, Hall | C5×C31×C71 ⋊ C2 |
| Hall | C31×C71 ⋊ D5 |
| Hall | C5×C71 ⋊ D31 |
| Hall | C5×C31 ⋊ D71 |
| Sylow | C71 ⋊ D155 |
| Sylow | C31 ⋊ D355 |
| Sylow | C5 ⋊ D2201 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C5×C31×C71 | C2 |
| LCS | G ▷ C5×C31×C71 | |
| UCS | C1 | |
16 conjugacy classes of subgroups.
| minimal degree | μ(G) = 107 |
| transitive degree | μt(G) = 11005 |
A minimal faithful permutation representation — 2 generators on 107 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);(2,5)(3,4)(7,36)(8,35)(9,34)(10,33)(11,32)(12,31)(13,30)(14,29)(15,28)(16,27)(17,26)(18,25)(19,24)(20,23)(21,22)(38,107)(39,106)(40,105)(41,104)(42,103)(43,102)(44,101)(45,100)(46,99)(47,98)(48,97)(49,96)(50,95)(51,94)(52,93)(53,92)(54,91)(55,90)(56,89)(57,88)(58,87)(59,86)(60,85)(61,84)(62,83)(63,82)(64,81)(65,80)(66,79)(67,78)(68,77)(69,76)(70,75)(71,74)(72,73)Polycyclic presentation:
2,5,31,71|b^a=b^4,c^a=c^30,d^a=d^705504 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(22010, 14);