Finite Group DatabasebetaFinite Group Databasebeta| order | 16170 |
| automorphism | ? |
| homology | Abelianization (H1) = C2 · Schur multiplier (H2) = C1 |
| dihedral | D8085 |
| unambiguous semidirect product | C3⋊D2695 |
| unambiguous semidirect product | C5⋊D1617 |
| unambiguous semidirect product | C11⋊D735 |
| unambiguous semidirect product | C49⋊D165 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, Fitting, cosocle, hypocenter | C3×C5×C11×C49 | C2 |
| Frattini | C7 | D1155 |
| socle | C3×C5×C7×C11 | D7 |
| LCS | G ▷ C3×C5×C11×C49 | |
| UCS | C1 | |
48 conjugacy classes of subgroups.
| minimal degree | μ(G) = 68 |
| transitive degree | μt(G) = 8085 |
A minimal faithful permutation representation — 2 generators on 68 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);(2,3)(5,8)(6,7)(10,57)(11,56)(12,55)(13,54)(14,53)(15,52)(16,51)(17,50)(18,49)(19,48)(20,47)(21,46)(22,45)(23,44)(24,43)(25,42)(26,41)(27,40)(28,39)(29,38)(30,37)(31,36)(32,35)(33,34)(59,68)(60,67)(61,66)(62,65)(63,64)Polycyclic presentation:
2,3,5,7,11,7|d^7=f^6,b^a=b^2,c^a=c^4,d^a=d^6f,e^a=e^10,f^a=f^64044 conjugacy classes. The character table is too large to display.
| maximal subgroup of | D11×D735, D49×D165, D15×D539, … |
| maximal normal subgroup of | D11×D735, D49×D165, D15×D539, … |
| maximal quotient of | Dic8085, C2×D8085 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(16170, 19);