Finite Group DatabasebetaFinite Group Databasebeta| order | 16770 |
| automorphism | ? |
| homology | Abelianization (H1) = C2 · Schur multiplier (H2) = C1 |
| dihedral | D8385 |
| unambiguous semidirect product | C3⋊D2795 |
| unambiguous semidirect product | C5⋊D1677 |
| unambiguous semidirect product | C13⋊D645 |
| unambiguous semidirect product | C43⋊D195 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C3×C5×C13×C43 | C2 |
| LCS | G ▷ C3×C5×C13×C43 | |
| UCS | C1 | |
32 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 8385 |
A minimal faithful permutation representation — 2 generators on 64 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);(2,3)(5,8)(6,7)(10,21)(11,20)(12,19)(13,18)(14,17)(15,16)(23,64)(24,63)(25,62)(26,61)(27,60)(28,59)(29,58)(30,57)(31,56)(32,55)(33,54)(34,53)(35,52)(36,51)(37,50)(38,49)(39,48)(40,47)(41,46)(42,45)(43,44)Polycyclic presentation:
2,3,5,13,43|b^a=b^2,c^a=c^4,d^a=d^12,e^a=e^424194 conjugacy classes. The character table is too large to display.
| maximal subgroup of | D43×D195, D13×D645, D15×D559, … |
| maximal normal subgroup of | D43×D195, D13×D645, D15×D559, … |
| maximal quotient of | Dic8385, C2×D8385, D25155 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(16770, 20);