Finite Group DatabasebetaFinite Group Databasebeta| order | 16170 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | C5⋊(C11⋊(D49⋊C3)) |
| unambiguous semidirect product | C11⋊(C5⋊(D49⋊C3)) |
| unambiguous semidirect product | (C49⋊C3)⋊D55 |
| unambiguous semidirect product | D2695⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, Fitting, cosocle, hypocenter | C5×C11×C49 | C2×C3 |
| Frattini | C7 | C5⋊(C11⋊AGL(1,𝔽7)) |
| socle | C5×C7×C11 | AGL(1,𝔽7) |
| LCS | G ▷ C5×C11×C49 | |
| UCS | C1 | |
48 conjugacy classes of subgroups.
| minimal degree | μ(G) = 65 |
| transitive degree | μt(G) = 2695 |
A minimal faithful permutation representation — 2 generators on 65 points:
(2,3)(4,5)(6,7)(8,9)(10,11)(13,21,23,15,16,24)(14,33,35,19,20,36)(17,45,32,26,27,54)(18,50,51,30,31,60)(22,29,28,34,39,47)(25,48,57,43,44,58)(37,49,52,55,38,41)(40,53,42,56,46,59)(62,63)(64,65);(1,2,4,6,8,10,11,9,7,5,3)(12,57,29,52,51,38,40,19,55,17,42,47,21,22,33,32,53,24,44,43,15,35,18,31,45,27,50,30,36,13,25,48,23,46,54,20,34,16,28,59,26,37,14,56,49,60,41,39,58)(61,62,64,65,63)Polycyclic presentation:
2,3,5,7,11,7|d^7=f^6,c^a=c^4,d^a=d^6f,e^a=e^10,f^a=f^6,d^b=d^2f^3,f^b=f^2527 conjugacy classes — view the full character table.
| maximal subgroup of | D11×C5⋊(D49⋊C3), D55×D49⋊C3, D5×C11⋊(D49⋊C3), … |
| maximal normal subgroup of | D11×C5⋊(D49⋊C3), D55×D49⋊C3, D5×C11⋊(D49⋊C3), … |
| maximal quotient of | C11⋊((C49⋊C3)⋊Dic5), C2×C5⋊(C11⋊(D49⋊C3)) |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(16170, 33);