Finite Group DatabasebetaFinite Group Databasebeta| order | 680 |
| isoclinism | isoclinic class of #66772 |
| automorphism | ? |
| homology | Abelianization (H1) = C8 · Schur multiplier (H2) = C1 |
No aliases for this group.
| derived, Hall | C5×C17 ⋊ C8 |
| Sylow | C17 ⋊ #168 |
| Sylow | C5 ⋊ #3624 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | #66772 |
| commutator, hypocenter | C5×C17 | C8 |
| Fitting, socle, prime radical | C2×C5×C17 | C4 |
| cosocle | C5×Dic17 | C2 |
| LCS | G ▷ C5×C17 | |
| UCS | C1 ◁ C2 | |
16 conjugacy classes of subgroups.
| minimal degree | μ(G) = 30 |
| transitive degree | μt(G) = 680 |
Polycyclic presentation:
2^3,5,17|a^2=b,b^2=c,d^a=d^4,e^a=e^4,e^b=e^1656 conjugacy classes — view the full character table.
| maximal subgroup of | #13062725, #13062726, D5×#3624, … |
| maximal normal subgroup of | #13062725, #13062726, D5×#3624, … |
| maximal quotient of | #13062638, C2×#10621746 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(680, 6);