Finite Group DatabasebetaFinite Group Databasebeta| order | 320 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | C5×#137 ⋊ C2 |
| Sylow | C5 ⋊ #471 |
104 conjugacy classes of subgroups; only the 37 normal subgroups are shown.
This group has 104 conjugacy classes of subgroups — too many to draw. Only its 37 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 21 |
| transitive degree | μt(G) = 80 |
A minimal faithful permutation representation — 3 generators on 21 points:
(1,2,4,5,3)(6,8,11,18)(7,13,14,21)(9,17,10,15)(12,20,19,16);(6,7)(8,16)(9,12)(10,19)(11,14)(13,17)(15,21)(18,20);(2,3)(4,5)(7,12)(8,15)(9,10)(14,19)(16,20)(17,18)Polycyclic presentation:
2^6,5|c^2=f,d^2=f,e^2=f,b^a=be,c^a=cd,d^a=df,e^a=ef,g^a=g^4,c^b=ce,e^b=ef,d^c=df,e^c=ef38 conjugacy classes — view the full character table.
| maximal subgroup of | #10612025, #10612027, #10612032, … |
| maximal normal subgroup of | #10612025, #10612027, #10612032, … |
| maximal quotient of | #10601092, #10601095, #10601096, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(320, 830);