Finite Group DatabasebetaFinite Group Databasebeta| order | 320 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| 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)(7,12)(8,16)(9,19)(13,14)(15,20)(17,18);(6,8)(7,11)(9,16)(10,18)(12,15)(13,20)(14,21)(17,19);(2,3)(4,5)(6,7,10,14)(8,15,18,20)(9,12,19,13)(11,16,21,17)Polycyclic presentation:
2^6,5|a^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 | #10612011, #10612012, #10612019, … |
| maximal normal subgroup of | #10612011, #10612012, #10612019, … |
| maximal quotient of | #10601092, #10601095, #10601097, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(320, 827);