Finite Group DatabasebetaFinite Group Databasebeta| order | 320 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Sylow | C5 ⋊ #421 |
154 conjugacy classes of subgroups; only the 67 normal subgroups are shown.
This group has 154 conjugacy classes of subgroups — too many to draw. Only its 67 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)(7,12)(9,17)(10,19)(11,16)(13,21)(14,20)(15,18);(8,16)(12,20)(15,21)(17,19);(2,3)(4,5)(6,7,9,13,11,14,10,18)(8,15,17,12,16,21,19,20)Polycyclic presentation:
2^6,5|a^2=d,d^2=f,e^2=f,c^a=ce,e^a=ef,g^a=g^4,c^b=cf,e^c=ef62 conjugacy classes — view the full character table.
| maximal subgroup of | #10600798, #10600805, #10600831, … |
| maximal normal subgroup of | #10600798, #10600805, #10600831, … |
| maximal quotient of | #10602367, #10602383, #10602390, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(320, 863);