Finite Group DatabasebetaFinite Group Databasebeta| order | 320 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Fitting | C2×C5×Q8 ⋊ C4 |
| Sylow | C5 ⋊ #419 |
150 conjugacy classes of subgroups; only the 50 normal subgroups are shown.
This group has 150 conjugacy classes of subgroups — too many to draw. Only its 50 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:
(1,2,3,5,4)(6,9,11,18)(7,13,14,21)(8,15,10,17)(12,20,19,16);(6,8,11,10)(7,12,14,19)(9,17,18,15)(13,16,21,20);(2,3,4,5)(6,7)(8,12)(9,16)(10,19)(11,14)(13,15)(17,21)(18,20)Polycyclic presentation:
2^6,5|a^2=d,b^2=f,c^2=f,e^2=f,c^a=ce,e^a=ef,g^a=g^2,c^b=cf,e^c=ef,g^d=g^432 conjugacy classes — view the full character table.
| maximal subgroup of | #10603933, #10604562, #10604570, … |
| maximal normal subgroup of | #10603933, #10604562, #10604570, … |
| maximal quotient of | #10604259, #10604261, #10604271, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(320, 1120);