Finite Group DatabasebetaFinite Group Databasebeta| order | 320 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| Fitting | C22×C4×C5 ⋊ C4 |
| Sylow | C5 ⋊ #410 |
158 conjugacy classes of subgroups; only the 48 normal subgroups are shown.
This group has 158 conjugacy classes of subgroups — too many to draw. Only its 48 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)(7,12)(8,14)(10,17)(11,18)(13,16)(15,21)(19,20);(6,8,10,15)(7,11,13,19)(9,14,17,21)(12,18,16,20);(2,3,4,5)(6,7)(8,11)(9,16,17,12)(10,13)(14,20,21,18)(15,19)Polycyclic presentation:
2^6,5|a^2=d,b^2=f,c^a=ce,e^a=ef,g^a=g^2,d^c=df,g^d=g^438 conjugacy classes — view the full character table.
| maximal subgroup of | #10598768, #10598769, #10598790, … |
| maximal normal subgroup of | #10598768, #10598769, #10598790, … |
| maximal quotient of | #10604159, #10604161, #10604162, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(320, 1083);