Finite Group DatabasebetaFinite Group Databasebeta| order | 320 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
| unambiguous semidirect product | D5⋊#99 |
| Fitting | C2×C5×D4 ⋊ C4 |
| Sylow | C5 ⋊ C2×#99 |
210 conjugacy classes of subgroups; only the 50 normal subgroups are shown.
This group has 210 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) = 13 |
| transitive degree | μt(G) = 40 |
A minimal faithful permutation representation — 3 generators on 13 points:
(1,2,3,5,4)(6,8)(7,10)(9,13)(11,12);(8,13)(10,12);(2,3,4,5)(6,7)(8,12,13,10)(9,11)Polycyclic presentation:
2^6,5|a^2=d,c^a=ce,e^a=ef,g^a=g^2,c^b=cf,d^c=df,g^d=g^432 conjugacy classes — view the full character table.
| maximal subgroup of | #10598568, #10598573, AGL(1,𝔽5)×#99, … |
| maximal normal subgroup of | #10598568, #10598573, AGL(1,𝔽5)×#99, … |
| maximal quotient of | D5⋊#323, #10604201, #10604202, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(320, 1108);