Finite Group DatabasebetaFinite Group Databasebeta| order | 320 |
| automorphism | Aut(G) = C22×D4×AGL(1,𝔽5) (order 640) · Inn(G) = D4×AGL(1,𝔽5) · Out(G) = C22 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | C5×SD16 ⋊ C4 |
| Sylow | C5 ⋊ #443 |
106 conjugacy classes of subgroups; only the 40 normal subgroups are shown.
This group has 106 conjugacy classes of subgroups — too many to draw. Only its 40 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)(6,9,12,13)(7,14,17,18)(8,16,19,15)(10,20,11,21);(6,8)(7,13)(9,17)(10,18)(11,14)(12,19)(15,21)(16,20);(2,3,4,5)(6,7,10,15,12,17,11,16)(8,13,18,21,19,9,14,20)Polycyclic presentation:
2^6,5|a^2=d,c^2=f,d^2=f,e^2=f,g^a=g^2,c^b=ce,e^b=ef,e^c=ef,g^d=g^435 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#65736, #10617431, D5⋊#443, … |
| maximal normal subgroup of | C2×#65736, #10617431, D5⋊#443, … |
| maximal quotient of | #10603935, #10603944, #10603945, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(320, 1074);