Finite Group DatabasebetaFinite Group Databasebeta| order | 320 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | C5×#131 ⋊ C2 |
| Sylow | C5 ⋊ #444 |
104 conjugacy classes of subgroups; only the 47 normal subgroups are shown.
This group has 104 conjugacy classes of subgroups — too many to draw. Only its 47 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)(6,9,10,13,12,20,11,19)(7,8,14,16,15,17,18,21);(6,8)(7,13)(9,18)(10,16)(11,21)(12,17)(14,20)(15,19);(2,3)(4,5)(6,7)(8,9)(10,14)(11,18)(12,15)(13,16)(17,20)(19,21)Polycyclic presentation:
2^6,5|c^2=d,d^2=f,e^2=f,b^a=be,e^a=ef,g^a=g^4,c^b=cf,e^b=ef62 conjugacy classes — view the full character table.
| maximal subgroup of | #10611620, #10611621, D5×#444, … |
| maximal normal subgroup of | #10611620, #10611621, D5×#444, … |
| maximal quotient of | #10600681, #10600683, #10600692, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(320, 772);