Finite Group DatabasebetaFinite Group Databasebeta| order | 320 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| Fitting | C5×#117 ⋊ C2 |
| Sylow | C5 ⋊ #410 |
158 conjugacy classes of subgroups; only the 59 normal subgroups are shown.
This group has 158 conjugacy classes of subgroups — too many to draw. Only its 59 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)(7,12,13,16)(8,14)(10,17)(11,18,19,20)(15,21);(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)(10,13)(12,17)(14,20)(15,19)(18,21)Polycyclic presentation:
2^6,5|b^2=f,c^2=d,c^a=ce,d^a=df,g^a=g^4,e^c=ef62 conjugacy classes — view the full character table.
| maximal subgroup of | #10599491, #10599497, #10599500, … |
| maximal normal subgroup of | #10599491, #10599497, #10599500, … |
| maximal quotient of | #10600224, #10600225, #10600226, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(320, 633);