Finite Group DatabasebetaFinite Group Databasebeta| order | 320 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Fitting | C5×#41 ⋊ C4 |
| Sylow | C5 ⋊ #416 |
158 conjugacy classes of subgroups; only the 50 normal subgroups are shown.
This group has 158 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) = 21 |
| transitive degree | μt(G) = 80 |
A minimal faithful permutation representation — 3 generators on 21 points:
(1,2,3,5,4)(7,13)(8,16)(9,10)(12,19)(14,15)(20,21);(6,8)(7,12)(9,17)(10,18)(11,16)(13,20)(14,19)(15,21);(2,3,4,5)(6,7,9,13,11,15,10,14)(8,12,17,20,16,21,18,19)Polycyclic presentation:
2^6,5|a^2=d,d^2=f,e^2=f,c^a=ce,g^a=g^2,c^b=cf,d^c=df,e^c=ef,g^d=g^438 conjugacy classes — view the full character table.
| maximal subgroup of | #10604005, #10604006, #10604777, … |
| maximal normal subgroup of | #10604005, #10604006, #10604777, … |
| maximal quotient of | #10604346, #10604349, #10604359, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(320, 1132);