Finite Group DatabasebetaFinite Group Databasebeta| order | 320 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C5 ⋊ #473 |
100 conjugacy classes of subgroups; only the 37 normal subgroups are shown.
This group has 100 conjugacy classes of subgroups — too many to draw. Only its 37 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 160 |
A minimal faithful permutation representation — 3 generators on 37 points:
(1,2,4,5,3)(6,9,12,26)(7,14,17,32)(8,19,21,35)(10,25,11,24)(13,28,29,37)(15,31,16,22)(18,34,33,30)(20,36,27,23);(6,8)(7,13)(9,23)(10,20)(11,27)(12,21)(14,30)(15,18)(16,33)(17,29)(19,25)(22,37)(24,35)(26,36)(28,31)(32,34);(2,3)(4,5)(6,7,12,17)(8,18,21,33)(9,22,26,31)(10,16,11,15)(13,27,29,20)(14,25,32,24)(19,28,35,37)(23,34,36,30)Polycyclic presentation:
2^6,5|a^2=f,c^2=f,d^2=f,e^2=f,b^a=be,c^a=cd,d^a=df,e^a=ef,g^a=g^4,c^b=ce,e^b=ef,d^c=df,e^c=ef38 conjugacy classes — view the full character table.
| maximal subgroup of | #10612026, #10612027, #10612032, … |
| maximal normal subgroup of | #10612026, #10612027, #10612032, … |
| maximal quotient of | #10601093, #10601094, #10601097, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(320, 831);