Finite Group DatabasebetaFinite Group Databasebeta| order | 320 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Fitting | C5×AGL(1,ℤ/8) ⋊ C2 |
| Sylow | C5 ⋊ #454 |
146 conjugacy classes of subgroups; only the 39 normal subgroups are shown.
This group has 146 conjugacy classes of subgroups — too many to draw. Only its 39 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)(7,13)(8,16)(9,10)(14,15)(17,18)(19,20);(6,8,11,18)(7,12,15,21)(9,17,10,16)(13,20,14,19);(2,3)(4,5)(6,7)(8,12)(9,13)(10,14)(11,15)(16,19)(17,20)(18,21)Polycyclic presentation:
2^6,5|b^2=f,d^2=f,e^2=f,c^a=cd,d^a=df,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 | #10612008, #10612010, D5×#454, … |
| maximal normal subgroup of | #10612008, #10612010, D5×#454, … |
| maximal quotient of | #10601091, #10601101, #10601102, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(320, 828);