Finite Group DatabasebetaFinite Group Databasebeta| order | 320 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Fitting | C5×AGL(1,ℤ/8) ⋊ C2 |
| Sylow | C5 ⋊ #576 |
268 conjugacy classes of subgroups; only the 99 normal subgroups are shown.
This group has 268 conjugacy classes of subgroups — too many to draw. Only its 99 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 — 4 generators on 21 points:
(1,2,4,5,3)(6,8)(7,12)(9,17)(10,14)(11,18)(13,21)(15,16)(19,20);(8,15)(9,16)(12,20)(13,19)(14,17)(18,21);(7,11)(8,14)(13,19)(15,17);(2,3)(4,5)(6,7)(8,12)(9,13)(10,11)(14,18)(15,20)(16,19)(17,21)Polycyclic presentation:
2^6,5|e^2=f,b^a=bf,g^a=g^4,d^b=df,d^c=de,e^c=ef,e^d=ef44 conjugacy classes — view the full character table.
| maximal subgroup of | #10612011, #10612013, #10612018, … |
| maximal normal subgroup of | #10612011, #10612013, #10612018, … |
| maximal quotient of | #10610599, #10610603, #10610605, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(320, 1446);