Finite Group DatabasebetaFinite Group Databasebeta| order | 320 |
| automorphism | ? |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Fitting | C5×C4⋊C4 ⋊ C4 |
| Sylow | C5 ⋊ #340 |
114 conjugacy classes of subgroups; only the 46 normal subgroups are shown.
This group has 114 conjugacy classes of subgroups — too many to draw. Only its 46 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 17 |
| transitive degree | μt(G) = 80 |
A minimal faithful permutation representation — 2 generators on 17 points:
(1,2,3,5,4)(6,7,8,9)(11,14)(12,17)(15,16);(2,3,4,5)(10,11)(12,14)(13,16)(15,17)Polycyclic presentation:
2^6,5|a^2=c,b^2=d,e^2=f,b^a=be,e^a=ef,g^a=g^2,e^b=ef,g^c=g^438 conjugacy classes — view the full character table.
| maximal subgroup of | AGL(1,𝔽5)×#102, #10603952, #10603953, … |
| maximal normal subgroup of | AGL(1,𝔽5)×#102, #10603952, #10603953, … |
| maximal quotient of | #10598504, #10598506, #10598508, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(320, 212);