Finite Group DatabasebetaFinite Group Databasebeta| order | 320 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Sylow | C5 ⋊ #413 |
150 conjugacy classes of subgroups; only the 48 normal subgroups are shown.
This group has 150 conjugacy classes of subgroups — too many to draw. Only its 48 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)(7,12)(8,14)(9,16)(17,21);(6,8)(7,11)(9,15)(10,14)(12,18)(13,17)(16,20)(19,21);(2,3,4,5)(6,7,9,13,10,12,16,19)(8,11,15,17,14,18,20,21)Polycyclic presentation:
2^6,5|a^2=d,d^2=f,c^a=ce,e^a=ef,g^a=g^2,c^b=cf,d^c=df,g^d=g^432 conjugacy classes — view the full character table.
| maximal subgroup of | #10598570, #10598574, #10603920, … |
| maximal normal subgroup of | #10598570, #10598574, #10603920, … |
| maximal quotient of | #10604200, #10604201, #10604203, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(320, 1115);