Finite Group DatabasebetaFinite Group Databasebeta| order | 320 |
| automorphism | ? |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Fitting | C5×C22⋊C4 ⋊ C4 |
| Sylow | C5 ⋊ #342 |
142 conjugacy classes of subgroups; only the 36 normal subgroups are shown.
This group has 142 conjugacy classes of subgroups — too many to draw. Only its 36 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 13 |
| transitive degree | μt(G) = 40 |
A minimal faithful permutation representation — 2 generators on 13 points:
(1,2,3,5,4)(6,8)(7,10,11,12)(9,13);(2,3,4,5)(6,7)(8,12)(9,11)(10,13)Polycyclic presentation:
2^6,5|a^2=c,b^2=d,b^a=be,d^a=df,g^a=g^2,e^b=ef,g^c=g^432 conjugacy classes — view the full character table.
| maximal subgroup of | #10603919, AGL(1,𝔽5)×#99, C2×#64866, … |
| maximal normal subgroup of | #10603919, AGL(1,𝔽5)×#99, C2×#64866, … |
| maximal quotient of | #10598503, #10598504, #10598506, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(320, 204);