Finite Group DatabasebetaFinite Group Databasebeta| order | 320 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
| unambiguous semidirect product | D5⋊#104 |
| Fitting | C5×#41 ⋊ C4 |
| Sylow | C5 ⋊ C2×#104 |
170 conjugacy classes of subgroups; only the 50 normal subgroups are shown.
This group has 170 conjugacy classes of subgroups — too many to draw. Only its 50 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 — 3 generators on 13 points:
(1,2,3,5,4)(6,7)(8,12)(9,11)(10,13);(7,11)(10,12);(2,3,4,5)(7,10,11,12)Polycyclic presentation:
2^6,5|a^2=d,e^2=f,c^a=ce,g^a=g^2,c^b=cf,d^c=df,e^c=ef,g^d=g^438 conjugacy classes — view the full character table.
| maximal subgroup of | AGL(1,𝔽5)×#104, #10604002, #10604777, … |
| maximal normal subgroup of | AGL(1,𝔽5)×#104, #10604002, #10604777, … |
| maximal quotient of | #10604346, #10604347, #10604350, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(320, 1130);