Finite Group DatabasebetaFinite Group Databasebeta| order | 320 |
| isoclinism | isoclinic class of #66297 |
| automorphism | Aut(G) = #66297 (order 320) · Inn(G) = G · Out(G) = C1 |
| homology | Abelianization (H1) = C4 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Fitting, socle | C24 ⋊ AGL(1,𝔽5) |
| derived | C24⋊C5 ⋊ C4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, hypocenter | C24⋊C5 | C4 |
| Fitting, socle | C24 | AGL(1,𝔽5) |
| prime radical, cosocle | (C24⋊C5)⋊C2 | C2 |
| LCS | G ▷ C24⋊C5 | |
| UCS | C1 | |
40 conjugacy classes of subgroups.
| minimal degree | μ(G) = 10 |
| transitive degree | μt(G) = 10 |
A minimal faithful permutation representation — 2 generators on 10 points:
(2,8,5,10)(4,9,7,6);(1,2,4,7,5)(3,6,8,10,9)Polycyclic presentation:
2^2,5,2^4|a^2=b,c^a=c^2,d^a=df,e^a=dfg,f^a=ef,g^a=d,c^b=c^4,d^b=de,f^b=deg,g^b=df,d^c=g,e^c=dg,f^c=deg,g^c=defg| C1 | C2 | C3 | C4 | C5 | C6 | C7 | C8 | C9 | C10 | C11 | |
|---|---|---|---|---|---|---|---|---|---|---|---|
| size | 1 | 5 | 10 | 20 | 20 | 40 | 40 | 40 | 40 | 40 | 64 |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 8 | 8 | 5 |
| χ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| χ2 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 |
| χ3 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | ||||
| χ4 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | ||||
| χ5 | 4 | 4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 |
| χ6 | 5 | -3 | 1 | 1 | 1 | -1 | 1 | 1 | -1 | -1 | 0 |
| χ7 | 5 | -3 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | 0 |
| χ8 | 5 | -3 | 1 | -1 | -1 | 1 | 0 | ||||
| χ9 | 5 | -3 | 1 | -1 | -1 | 1 | 0 | ||||
| χ10 | 10 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 |
| χ11 | 10 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 |
Here i is the imaginary unit (i² = −1).
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(320, 1635);