Finite Group DatabasebetaFinite Group Databasebeta| order | 160 |
| isoclinism | isoclinic class of (C24⋊C5)⋊C2 |
| automorphism | Aut(G) = AΓL(1,𝔽16) (order 960) · Inn(G) = G · Out(G) = D3 |
| homology | Abelianization (H1) = C2 · Schur multiplier (H2) = C22 |
| unambiguous semidirect product | (C24⋊C5)⋊C2 |
| Fitting, socle | C24 ⋊ D5 |
| derived | C24⋊C5 ⋊ C2 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | C24⋊C5 | C2 |
| Fitting, socle | C24 | D5 |
| LCS | G ▷ C24⋊C5 | |
| UCS | C1 | |
43 conjugacy classes of subgroups.
| minimal degree | μ(G) = 10 |
| transitive degree | μt(G) = 10 |
A minimal faithful permutation representation — 2 generators on 10 points:
(2,4,6,7)(5,10,9,8);(1,2,5,8,4)(3,6,9,10,7)Polycyclic presentation:
2,5,2^4|b^a=b^4,c^a=cd,e^a=cdf,f^a=ce,c^b=cd,d^b=de,e^b=ef,f^b=c| C1 | C2 | C3 | C4 | C5 | C6 | C7 | C8 | C9 | C10 | |
|---|---|---|---|---|---|---|---|---|---|---|
| size | 1 | 5 | 5 | 5 | 20 | 20 | 20 | 20 | 32 | 32 |
| order | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 5 | 5 |
| χ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| χ2 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 |
| χ3 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | ||
| χ4 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | ||
| χ5 | 5 | 1 | 1 | -3 | 1 | 1 | -1 | -1 | 0 | 0 |
| χ6 | 5 | 1 | 1 | -3 | -1 | -1 | 1 | 1 | 0 | 0 |
| χ7 | 5 | 1 | -3 | 1 | 1 | -1 | 1 | -1 | 0 | 0 |
| χ8 | 5 | 1 | -3 | 1 | -1 | 1 | -1 | 1 | 0 | 0 |
| χ9 | 5 | -3 | 1 | 1 | 1 | -1 | -1 | 1 | 0 | 0 |
| χ10 | 5 | -3 | 1 | 1 | -1 | 1 | 1 | -1 | 0 | 0 |
| maximal subgroup of | #66297, C2×(C24⋊C5)⋊C2, ((C24⋊C5)⋊C2)⋊C3, … |
| maximal normal subgroup of | #66297, C2×(C24⋊C5)⋊C2, ((C24⋊C5)⋊C2)⋊C3, … |
| maximal quotient of | #66243, #66244, #66245, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(160, 234);