Finite Group DatabasebetaFinite Group Databasebeta| order | 160 |
| isoclinism | isoclinic class of #700 |
| automorphism | Aut(G) = #41×AGL(1,𝔽5) (order 320) · Inn(G) = C2×AGL(1,𝔽5) · Out(G) = C23 |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C1 |
No aliases for this group.
| Sylow | C5 ⋊ M32 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C4 | C2×AGL(1,𝔽5) |
| commutator, socle | C2×C5 | C2×C8 |
| Fitting | C5×C8 | C4 |
| prime radical | C2×D5 | C8 |
| cosocle | #168 | C22 |
| hypercenter | C8 | AGL(1,𝔽5) |
| hypocenter | C5 | M32 |
| LCS | G ▷ C2×C5 ▷ C5 | |
| UCS | C1 ◁ C4 ◁ C8 | |
26 conjugacy classes of subgroups.
| minimal degree | μ(G) = 21 |
| transitive degree | μt(G) = 80 |
A minimal faithful permutation representation — 2 generators on 21 points:
(1,2,3,5,4)(6,8,10,17,11,9,16,18)(7,12,13,19,14,15,20,21);(2,3,4,5)(6,7,9,12,10,13,18,19,11,14,8,15,16,20,17,21)Polycyclic presentation:
2^5,5|a^2=c,b^2=d,c^2=d,d^2=e,b^a=be,f^a=f^2,f^c=f^428 conjugacy classes — view the full character table.
| maximal subgroup of | #64845, #64846, #64903, … |
| maximal normal subgroup of | #64845, #64846, #64903, … |
| maximal quotient of | #64886, #64887, #64888, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(160, 65);