Finite Group DatabasebetaFinite Group Databasebeta| order | 160 |
| isoclinism | isoclinic class of #700 |
| automorphism | Aut(G) = C2×D4×AGL(1,𝔽5) (order 320) · Inn(G) = C2×AGL(1,𝔽5) · Out(G) = C23 |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Fitting | C5×C8 ⋊ C4 |
| Sylow | C5 ⋊ #97 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C4 | C2×AGL(1,𝔽5) |
| commutator, socle | C2×C5 | C42 |
| Fitting | C5×C8 | C4 |
| prime radical | C2×D5 | C2×C4 |
| cosocle | C4×D5 | C22 |
| hypercenter | C8 | AGL(1,𝔽5) |
| hypocenter | C5 | #97 |
| LCS | G ▷ C2×C5 ▷ C5 | |
| UCS | C1 ◁ C4 ◁ C8 | |
40 conjugacy classes of subgroups.
| 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)(6,7,8,11,9,10,12,13);(2,3,4,5)(7,10)(11,13)Polycyclic presentation:
2^5,5|a^2=c,b^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, C2×#3992, … |
| maximal normal subgroup of | #64845, #64846, C2×#3992, … |
| maximal quotient of | #64845, #64846, #64879, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(160, 67);