Finite Group DatabasebetaFinite Group Databasebeta| order | 160 |
| automorphism | Aut(G) = C2×AGL(1,𝔽5)×S4 (order 960) · Inn(G) = C22×AGL(1,𝔽5) · Out(G) = C2×D3 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C5 ⋊ #131 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C22×AGL(1,𝔽5) |
| commutator, socle | C2×C5 | C22×C4 |
| Fitting | C5×Q8 | C4 |
| prime radical | #713 | C2 |
| cosocle | Dic5 | C23 |
| hypercenter | Q8 | AGL(1,𝔽5) |
| hypocenter | C5 | #131 |
| LCS | G ▷ C2×C5 ▷ C5 | |
| UCS | C1 ◁ C2 ◁ Q8 | |
62 conjugacy classes of subgroups.
| minimal degree | μ(G) = 21 |
| transitive degree | μt(G) = 80 |
A minimal faithful permutation representation — 3 generators on 21 points:
(1,2,3,5,4)(6,9,11,10)(7,13,15,14)(8,16,17,18)(12,19,20,21);(6,8,11,17)(7,12,15,20)(9,18,10,16)(13,21,14,19);(2,3,4,5)(6,7,10,14,11,15,9,13)(8,12,16,19,17,20,18,21)Polycyclic presentation:
2^5,5|a^2=d,b^2=e,c^2=e,d^2=e,f^a=f^2,c^b=ce,f^d=f^425 conjugacy classes — view the full character table.
| maximal subgroup of | #65736, #65737, #65740, … |
| maximal normal subgroup of | #65736, #65737, #65740, … |
| maximal quotient of | #65701, #65702, #65704, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(160, 208);