Finite Group DatabasebetaFinite Group Databasebeta| order | 160 |
| automorphism | Aut(G) = C22×D4×AGL(1,𝔽5) (order 640) · Inn(G) = C22×D5 · Out(G) = C24 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | C5×M16 ⋊ C2 |
| Sylow | C5 ⋊ #131 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C4 | C22×D5 |
| commutator, socle | C2×C5 | C22×C4 |
| Fitting | C5×M16 | C2 |
| prime radical | #709 | C2 |
| cosocle | C4×C5 | C23 |
| hypercenter | M16 | D5 |
| hypocenter | C5 | #131 |
| LCS | G ▷ C2×C5 ▷ C5 | |
| UCS | C1 ◁ C4 ◁ M16 | |
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:
(1,2,4,5,3)(6,8,9,15,10,14,16,20)(7,12,13,19,11,17,18,21);(7,11)(8,14)(13,18)(15,20);(2,3)(4,5)(6,7)(8,12)(9,13)(10,11)(14,17)(15,19)(16,18)(20,21)Polycyclic presentation:
2^5,5|c^2=d,d^2=e,b^a=be,f^a=f^4,c^b=ce40 conjugacy classes — view the full character table.
| maximal subgroup of | #64898, #65037, #65038, … |
| maximal normal subgroup of | #64898, #65037, #65038, … |
| maximal quotient of | #64990, #64995, #64997, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(160, 128);