Finite Group DatabasebetaFinite Group Databasebeta| order | 160 |
| automorphism | Aut(G) = C22×AGL(1,𝔽5)×S4 (order 1920) · Inn(G) = C22×D5 · Out(G) = C23×D3 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C5 ⋊ #131 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C4 | C22×D5 |
| commutator, socle | C2×C5 | C22×C4 |
| Fitting, prime radical | C5×#41 | C2 |
| cosocle | C4×C5 | C23 |
| hypercenter | #41 | D5 |
| hypocenter | C5 | #131 |
| LCS | G ▷ C2×C5 ▷ C5 | |
| UCS | C1 ◁ C4 ◁ #41 | |
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)(8,14)(11,17)(15,20)(18,21);(6,8)(7,11)(9,15)(10,14)(12,18)(13,17)(16,20)(19,21);(2,3)(4,5)(6,7,9,12,10,13,16,19)(8,11,15,18,14,17,20,21)Polycyclic presentation:
2^5,5|a^2=d,d^2=e,f^a=f^4,c^b=ce40 conjugacy classes — view the full character table.
| maximal subgroup of | #64932, #65112, #65113, … |
| maximal normal subgroup of | #64932, #65112, #65113, … |
| maximal quotient of | #65286, #65288, #65289, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(160, 169);