Finite Group DatabasebetaFinite Group Databasebeta| order | 160 |
| automorphism | Aut(G) = C26×AGL(1,𝔽5) (order 1280) · Inn(G) = C22×D5 · Out(G) = C23×C4 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | C5×C4⋊C4 ⋊ C2 |
| Sylow | C5 ⋊ #123 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C22×D5 |
| commutator, socle, cosocle | C22×C5 | C23 |
| Fitting | C5×C4⋊C4 | C2 |
| prime radical | C2×D20 | C2 |
| hypercenter | C4⋊C4 | D5 |
| hypocenter | C5 | #123 |
| LCS | G ▷ C22×C5 ▷ C5 | |
| UCS | C1 ◁ C22 ◁ C4⋊C4 | |
78 conjugacy classes of subgroups.
| minimal degree | μ(G) = 17 |
| transitive degree | μt(G) = 80 |
A minimal faithful permutation representation — 3 generators on 17 points:
(1,2,4,5,3)(6,7,8,9)(11,15)(14,17);(7,9)(10,12,13,16)(11,14,15,17);(2,3)(4,5)(10,11)(12,14)(13,15)(16,17)Polycyclic presentation:
2^5,5|b^2=d,c^2=e,c^a=cd,f^a=f^4,c^b=ce34 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#4040, #65841, #65848, … |
| maximal normal subgroup of | C2×#4040, #65841, #65848, … |
| maximal quotient of | #64944, #64949, #64953, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(160, 115);