Finite Group DatabasebetaFinite Group Databasebeta| order | 160 |
| automorphism | Aut(G) = AGL(1,𝔽5)×#23900 (order 5120) · Inn(G) = C22×D5 · Out(G) = C2×#525 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| Fitting | C42×C5 ⋊ C2 |
| Sylow | C5 ⋊ #124 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C22×D5 |
| commutator, socle, cosocle | C22×C5 | C23 |
| Fitting | C42×C5 | C2 |
| prime radical | C2×D20 | C2 |
| hypercenter | C42 | D5 |
| hypocenter | C5 | #124 |
| LCS | G ▷ C22×C5 ▷ C5 | |
| UCS | C1 ◁ C22 ◁ C42 | |
76 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)(6,8,9,7)(11,15)(14,17);(6,7,9,8)(10,12,13,16)(11,14,15,17);(2,3)(4,5)(7,8)(10,11)(12,14)(13,15)(16,17)Polycyclic presentation:
2^5,5|b^2=d,c^2=e,b^a=be,c^a=cd,f^a=f^446 conjugacy classes — view the full character table.
| maximal subgroup of | #64684, #64854, #64856, … |
| maximal normal subgroup of | #64684, #64854, #64856, … |
| maximal quotient of | #64948, #64961, #64962, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(160, 96);