Finite Group DatabasebetaFinite Group Databasebeta| order | 160 |
| automorphism | Aut(G) = AGL(1,𝔽5)×#23900 (order 5120) · Inn(G) = C22×D5 · Out(G) = D4×#41 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| Fitting | C2×C5×Q8 ⋊ C2 |
| Sylow | C5 ⋊ #124 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C22×D5 |
| commutator, socle, cosocle | C22×C5 | C23 |
| Fitting | C2×C5×Q8 | C2 |
| prime radical | C2×D20 | C2 |
| hypercenter | C2×Q8 | D5 |
| hypocenter | C5 | #124 |
| LCS | G ▷ C22×C5 ▷ C5 | |
| UCS | C1 ◁ C22 ◁ C2×Q8 | |
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)(7,9)(10,12,13,17)(11,15,14,16);(10,11,13,14)(12,16,17,15);(2,3)(4,5)(6,7)(8,9)(11,14)(15,16)Polycyclic presentation:
2^5,5|b^2=e,c^2=e,b^a=be,c^a=cd,f^a=f^4,c^b=ce34 conjugacy classes — view the full character table.
| maximal subgroup of | #64697, #64927, #64928, … |
| maximal normal subgroup of | #64697, #64927, #64928, … |
| maximal quotient of | #65266, #65275, #65277, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(160, 168);