Finite Group DatabasebetaFinite Group Databasebeta| order | 160 |
| automorphism | Aut(G) = AGL(1,𝔽5)×#2846 (order 2560) · Inn(G) = C22×D5 · Out(G) = C2×C4×D4 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C5 ⋊ #123 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C22×D5 |
| commutator, socle, cosocle | C22×C5 | C23 |
| Fitting, prime radical | C2×C5×D4 | C2 |
| hypercenter | C2×D4 | D5 |
| hypocenter | C5 | #123 |
| LCS | G ▷ C22×C5 ▷ C5 | |
| UCS | C1 ◁ C22 ◁ C2×D4 | |
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)(7,9)(12,16)(14,17);(10,12)(11,14)(13,16)(15,17);(2,3)(4,5)(6,7)(8,9)(10,11,13,15)(12,14,16,17)Polycyclic presentation:
2^5,5|a^2=e,c^a=cd,f^a=f^4,c^b=ce34 conjugacy classes — view the full character table.
| maximal subgroup of | #64692, #64696, #65031, … |
| maximal normal subgroup of | #64692, #64696, #65031, … |
| maximal quotient of | #65232, #65233, #65235, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(160, 156);