Finite Group DatabasebetaFinite Group Databasebeta| order | 160 |
| automorphism | Aut(G) = C22×AGL(1,𝔽5)×#120 (order 2560) · Inn(G) = C22×D5 · Out(G) = C23×D4 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C5 ⋊ C4×Q8 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C22×D5 |
| commutator | C2×C5 | C22×C4 |
| Fitting | C5×C4⋊C4 | C2 |
| socle, prime radical, cosocle | C22×C5 | C23 |
| hypercenter | C4⋊C4 | D5 |
| hypocenter | C5 | C4×Q8 |
| LCS | G ▷ C2×C5 ▷ C5 | |
| UCS | C1 ◁ C22 ◁ C4⋊C4 | |
70 conjugacy classes of subgroups.
| minimal degree | μ(G) = 17 |
| transitive degree | μt(G) = 160 |
A minimal faithful permutation representation — 3 generators on 17 points:
(1,2,4,5,3)(6,8,9,7)(12,16)(14,17);(10,12,13,16)(11,14,15,17);(2,3)(4,5)(6,7,9,8)(10,11,13,15)(12,14,16,17)Polycyclic presentation:
2^5,5|a^2=e,b^2=d,c^2=de,f^a=f^4,c^b=cd40 conjugacy classes — view the full character table.
| maximal subgroup of | #64872, #65046, #65047, … |
| maximal normal subgroup of | #64872, #65046, #65047, … |
| maximal quotient of | #64935, #64938, #64941, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(160, 108);