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) = C2 |
No aliases for this group.
| Fitting | C5×C4⋊C4 ⋊ C2 |
| Sylow | C5 ⋊ #126 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C22×D5 |
| commutator, socle, cosocle | C22×C5 | C23 |
| Fitting | C5×C4⋊C4 | C2 |
| prime radical | C22×D5 | C22 |
| hypercenter | C4⋊C4 | D5 |
| hypocenter | C5 | #126 |
| LCS | G ▷ C22×C5 ▷ C5 | |
| UCS | C1 ◁ C22 ◁ C4⋊C4 | |
60 conjugacy classes of subgroups.
| minimal degree | μ(G) = 21 |
| transitive degree | μt(G) = 80 |
A minimal faithful permutation representation — 3 generators on 21 points:
(1,2,4,5,3)(7,11)(10,13)(14,16,17,20)(15,19,18,21);(6,8,9,12)(7,10,11,13)(15,18)(16,20);(2,3)(4,5)(6,7)(8,10)(9,11)(12,13)(14,15)(16,19)(17,18)(20,21)Polycyclic presentation:
2^5,5|b^2=d,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 | C2×#4044, #65847, #65848, … |
| maximal normal subgroup of | C2×#4044, #65847, #65848, … |
| maximal quotient of | #64940, #64943, #64949, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(160, 119);