Finite Group DatabasebetaFinite Group Databasebeta| order | 160 |
| isoclinism | isoclinic class of #700 |
| automorphism | Aut(G) = #41×AGL(1,𝔽5) (order 320) · Inn(G) = C2×AGL(1,𝔽5) · Out(G) = C23 |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C1 |
No aliases for this group.
| Sylow | C5 ⋊ M32 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C4 | C2×AGL(1,𝔽5) |
| commutator, socle | C2×C5 | C2×C8 |
| Fitting | C2×C4×C5 | C4 |
| prime radical | C22×C5 | C8 |
| cosocle | #168 | C22 |
| hypercenter | C2×C4 | AGL(1,𝔽5) |
| hypocenter | C5 | M32 |
| LCS | G ▷ C2×C5 ▷ C5 | |
| UCS | C1 ◁ C4 ◁ C2×C4 | |
26 conjugacy classes of subgroups.
| minimal degree | μ(G) = 21 |
| transitive degree | μt(G) = 80 |
A minimal faithful permutation representation — 2 generators on 21 points:
(1,2,3,5,4)(7,11)(12,17)(13,18)(19,21);(2,3,4,5)(6,7,8,12,9,13,14,19,10,11,15,17,16,18,20,21)Polycyclic presentation:
2^5,5|a^2=c,c^2=d,d^2=e,b^a=be,f^a=f^2,f^c=f^428 conjugacy classes — view the full character table.
| maximal subgroup of | #64860, #64861, #64889, … |
| maximal normal subgroup of | #64860, #64861, #64889, … |
| maximal quotient of | C4⋊#674, #64859, C22⋊#674, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(160, 73);