Finite Group DatabasebetaFinite Group Databasebeta| order | 320 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| Fitting | C5×#127 ⋊ C2 |
| Sylow | C5 ⋊ D42 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C23×D5 |
| commutator, socle, cosocle | C22×C5 | C24 |
| Fitting | C5×#127 | C2 |
| hypercenter | #127 | D5 |
| hypocenter | C5 | D42 |
| LCS | G ▷ C22×C5 ▷ C5 | |
| UCS | C1 ◁ C22 ◁ #127 | |
428 conjugacy classes of subgroups.
| minimal degree | μ(G) = 13 |
| transitive degree | μt(G) = 80 |
A minimal faithful permutation representation — 4 generators on 13 points:
(1,2,4,5,3)(6,7,8,9)(10,12,13,11);(7,9)(10,11)(12,13);(7,9)(11,12);(2,3)(4,5)(7,9)Polycyclic presentation:
2^6,5|d^2=f,d^a=def,g^a=g^4,c^b=ce,d^b=df,d^c=df53 conjugacy classes — view the full character table.
| maximal subgroup of | #10599183, #10599184, #10599185, … |
| maximal normal subgroup of | #10599183, #10599184, #10599185, … |
| maximal quotient of | #10606057, #10606066, #10606091, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(320, 1389);