Finite Group DatabasebetaFinite Group Databasebeta| order | 320 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #99 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C23 |
| direct product | C2×C5×#101 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C22⋊C4 |
| commutator | C22 | C22×C4×C5 |
| Frattini, cosocle | C2×C4 | C23×C5 |
| prime radical | C23×C5 | C2×C4 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C22×C4×C5 ◁ G | |
146 conjugacy classes of subgroups.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 160 |
A minimal faithful permutation representation — 3 generators on 23 points:
(1,2)(3,4,5,6,7);(8,9,11,15,12,14,20,22)(10,16,17,23,18,13,19,21);(8,10,12,18)(9,13,14,16)(11,19,20,17)(15,23,22,21)110 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(320, 913);