Finite Group DatabasebetaFinite Group Databasebeta| order | 320 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #471 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C5 · Schur multiplier (H2) = C22 |
| direct product | C5×#472 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C5 | #121 |
| commutator, Frattini, cosocle | C2×C4 | C23×C5 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C5 ◁ C2×C4×C5 ◁ G | |
108 conjugacy classes of subgroups.
| minimal degree | μ(G) = 21 |
| transitive degree | μt(G) = 80 |
A minimal faithful permutation representation — 3 generators on 21 points:
(2,6)(3,10)(4,7)(9,14)(11,12)(13,15)(17,18,19,20,21);(1,2,7,14,5,9,4,6)(3,8,10,15,12,16,11,13);(1,3)(2,8)(4,11)(5,12)(6,13)(7,10)(9,16)(14,15)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(320, 973);