Finite Group DatabasebetaFinite Group Databasebeta| order | 320 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #453 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C5 · Schur multiplier (H2) = C23 |
| direct product | C5×#456 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C5 | #120 |
| commutator, Frattini, cosocle | C2×C4 | C23×C5 |
| prime radical | C5×2−1+4 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C5 ◁ C2×C4×C5 ◁ G | |
142 conjugacy classes of subgroups.
| minimal degree | μ(G) = 21 |
| transitive degree | μt(G) = 80 |
A minimal faithful permutation representation — 3 generators on 21 points:
(1,2,6,4)(3,9,12,16)(5,13,15,8)(7,14,10,11)(17,18,19,20,21);(1,3)(2,7)(4,10)(5,14)(6,12)(8,16)(9,13)(11,15);(1,4,6,2)(3,10,12,7)(5,8,15,13)(9,14,16,11)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(320, 957);