Finite Group DatabasebetaFinite Group Databasebeta| order | 320 |
| isoclinism | isoclinic class of #64847 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C1 |
| general semilinear | ΓL(1,𝔽81) |
| Fitting | C5×C16 ⋊ C4 |
| Sylow | C5 ⋊ #365 |
42 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)(6,7,12,17,9,14,19,21,10,13,8,11,16,20,15,18);(2,3,4,5)(7,11,13,17)(8,15)(9,16)(12,19)(14,21,20,18)Polycyclic presentation:
2^6,5|a^2=c,b^2=df,d^2=e,e^2=f,b^a=bd,d^a=de,e^a=ef,g^a=g^2,c^b=cf,g^c=g^432 conjugacy classes — view the full character table.
| maximal subgroup of | C2×ΓL(1,𝔽81), #10604104, D5⋊#365, … |
| maximal normal subgroup of | C2×ΓL(1,𝔽81), #10604104, D5⋊#365, … |
| maximal quotient of | #10598642, #10598643, #10598663, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(320, 186);