Finite Group DatabasebetaFinite Group Databasebeta| order | 320 |
| isoclinism | isoclinic class of #64845 |
| automorphism | Aut(G) = C4×D4×AGL(1,𝔽5) (order 640) · Inn(G) = C4×AGL(1,𝔽5) · Out(G) = C2×C4 |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C1 |
No aliases for this group.
| Fitting | C5×C16 ⋊ C4 |
| Sylow | C5 ⋊ #347 |
46 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)(6,7,8,13,15,17,21,20,10,12,14,19,9,11,16,18);(2,3,4,5)(7,11,12,17)(8,14)(13,20,19,18)(16,21)Polycyclic presentation:
2^6,5|a^2=c,b^2=d,d^2=ef,e^2=f,b^a=be,d^a=df,g^a=g^2,c^b=cf,g^c=g^438 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#64846, #10604101, #10604114, … |
| maximal normal subgroup of | C2×#64846, #10604101, #10604114, … |
| maximal quotient of | #10598638, #10598640, #10598660, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(320, 184);