Finite Group DatabasebetaFinite Group Databasebeta| order | 320 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| Fitting | C42×C5 ⋊ C4 |
| Sylow | C5 ⋊ #384 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C22×AGL(1,𝔽5) |
| commutator, socle | C22×C5 | C22×C4 |
| Fitting | C42×C5 | C4 |
| prime radical, cosocle | C22×D5 | C23 |
| hypercenter | C42 | AGL(1,𝔽5) |
| hypocenter | C5 | #384 |
| LCS | G ▷ C22×C5 ▷ C5 | |
| UCS | C1 ◁ C22 ◁ C42 | |
186 conjugacy classes of subgroups.
| minimal degree | μ(G) = 13 |
| transitive degree | μt(G) = 80 |
A minimal faithful permutation representation — 3 generators on 13 points:
(1,2,3,5,4)(10,11,12,13);(6,7,8,9);(2,3,4,5)(7,9)(11,13)Polycyclic presentation:
2^6,5|a^2=d,b^2=f,c^2=e,b^a=bf,c^a=ce,g^a=g^2,g^d=g^444 conjugacy classes — view the full character table.
| maximal subgroup of | #10598586, #10603845, #10603847, … |
| maximal normal subgroup of | #10598586, #10603845, #10603847, … |
| maximal quotient of | #10603784, #10603804, #10603847, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(320, 1026);