Finite Group DatabasebetaFinite Group Databasebeta| order | 320 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Fitting | C42×C5 ⋊ C4 |
| Sylow | C5 ⋊ #383 |
| 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 | #383 |
| LCS | G ▷ C22×C5 ▷ C5 | |
| UCS | C1 ◁ C22 ◁ C42 | |
138 conjugacy classes of subgroups.
| minimal degree | μ(G) = 21 |
| transitive degree | μt(G) = 80 |
A minimal faithful permutation representation — 3 generators on 21 points:
(1,2,3,5,4)(7,11)(10,13)(14,16,17,21)(15,19,18,20);(6,8,9,12)(7,10,11,13)(15,18)(19,20);(2,3,4,5)(6,7)(8,10)(9,11)(12,13)(14,15)(16,20)(17,18)(19,21)Polycyclic presentation:
2^6,5|a^2=d,b^2=e,c^2=f,b^a=bf,c^a=cef,g^a=g^2,g^d=g^444 conjugacy classes — view the full character table.
| maximal subgroup of | #10598528, #10598546, #10604393, … |
| maximal normal subgroup of | #10598528, #10598546, #10604393, … |
| maximal quotient of | #10603783, #10603803, #10603853, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(320, 1028);