Finite Group DatabasebetaFinite Group Databasebeta| order | 320 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| Fitting | C5×M16 ⋊ C4 |
| Sylow | C5 ⋊ #405 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C4 | C22×AGL(1,𝔽5) |
| commutator, socle | C2×C5 | C2×C42 |
| Fitting | C5×M16 | C4 |
| prime radical | C22×D5 | C2×C4 |
| cosocle | C4×D5 | C23 |
| hypercenter | M16 | AGL(1,𝔽5) |
| hypocenter | C5 | #405 |
| LCS | G ▷ C2×C5 ▷ C5 | |
| UCS | C1 ◁ C4 ◁ M16 | |
130 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)(6,8,10,18,11,16,9,17)(7,12,14,21,15,19,13,20);(8,16)(12,19)(17,18)(20,21);(2,3,4,5)(6,7,9,13,11,15,10,14)(8,12,17,20,16,19,18,21)Polycyclic presentation:
2^6,5|a^2=d,c^2=e,d^2=f,e^2=f,g^a=g^2,c^b=cf,g^d=g^450 conjugacy classes — view the full character table.
| maximal subgroup of | #10603926, #10603927, #10603932, … |
| maximal normal subgroup of | #10603926, #10603927, #10603932, … |
| maximal quotient of | #10603861, #10603862, #10603863, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(320, 1066);