Finite Group DatabasebetaFinite Group Databasebeta| order | 320 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Fitting | C5×#123 ⋊ C2 |
| Sylow | C5 ⋊ #529 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C23×D5 |
| commutator, socle, cosocle | C22×C5 | C24 |
| Fitting | C5×#123 | C2 |
| prime radical | #4085 | C2 |
| hypercenter | #123 | D5 |
| hypocenter | C5 | #529 |
| LCS | G ▷ C22×C5 ▷ C5 | |
| UCS | C1 ◁ C22 ◁ #123 | |
234 conjugacy classes of subgroups; only the 99 normal subgroups are shown.
This group has 234 conjugacy classes of subgroups — too many to draw. Only its 99 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 21 |
| transitive degree | μt(G) = 160 |
A minimal faithful permutation representation — 4 generators on 21 points:
(1,2,4,5,3)(8,12)(10,13)(15,19)(18,21);(6,8)(7,10)(9,12)(11,13)(14,16,17,20)(15,18,19,21);(14,15)(16,18)(17,19)(20,21);(2,3)(4,5)(6,7,9,11)(8,10,12,13)Polycyclic presentation:
2^6,5|a^2=f,c^2=e,g^a=g^4,d^b=de,d^c=df56 conjugacy classes — view the full character table.
| maximal subgroup of | #10617292, #10617293, #10617294, … |
| maximal normal subgroup of | #10617292, #10617293, #10617294, … |
| maximal quotient of | #10605782, #10605783, #10605796, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(320, 1321);