Finite Group DatabasebetaFinite Group Databasebeta| order | 320 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Fitting | C5×#135 ⋊ C2 |
| Sylow | C5 ⋊ #576 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | C2×D4×D5 |
| commutator, cosocle | C4×C5 | C24 |
| Frattini | C4 | C23×D5 |
| Fitting | C5×#135 | C2 |
| socle | C2×C5 | C22×D4 |
| hypercenter | #135 | D5 |
| hypocenter | C5 | #576 |
| LCS | G ▷ C4×C5 ▷ C2×C5 ▷ C5 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ #135 | |
268 conjugacy classes of subgroups; only the 99 normal subgroups are shown.
This group has 268 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) = 80 |
A minimal faithful permutation representation — 4 generators on 21 points:
(1,2,4,5,3)(7,11)(8,15)(9,18)(12,20)(14,21)(16,17);(6,8)(7,12)(9,15)(10,17)(11,14)(13,21)(16,18)(19,20);(6,7)(8,14)(9,13)(10,11)(12,17)(15,20)(16,21)(18,19);(2,3)(4,5)(7,11)(12,14)(13,19)(20,21)Polycyclic presentation:
2^6,5|e^2=f,b^a=bf,g^a=g^4,c^b=cf,d^b=df,d^c=de,e^c=ef,e^d=ef50 conjugacy classes — view the full character table.
| maximal subgroup of | #10600035, #10600037, #10602310, … |
| maximal normal subgroup of | #10600035, #10600037, #10602310, … |
| maximal quotient of | #10610551, #10610554, #10610558, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(320, 1441);