Finite Group DatabasebetaFinite Group Databasebeta| order | 1600 |
| isoclinism | isoclinic class of D5 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42×C5 · Schur multiplier (H2) = C25×C4 |
| direct product | C22×C4×C5×Dic5 |
| derived | C5 ⋊ C22×C42×C5 |
| Sylow | C52 ⋊ C22×C42 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C23×C4×C5 | D5 |
| commutator, hypocenter | C5 | C22×C42×C5 |
| Frattini | C22 | C23×C5×D5 |
| Fitting | C23×C4×C52 | C2 |
| socle, prime radical | C24×C52 | C22 |
| cosocle | C22×C5 | C24×C5 |
| LCS | G ▷ C5 | |
| UCS | C1 ◁ C23×C4×C5 | |
1232 conjugacy classes of subgroups; only the 734 normal subgroups are shown.
This group has 1232 conjugacy classes of subgroups — too many to draw. Only its 734 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 320 |
640 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C22×C4×#6036 |
| maximal normal subgroup of | C22×C4×#6036 |
| maximal quotient of | C22×C5×#3935, C5×#10614242 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1600, 9288);