Finite Group DatabasebetaFinite Group Databasebeta| order | 5248 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1468 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C41 · Schur multiplier (H2) = C22×C4 |
| direct product | C41×#1996 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C41 | C2×C22⋊C4 |
| commutator | C23 | C22×C4×C41 |
| Frattini, cosocle | C24 | C23×C41 |
| prime radical | C24×C41 | C23 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C41 ◁ C23×C4×C41 ◁ G | |
324 conjugacy classes of subgroups; only the 116 normal subgroups are shown.
This group has 324 conjugacy classes of subgroups — too many to draw. Only its 116 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
1312 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C41×#8467, C41×#8468, C41×#8480, … |
| maximal normal subgroup of | C41×#8467, C41×#8468, C41×#8480, … |
| maximal quotient of | C41×#9878, C41×#9879, C41×#9880, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(5248, 728);