Finite Group DatabasebetaFinite Group Databasebeta| order | 5248 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C41 · Schur multiplier (H2) = C23×C4 |
| direct product | C4×#102×C41 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4×C41 | D4 |
| commutator | C4 | C2×C42×C41 |
| Frattini, cosocle | C22×C4 | C23×C41 |
| socle | C23×C41 | C2×D4 |
| prime radical | C22×D4×C41 | C22 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C41 ◁ C2×C42×C41 ◁ G | |
396 conjugacy classes of subgroups; only the 184 normal subgroups are shown.
This group has 396 conjugacy classes of subgroups — too many to draw. Only its 184 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
2296 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C8×#102×C41, C41×#8200, C41×#8202, … |
| maximal normal subgroup of | C8×#102×C41, C41×#8200, C41×#8202, … |
| maximal quotient of | C4×C41×D4⋊C8, C41×#8174, C41×#8177, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(5248, 492);