Finite Group DatabasebetaFinite Group Databasebeta| order | 5248 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C41 · Schur multiplier (H2) = C22×C4 |
| direct product | C41×#1973 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C41 | C2×D4 |
| commutator | C2×C4 | C22×C4×C41 |
| Frattini, cosocle | C22×C4 | C23×C41 |
| prime radical | C22×D4×C41 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C41 ◁ C2×C42×C41 ◁ G | |
340 conjugacy classes of subgroups; only the 128 normal subgroups are shown.
This group has 340 conjugacy classes of subgroups — too many to draw. Only its 128 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 | C2×C41×#1973, C41×#21228, C41×#21230, … |
| maximal normal subgroup of | C2×C41×#1973, C41×#21228, C41×#21230, … |
| maximal quotient of | C41×#9660, C41×#9666, C41×#9722, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(5248, 705);