Finite Group DatabasebetaFinite Group Databasebeta| order | 5248 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C42×C41 · Schur multiplier (H2) = C22 |
| direct product | C41×#1389 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C41 | #102 |
| commutator | C8 | C42×C41 |
| Frattini, cosocle | C22×C8 | C22×C41 |
| socle | C2×C41 | #340 |
| prime radical | C23×C41 | SD16 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C41 ◁ C22×C4×C41 ◁ C22×C8×C41 ◁ G | |
128 conjugacy classes of subgroups; only the 76 normal subgroups are shown.
This group has 128 conjugacy classes of subgroups — too many to draw. Only its 76 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×#1389, C41×#12646, C41×#12649, … |
| maximal normal subgroup of | C2×C41×#1389, C41×#12646, C41×#12649, … |
| maximal quotient of | C41×#7159, C41×#7175, C41×#7177, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(5248, 121);