Finite Group DatabasebetaFinite Group Databasebeta| order | 5248 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C42×C41 · Schur multiplier (H2) = C2×C4 |
| direct product | C41×#1391 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C41 | #99 |
| commutator | C23 | C42×C41 |
| Frattini, cosocle | C2×#41 | C22×C41 |
| socle | C2×C41 | #342 |
| prime radical | C2×#41×C41 | C22 |
| LCS | G ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C41 ◁ C2×C4×C41 ◁ C2×#41×C41 ◁ G | |
166 conjugacy classes of subgroups; only the 60 normal subgroups are shown.
This group has 166 conjugacy classes of subgroups — too many to draw. Only its 60 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
1066 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C41×#7398, C41×#7399, C41×#7406, … |
| maximal normal subgroup of | C41×#7398, C41×#7399, C41×#7406, … |
| maximal quotient of | C41×#7188, C41×#7204, C41×#7214, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(5248, 123);