Finite Group DatabasebetaFinite Group Databasebeta| order | 5248 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C42×C41 · Schur multiplier (H2) = C23 |
| direct product | C41×#1390 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C41 | #99 |
| commutator | C23 | C42×C41 |
| Frattini, cosocle | C22×D4 | C22×C41 |
| prime radical | C22×D4×C41 | C22 |
| LCS | G ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C41 ◁ C23×C41 ◁ C22×D4×C41 ◁ G | |
234 conjugacy classes of subgroups; only the 64 normal subgroups are shown.
This group has 234 conjugacy classes of subgroups — too many to draw. Only its 64 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 | C2×C41×#1390, C41×#12573, C41×#12574, … |
| maximal normal subgroup of | C2×C41×#1390, C41×#12573, C41×#12574, … |
| maximal quotient of | C41×#7187, C41×#7203, C41×#7204, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(5248, 122);