Finite Group DatabasebetaFinite Group Databasebeta| order | 5248 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of AΣL(1,𝔽16) |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C41 · Schur multiplier (H2) = C23 |
| direct product | C2×C41×#352 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C41 | #99 |
| commutator | C23 | C22×C4×C41 |
| Frattini, cosocle | C2×D4 | C23×C41 |
| prime radical | C22×D4×C41 | C22 |
| LCS | G ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C41 ◁ C23×C41 ◁ C22×D4×C41 ◁ G | |
278 conjugacy classes of subgroups; only the 88 normal subgroups are shown.
This group has 278 conjugacy classes of subgroups — too many to draw. Only its 88 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 | C4×C41×#352, C41×#12621, C41×#12688, … |
| maximal normal subgroup of | C4×C41×#352, C41×#12621, C41×#12688, … |
| maximal quotient of | C2×C41×#1317, C2×C41×#1326, C41×#11472, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(5248, 851);