Finite Group DatabasebetaFinite Group Databasebeta| order | 5248 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C41 · Schur multiplier (H2) = C24 |
| direct product | C41×#1891 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C41 | #120 |
| commutator | C23 | C22×C4×C41 |
| Frattini, cosocle | C22×C4 | C23×C41 |
| socle | C2×C41 | #386 |
| prime radical | C41×#525 | C2 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C41 ◁ C22×C4×C41 ◁ G | |
342 conjugacy classes of subgroups; only the 108 normal subgroups are shown.
This group has 342 conjugacy classes of subgroups — too many to draw. Only its 108 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 | C41×#12755, C41×#12756, C41×#12757, … |
| maximal normal subgroup of | C41×#12755, C41×#12756, C41×#12757, … |
| maximal quotient of | C41×#8965, C41×#8966, C41×#8967, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(5248, 623);