Finite Group DatabasebetaFinite Group Databasebeta| order | 5248 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1655 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C41 · Schur multiplier (H2) = C2×C4 |
| direct product | C41×#1659 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C41 | #120 |
| commutator, Frattini, cosocle | C42 | C23×C41 |
| prime radical | C41×H(3,𝔽4) | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C41 ◁ C42×C41 ◁ G | |
228 conjugacy classes of subgroups; only the 60 normal subgroups are shown.
This group has 228 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.
820 conjugacy classes — view the full character table.
| maximal subgroup of | C41×#13678, C2×C41×#1659, C41×#18984, … |
| maximal normal subgroup of | C41×#13678, C2×C41×#1659, C41×#18984, … |
| maximal quotient of | C41×#7750, C41×#7755, C41×#7758, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(5248, 391);