Finite Group DatabasebetaFinite Group Databasebeta| order | 1248 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C22 |
| unambiguous semidirect product | (C22⋊D13)⋊A4 |
| Fitting | C24×C13 ⋊ C2×C3 |
| Hall | C22×C22⋊D13 ⋊ C3 |
| Sylow | C13 ⋊ D4×A4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C2×D13⋊A4 |
| commutator, socle, cosocle | C23×C13 | C22×C3 |
| Fitting | C24×C13 | C2×C3 |
| hypercenter | C22 | D13⋊A4 |
| hypocenter | C22×C13 | C3×D4 |
| LCS | G ▷ C23×C13 ▷ C22×C13 | |
| UCS | C1 ◁ C2 ◁ C22 | |
132 conjugacy classes of subgroups; only the 24 normal subgroups are shown.
This group has 132 conjugacy classes of subgroups — too many to draw. Only its 24 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 156 |
Polycyclic presentation:
2^2,3,2^3,13|b^a=bd,g^a=g^12,e^c=ef,f^c=e,g^c=g^352 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1248, 1278);