Finite Group DatabasebetaFinite Group Databasebeta| order | 1248 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C24 |
| unambiguous semidirect product | #88136⋊C3 |
| Fitting | C13×C22⋊C4 ⋊ C2×C3 |
| Hall | #88136 ⋊ C3 |
| Hall | C13⋊C3 ⋊ #120 |
| Sylow | C13 ⋊ C3×#120 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C22×D13⋊C3 |
| commutator, socle, cosocle | C22×C13 | C23×C3 |
| Fitting | C13×C22⋊C4 | C2×C3 |
| hypercenter | C22⋊C4 | D13⋊C3 |
| hypocenter | C13 | C3×#120 |
| LCS | G ▷ C22×C13 ▷ C13 | |
| UCS | C1 ◁ C22 ◁ C22⋊C4 | |
260 conjugacy classes of subgroups; only the 63 normal subgroups are shown.
This group has 260 conjugacy classes of subgroups — too many to draw. Only its 63 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 104 |
Polycyclic presentation:
2^3,3,2^2,13|c^2=f,c^a=cf,g^a=g^12,c^b=ce,g^d=g^962 conjugacy classes — view the full character table.
| maximal subgroup of | #120×D13⋊C3, #72242099, #72259180 |
| maximal normal subgroup of | #120×D13⋊C3, #72259180 |
| maximal quotient of | #72259180, #72259181, #72259186 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1248, 360);