Finite Group DatabasebetaFinite Group Databasebeta| order | 1248 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C22×C4 |
| unambiguous semidirect product | #88194⋊C3 |
| Fitting | C2×D4×C13 ⋊ C2×C3 |
| Hall | #88194 ⋊ C3 |
| Hall | C13⋊C3 ⋊ #127 |
| Sylow | C13 ⋊ C3×#127 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C22×D13⋊C3 |
| commutator, socle, cosocle | C22×C13 | C23×C3 |
| Fitting | C2×D4×C13 | C2×C3 |
| hypercenter | C2×D4 | D13⋊C3 |
| hypocenter | C13 | C3×#127 |
| LCS | G ▷ C22×C13 ▷ C13 | |
| UCS | C1 ◁ C22 ◁ C2×D4 | |
216 conjugacy classes of subgroups.
| transitive degree | μt(G) = 208 |
Polycyclic presentation:
2^3,3,2^2,13|a^2=f,b^a=bf,c^a=cf,g^a=g^12,c^b=ce,g^d=g^962 conjugacy classes — view the full character table.
| maximal subgroup of | #127×D13⋊C3, #72229149, #72259178, … |
| maximal normal subgroup of | #127×D13⋊C3, #72229149, #72259178, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1248, 410);