Finite Group DatabasebetaFinite Group Databasebeta| order | 1216 |
| isoclinism | isoclinic class of D19 |
| automorphism | ? |
| homology | Abelianization (H1) = C64 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | C19⋊C64 |
| derived, Sylow | C19 ⋊ C64 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C32 | D19 |
| commutator, hypocenter | C19 | C64 |
| Fitting, cosocle | C19×C32 | C2 |
| socle, prime radical | C2×C19 | C32 |
| LCS | G ▷ C19 | |
| UCS | C1 ◁ C32 | |
14 conjugacy classes of subgroups.
| minimal degree | μ(G) = 83 |
| transitive degree | μt(G) = 1216 |
Polycyclic presentation:
2^6,19|a^2=b,b^2=c,c^2=d,d^2=e,e^2=f,g^a=g^18352 conjugacy classes — view the full character table.
| maximal subgroup of | D19×C64, C2×C19⋊C64, #14561845, … |
| maximal normal subgroup of | D19×C64, C2×C19⋊C64, #14561845, … |
| maximal quotient of | C19⋊C128, C2×C19⋊C64 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1216, 268);