Finite Group DatabasebetaFinite Group Databasebeta| order | 1688 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | C22⋊D211 |
| Fitting | C22×C211 ⋊ C2 |
| Sylow | C211 ⋊ D4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C2×D211 |
| commutator, socle, cosocle | C2×C211 | C22 |
| Fitting | C22×C211 | C2 |
| hypercenter | C22 | D211 |
| hypocenter | C211 | D4 |
| LCS | G ▷ C2×C211 ▷ C211 | |
| UCS | C1 ◁ C2 ◁ C22 | |
16 conjugacy classes of subgroups.
| minimal degree | μ(G) = 215 |
| transitive degree | μt(G) = 844 |
Polycyclic presentation:
2^3,211|b^a=bc,d^b=d^210425 conjugacy classes — view the full character table.
| maximal subgroup of | D4×D211, C2×C22⋊D211, #16809961, … |
| maximal normal subgroup of | D4×D211, C2×C22⋊D211, #16809961, … |
| maximal quotient of | (C2×D211)⋊C4, C22⋊Dic211, Dic211⋊C4, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1688, 9);