Finite Group DatabasebetaFinite Group Databasebeta| order | 1688 |
| isoclinism | isoclinic class of D211 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
| direct product | C22×D211 |
| unambiguous semidirect product | C211⋊C23 |
| unambiguous semidirect product | (C2×C211)⋊C22 |
| Fitting, socle | C22×C211 ⋊ C2 |
| derived, Sylow | C211 ⋊ C23 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22 | D211 |
| commutator, cosocle, hypocenter | C211 | C23 |
| Frattini | C1 | G |
| Fitting, socle | C22×C211 | C2 |
| LCS | G ▷ C211 | |
| UCS | C1 ◁ C22 | |
32 conjugacy classes of subgroups.
| transitive degree | μt(G) = 844 |
428 conjugacy classes. The character table is too large to display.
| maximal subgroup of | (C2×D211)⋊C4, C2×C4×D211, D4×D211, … |
| maximal normal subgroup of | (C2×D211)⋊C4, C2×C4×D211, D4×D211, … |
| maximal quotient of | C2×C4×D211, C22×Dic211, D4×D211, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1688, 12);