Finite Group DatabasebetaFinite Group Databasebeta| order | 648 |
| isoclinism | isoclinic class of Dic6 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C33 · Schur multiplier (H2) = C2×C33 |
| direct product | C33×#67 |
| Fitting | C22×C34 ⋊ C2 |
| Sylow | C34 ⋊ D4 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C33 | C2×D3 |
| commutator, cosocle | C2×C3 | C22×C33 |
| Frattini | C2 | C2×C33×D3 |
| Fitting | C22×C34 | C2 |
| socle | C2×C34 | C22 |
| hypercenter | C22×C33 | D3 |
| hypocenter | C3 | C33×D4 |
| LCS | G ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C2×C33 ◁ C22×C33 | |
838 conjugacy classes of subgroups.
| transitive degree | μt(G) = 108 |
243 conjugacy classes — view the full character table.
| maximal subgroup of | C32×#3804, C32×#3805, C32×D3×#67, … |
| maximal normal subgroup of | C32×#3804, C32×#3805, C32×D3×#67, … |
| maximal quotient of | C33×Dic3⋊C4, C33×(C2×D3)⋊C4, C33×#213, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(648, 683);