Finite Group DatabasebetaFinite Group Databasebeta| order | 648 |
| isoclinism | isoclinic class of C3⋊(C3⋊(C3⋊D3)) |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C2×C36 |
| direct product | C4×C3⋊(C3⋊(C3⋊D3)) |
| Fitting | C34×C4 ⋊ C2 |
| derived, Sylow | C34 ⋊ C2×C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C4 | C3⋊(C3⋊(C3⋊D3)) |
| commutator, hypocenter | C34 | C2×C4 |
| Frattini | C2 | C2×C3⋊(C3⋊(C3⋊D3)) |
| Fitting | C34×C4 | C2 |
| socle, cosocle | C2×C34 | C22 |
| prime radical | C2×C3⋊(C3⋊(C3⋊D3)) | C2 |
| LCS | G ▷ C34 | |
| UCS | C1 ◁ C4 | |
1696 conjugacy classes of subgroups.
| transitive degree | μt(G) = 324 |
168 conjugacy classes — view the full character table.
| maximal subgroup of | #13049043, #13049046, #13049067, … |
| maximal normal subgroup of | #13049043, #13049046, #13049067, … |
| maximal quotient of | C8×C3⋊(C3⋊(C3⋊D3)), #13049148, C4×C3⋊(C3⋊(C3⋊Dic3)), … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(648, 695);