Finite Group DatabasebetaFinite Group Databasebeta| order | 2268 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C32 · Schur multiplier (H2) = C2×C32 |
| direct product | C3×D7⋊((C3⋊D3)⋊C3) |
| Fitting | C33×C7 ⋊ C22×C3 |
| derived | C32×C7 ⋊ C22×C32 |
| Hall | C3×C7⋊H(3,𝔽3) ⋊ C22 |
| Sylow | C7 ⋊ C2×C3×(C3⋊D3)⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3 | D7⋊((C3⋊D3)⋊C3) |
| commutator, cosocle, hypocenter | C32×C7 | C22×C32 |
| Frattini | C3 | C3×D3×AGL(1,𝔽7) |
| Fitting | C33×C7 | C22×C3 |
| socle | C32×C7 | C2×C3×D3 |
| LCS | G ▷ C32×C7 | |
| UCS | C1 ◁ C3 | |
272 conjugacy classes of subgroups; only the 54 normal subgroups are shown.
This group has 272 conjugacy classes of subgroups — too many to draw. Only its 54 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 126 |
78 conjugacy classes — view the full character table.
| maximal subgroup of | C3×AGL(1,𝔽7)×(C3⋊D3)⋊C3, #72269680, #72269700, … |
| maximal normal subgroup of | C3×AGL(1,𝔽7)×(C3⋊D3)⋊C3 |
Past the SmallGroups library coverage — no SmallGroups id.