Finite Group DatabasebetaFinite Group Databasebeta| order | 2268 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C32 · Schur multiplier (H2) = C2×C32 |
| direct product | C2×ΓL(1,𝔽8)⋊((C3⋊D3)⋊C3) |
| Fitting | C2×C7×H(3,𝔽3) ⋊ C2×C3 |
| derived | C32×C7 ⋊ C22×C32 |
| Hall | ΓL(1,𝔽8)×H(3,𝔽3) ⋊ C22 |
| Sylow | C7 ⋊ C2×C3×(C3⋊D3)⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | ΓL(1,𝔽8)⋊((C3⋊D3)⋊C3) |
| commutator, cosocle, hypocenter | C32×C7 | C22×C32 |
| Frattini | C3 | C2×C3×C3⋊AGL(1,𝔽7) |
| Fitting | C2×C7×H(3,𝔽3) | C2×C3 |
| socle | C2×C3×C7 | C32×D3 |
| LCS | G ▷ C32×C7 | |
| UCS | C1 ◁ C2 | |
272 conjugacy classes of subgroups; only the 56 normal subgroups are shown.
This group has 272 conjugacy classes of subgroups — too many to draw. Only its 56 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 126 |
82 conjugacy classes — view the full character table.
| maximal subgroup of | #72230279, C2×AGL(1,𝔽7)×(C3⋊D3)⋊C3, C2×#72224671, … |
| maximal normal subgroup of | C2×AGL(1,𝔽7)×(C3⋊D3)⋊C3, C2×#72224677, C2×#72224685 |
| maximal quotient of | #72230279, C2×#72224671 |
Past the SmallGroups library coverage — no SmallGroups id.