Finite Group DatabasebetaFinite Group Databasebeta| order | 4536 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C23 |
| direct product | C2×#72224677 |
| Fitting | C2×C7×H(3,𝔽3) ⋊ C22×C3 |
| derived | C7×H(3,𝔽3) ⋊ C23×C3 |
| Hall | ΓL(1,𝔽8)×H(3,𝔽3) ⋊ C23 |
| Sylow | C7 ⋊ C2×C3×((C3⋊D3)⋊C3)⋊C2 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | #72224677 |
| commutator, cosocle, hypocenter | C7×H(3,𝔽3) | C23×C3 |
| Frattini | C3 | C2×D3×C3⋊AGL(1,𝔽7) |
| Fitting | C2×C7×H(3,𝔽3) | C22×C3 |
| socle | C2×C3×C7 | C3×D32 |
| LCS | G ▷ C7×H(3,𝔽3) | |
| UCS | C1 ◁ C2 | |
610 conjugacy classes of subgroups; only the 76 normal subgroups are shown.
This group has 610 conjugacy classes of subgroups — too many to draw. Only its 76 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 126 |
86 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#72224678, C2×AGL(1,𝔽7)×((C3⋊D3)⋊C3)⋊C2 |
| maximal normal subgroup of | C2×AGL(1,𝔽7)×((C3⋊D3)⋊C3)⋊C2 |
Past the SmallGroups library coverage — no SmallGroups id.