Finite Group DatabasebetaFinite Group Databasebeta| order | 6048 |
| isoclinism | isoclinic class of AGL(2,𝔽3) |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C7 · Schur multiplier (H2) = C2 |
| direct product | C2×C7×AGL(2,𝔽3) |
| Fitting, socle | C2×C32×C7 ⋊ GL(2,𝔽3) |
| derived | PSU(3,𝔽2)⋊C3 ⋊ C22×C7 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C7 | AGL(2,𝔽3) |
| commutator, cosocle, hypocenter | PSU(3,𝔽2)⋊C3 | C22×C7 |
| Frattini | C1 | G |
| Fitting, socle | C2×C32×C7 | GL(2,𝔽3) |
| LCS | G ▷ PSU(3,𝔽2)⋊C3 | |
| UCS | C1 ◁ C2×C7 | |
288 conjugacy classes of subgroups; only the 28 normal subgroups are shown.
This group has 288 conjugacy classes of subgroups — too many to draw. Only its 28 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 126 |
154 conjugacy classes — view the full character table.
| maximal subgroup of | C2×ΓL(1,𝔽8)×AGL(2,𝔽3), C2×D7×AGL(2,𝔽3) |
| maximal normal subgroup of | C2×ΓL(1,𝔽8)×AGL(2,𝔽3), C2×D7×AGL(2,𝔽3) |
Past the SmallGroups library coverage — no SmallGroups id.