Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| isoclinism | isoclinic class of #226 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C22×C4 |
| direct product | C42×GL(2,𝔽3) |
| Fitting | C42×Q8 ⋊ D3 |
| derived | SL(2,𝔽3) ⋊ C2×C42 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C42 | S4 |
| commutator, hypocenter | SL(2,𝔽3) | C2×C42 |
| Frattini, socle | C23 | C22×S4 |
| Fitting | C42×Q8 | D3 |
| prime radical | C22×GL(2,𝔽3) | C22 |
| cosocle | C22×SL(2,𝔽3) | C23 |
| LCS | G ▷ SL(2,𝔽3) | |
| UCS | C1 ◁ C2×C42 | |
827 conjugacy classes of subgroups; only the 123 normal subgroups are shown.
This group has 827 conjugacy classes of subgroups — too many to draw. Only its 123 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 128 |
128 conjugacy classes — view the full character table.
| maximal subgroup of | C42×AGL(2,𝔽3), #127×GL(2,𝔽3), #97×GL(2,𝔽3) |
| maximal normal subgroup of | #127×GL(2,𝔽3), #97×GL(2,𝔽3) |
| maximal quotient of | C42×AGL(2,𝔽3), #97×GL(2,𝔽3) |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1086955);