Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| isoclinism | isoclinic class of #68001 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C22 |
| direct product | C2×#68001 |
| derived | C22⋊SL(2,𝔽3) ⋊ C2×C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22 | #4625 |
| commutator, hypocenter | C22⋊SL(2,𝔽3) | C2×C4 |
| Frattini | C24 | C2×S4 |
| Fitting | C2×#393 | D3 |
| socle | C24 | SL(2,ℤ/4) |
| prime radical | C2×#393⋊C3 | C2 |
| cosocle | #393⋊C3 | C22 |
| LCS | G ▷ C22⋊SL(2,𝔽3) | |
| UCS | C1 ◁ C22 | |
293 conjugacy classes of subgroups; only the 34 normal subgroups are shown.
This group has 293 conjugacy classes of subgroups — too many to draw. Only its 34 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 48 |
42 conjugacy classes — view the full character table.
| maximal subgroup of | #72209919, #72268903, #72268914, … |
| maximal normal subgroup of | #72268903, #72268914 |
| maximal quotient of | #72268914, C2×#11710965 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1085741);