Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C23 |
| direct product | C2×#85554 |
| Fitting | C2×Q82 ⋊ D3 |
| derived | Q8⋊SL(2,𝔽3) ⋊ C22 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #5934 |
| commutator, cosocle, hypocenter | Q8⋊SL(2,𝔽3) | C22 |
| Frattini | C22 | C2×#1020 |
| Fitting | C2×Q82 | D3 |
| hypercenter | C23 | #1020 |
| LCS | G ▷ Q8⋊SL(2,𝔽3) | |
| UCS | C1 ◁ C22 ◁ C23 | |
482 conjugacy classes of subgroups; only the 21 normal subgroups are shown.
This group has 482 conjugacy classes of subgroups — too many to draw. Only its 21 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 32 |
34 conjugacy classes — view the full character table.
| maximal subgroup of | #72228024, #72228026, C2×#72209898, … |
| maximal normal subgroup of | #72228026, #72261377, C22×#85554 |
| maximal quotient of | C2×#72209705, C22×#85554, C2×#11714277, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1090137);