Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
| direct product | C2×#72999 |
| Fitting | C2×#393 ⋊ D3 |
| derived | C22⋊SL(2,𝔽3) ⋊ C23 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22 | #1020⋊C2 |
| commutator, cosocle, hypocenter | C22⋊SL(2,𝔽3) | C23 |
| Frattini | C23 | C22×S4 |
| Fitting | C2×#393 | D3 |
| socle | C24 | C2×S4 |
| LCS | G ▷ C22⋊SL(2,𝔽3) | |
| UCS | C1 ◁ C22 | |
827 conjugacy classes of subgroups; only the 42 normal subgroups are shown.
This group has 827 conjugacy classes of subgroups — too many to draw. Only its 42 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 | C2×#72209655 |
| maximal quotient of | C2×#11711401 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1086302);