Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| isoclinism | isoclinic class of #226 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C8 · Schur multiplier (H2) = C4 |
| direct product | C8×GU(2,𝔽3) |
| derived | SL(2,𝔽3) ⋊ C4×C8 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C4×C8 | S4 |
| commutator, hypocenter | SL(2,𝔽3) | C4×C8 |
| Frattini | C42 | C2×S4 |
| Fitting | C8×#41 | D3 |
| socle | C22 | C4×SL(2,ℤ/4) |
| prime radical | C2×#41⋊C3 | C2×C4 |
| cosocle | C4×#41⋊C3 | C22 |
| LCS | G ▷ SL(2,𝔽3) | |
| UCS | C1 ◁ C4×C8 | |
271 conjugacy classes of subgroups; only the 55 normal subgroups are shown.
This group has 271 conjugacy classes of subgroups — too many to draw. Only its 55 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 192 |
128 conjugacy classes — view the full character table.
| maximal subgroup of | C8×GL(2,𝔽5), SD16×GU(2,𝔽3) |
| maximal normal subgroup of | SD16×GU(2,𝔽3) |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1085375);