Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
| direct product | Dic3×B(4,𝔽2) |
| Sylow | C3 ⋊ C4×B(4,𝔽2) |
1910 conjugacy classes of subgroups; only the 297 normal subgroups are shown.
This group has 1910 conjugacy classes of subgroups — too many to draw. Only its 297 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 96 |
96 conjugacy classes — view the full character table.
| maximal subgroup of | ΓL(1,𝔽16)×B(4,𝔽2), C4×D3×B(4,𝔽2), #72260356, … |
| maximal normal subgroup of | C4×D3×B(4,𝔽2), #67×B(4,𝔽2), #72271038, … |
| maximal quotient of | ΓL(1,𝔽16)×B(4,𝔽2), #72260356 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 175551);