Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| isoclinism | isoclinic class of #127⋊C3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C26×C4 |
| direct product | C23×#127⋊C3 |
| Fitting, Sylow | C23×#127 ⋊ C3 |
| derived | C42 ⋊ C24×C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C23 | #127⋊C3 |
| commutator, cosocle, hypocenter | C42 | C24×C3 |
| Frattini | C22 | C24×A4 |
| Fitting | C23×#127 | C3 |
| socle | C25 | C2×A4 |
| LCS | G ▷ C42 | |
| UCS | C1 ◁ C23 | |
4340 conjugacy classes of subgroups; only the 217 normal subgroups are shown.
This group has 4340 conjugacy classes of subgroups — too many to draw. Only its 217 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 48 |
80 conjugacy classes — view the full character table.
| maximal subgroup of | #72225966, C22×C4⋊(#127⋊C3), C2×A4×#127⋊C3, … |
| maximal normal subgroup of | #72225966, C22×C4⋊(#127⋊C3), C2×A4×#127⋊C3, … |
| maximal quotient of | #72225958, C22×C4⋊(#127⋊C3), C2×D4×#127⋊C3, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1085000);