Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| isoclinism | isoclinic class of #85647 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C24×C4 |
| direct product | C2×#85647 |
| Fitting, Sylow | C23×#127 ⋊ C3 |
| derived | C22×C42 ⋊ C22×C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | #85647 |
| commutator, cosocle, hypocenter | C22×C42 | C22×C3 |
| Frattini | C22 | C22×C22⋊A4 |
| Fitting | C23×#127 | C3 |
| socle | C25 | C2×A4 |
| LCS | G ▷ C22×C42 | |
| UCS | C1 ◁ C2 | |
4105 conjugacy classes of subgroups; only the 50 normal subgroups are shown.
This group has 4105 conjugacy classes of subgroups — too many to draw. Only its 50 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 24 |
48 conjugacy classes — view the full character table.
| maximal subgroup of | #72222672, C2×#72210013, C2×A4×#127⋊C3, … |
| maximal normal subgroup of | C2×A4×#127⋊C3, C2×#11710644, C2×#15258⋊C3, … |
| maximal quotient of | C2×#72210013, #72228270, #72228378, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1085001);