Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C25 |
| direct product | C2×#73210 |
| Fitting, Sylow | C2×#2840 ⋊ C3 |
| derived | C22×Q8 ⋊ C23×C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22 | #5450 |
| commutator, cosocle, hypocenter | C22×Q8 | C23×C3 |
| Frattini | C23 | C23×A4 |
| Fitting | C2×#2840 | C3 |
| socle | C24 | C22×A4 |
| LCS | G ▷ C22×Q8 | |
| UCS | C1 ◁ C22 | |
845 conjugacy classes of subgroups; only the 104 normal subgroups are shown.
This group has 845 conjugacy classes of subgroups — too many to draw. Only its 104 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 48 |
60 conjugacy classes — view the full character table.
| maximal subgroup of | #72268767, #72268788, C2×#72216883, … |
| maximal normal subgroup of | #72340873 |
| maximal quotient of | C2×#32873⋊C3 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1084415);