Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26×C4 |
| direct product | C2×#85496 |
| Fitting | C24×D4 ⋊ D3 |
| subgroup | G/H | |
|---|---|---|
| center | C23 | C22×S4 |
| commutator, cosocle | C22×A4 | C24 |
| Frattini | C22 | C23×S4 |
| Fitting | C24×D4 | D3 |
| socle | C25 | C22×D3 |
| hypercenter | C22×D4 | S4 |
| hypocenter | A4 | C2×#127 |
| LCS | G ▷ C22×A4 ▷ A4 | |
| UCS | C1 ◁ C23 ◁ C22×D4 | |
3529 conjugacy classes of subgroups; only the 181 normal subgroups are shown.
This group has 3529 conjugacy classes of subgroups — too many to draw. Only its 181 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 96 |
76 conjugacy classes — view the full character table.
| maximal subgroup of | C2×D4×GL(2,ℤ/4), #72228332, C2×#11714540, … |
| maximal normal subgroup of | C2×D4×GL(2,ℤ/4), #72228332, C2×#11714540, … |
| maximal quotient of | #72228332, #72228394, #72258634, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1088722);