Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C25 |
| direct product | C2×#73081 |
| Fitting | C22×#120 ⋊ D3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22 | #1020⋊C2 |
| commutator, cosocle | C2×C22⋊A4 | C23 |
| Frattini | C23 | C22×S4 |
| Fitting | C22×#120 | D3 |
| socle | C24 | C2×S4 |
| hypocenter | C22⋊A4 | C2×D4 |
| LCS | G ▷ C2×C22⋊A4 ▷ C22⋊A4 | |
| UCS | C1 ◁ C22 | |
1360 conjugacy classes of subgroups; only the 40 normal subgroups are shown.
This group has 1360 conjugacy classes of subgroups — too many to draw. Only its 40 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 24 |
44 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#11713200, #72215795, #72218466, … |
| maximal normal subgroup of | C2×#11713200, #72215795, #72220509, … |
| maximal quotient of | #72215795, #72220591, D3×#73081, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1087526);