Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
| direct product | D4×#857 |
| Fitting | C42×D4 ⋊ D3 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | C22×#857 |
| commutator, cosocle | C2×C42⋊C3 | C23 |
| Frattini, socle | C23 | C22×S4 |
| Fitting | C42×D4 | D3 |
| hypercenter | D4 | #857 |
| hypocenter | C42⋊C3 | C2×D4 |
| LCS | G ▷ C2×C42⋊C3 ▷ C42⋊C3 | |
| UCS | C1 ◁ C2 ◁ D4 | |
739 conjugacy classes of subgroups; only the 37 normal subgroups are shown.
This group has 739 conjugacy classes of subgroups — too many to draw. Only its 37 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 24 |
50 conjugacy classes — view the full character table.
| maximal subgroup of | #41×#857, S4×#857, #72226084, … |
| maximal normal subgroup of | #41×#857, #72226084, D4×#857⋊C2, … |
| maximal quotient of | C22⋊C4×#857, #72226107 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1087298);