Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C8 |
| direct product | D3×#1712 |
| Fitting | C3×#1712 ⋊ C2 |
| Sylow | C3 ⋊ C2×#1712 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | D3×#127 |
| commutator, cosocle | C3×C42 | C24 |
| Frattini | C42 | C23×D3 |
| Fitting | C3×#1712 | C2 |
| socle | C22×C3 | C2×#127 |
| hypercenter | #1712 | D3 |
| hypocenter | C3 | C2×#1712 |
| LCS | G ▷ C3×C42 ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ #1712 | |
1342 conjugacy classes of subgroups; only the 239 normal subgroups are shown.
This group has 1342 conjugacy classes of subgroups — too many to draw. Only its 239 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 192 |
114 conjugacy classes — view the full character table.
| maximal subgroup of | C3⋊D3×#1712, D3×#19745, D3×#12310, … |
| maximal normal subgroup of | D3×#19745, D3×#12310, D3×#12517, … |
| maximal quotient of | C3⋊D3×#1712, D3×#12517, C2×D3×#1712, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 157257);