Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C27 · Schur multiplier (H2) = C220 |
| direct product | D3×H(7,𝔽2) |
| Fitting | C3×H(7,𝔽2) ⋊ C2 |
| Sylow | C3 ⋊ C2×H(7,𝔽2) |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C26×D3 |
| commutator, socle, cosocle | C2×C3 | C27 |
| Fitting | C3×H(7,𝔽2) | C2 |
| hypercenter | H(7,𝔽2) | D3 |
| hypocenter | C3 | C2×H(7,𝔽2) |
| LCS | G ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C2 ◁ H(7,𝔽2) | |
60570 conjugacy classes of subgroups; only the 32041 normal subgroups are shown.
This group has 60570 conjugacy classes of subgroups — too many to draw. Only its 32041 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 48 |
195 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#87547, D3×#60388, D3×#60399, … |
| maximal normal subgroup of | D3×#87547, D3×#60388, D3×#60399, … |
| maximal quotient of | D3×#62892, D3×D4×H(5,𝔽2), D3×#63041, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1083463);