Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C23 |
| direct product | C22×#5465 |
| Fitting, Sylow | C22×H(3,𝔽4) ⋊ C3 |
| derived | H(3,𝔽4) ⋊ C22×C3 |
| subgroup | G/H | |
|---|---|---|
| center, socle, hypercenter | C24 | C22⋊A4 |
| commutator, cosocle, hypocenter | H(3,𝔽4) | C22×C3 |
| Frattini | C22 | C22×C22⋊A4 |
| Fitting | C22×H(3,𝔽4) | C3 |
| LCS | G ▷ H(3,𝔽4) | |
| UCS | C1 ◁ C24 | |
2131 conjugacy classes of subgroups; only the 102 normal subgroups are shown.
This group has 2131 conjugacy classes of subgroups — too many to draw. Only its 102 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 64 |
68 conjugacy classes — view the full character table.
| maximal subgroup of | #72227612, #72227615, #72268238, … |
| maximal normal subgroup of | #72227615, #72268241, #72268459, … |
| maximal quotient of | C2×D3×#5465 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1085026);