Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C25 |
| direct product | C2×#87510 |
| Fitting | C23×#41 ⋊ D3 |
| derived | C22⋊SL(2,𝔽3) ⋊ C23 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | C2×#1020 |
| commutator, cosocle, hypocenter | C22⋊SL(2,𝔽3) | C23 |
| Frattini | C2 | C22×#1020 |
| Fitting | C23×#41 | D3 |
| socle | C24 | C2×S4 |
| hypercenter | C2×C4 | #1020 |
| LCS | G ▷ C22⋊SL(2,𝔽3) | |
| UCS | C1 ◁ C22 ◁ C2×C4 | |
1745 conjugacy classes of subgroups; only the 47 normal subgroups are shown.
This group has 1745 conjugacy classes of subgroups — too many to draw. Only its 47 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 48 |
48 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#11715702, #72227579, C2×#72257993, … |
| maximal normal subgroup of | C2×#11715702, #72227579 |
| maximal quotient of | #72227579, C2×#72211543, C2×#11715653, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1090195);