Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C25 |
| direct product | C22×#5930 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C23 | #1020 |
| commutator, cosocle, hypocenter | C22⋊SL(2,𝔽3) | C23 |
| Frattini | C2 | C22×#1020 |
| Fitting | C24×Q8 | D3 |
| socle | C25 | S4 |
| prime radical | C22×C22⋊SL(2,𝔽3) | C2 |
| LCS | G ▷ C22⋊SL(2,𝔽3) | |
| UCS | C1 ◁ C23 | |
1573 conjugacy classes of subgroups; only the 63 normal subgroups are shown.
This group has 1573 conjugacy classes of subgroups — too many to draw. Only its 63 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 192 |
60 conjugacy classes — view the full character table.
| maximal subgroup of | #72227971, #72227985 |
| maximal normal subgroup of | #72227985 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1090192);