Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C25 |
| direct product | C2×#87507 |
| Fitting | C24×Q8 ⋊ D3 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | C2×#1020 |
| commutator, cosocle, hypocenter | C22⋊SL(2,𝔽3) | C23 |
| Frattini | C2 | C22×#1020 |
| Fitting | C24×Q8 | D3 |
| socle | C24 | C2×S4 |
| hypercenter | C23 | #1020 |
| LCS | G ▷ C22⋊SL(2,𝔽3) | |
| UCS | C1 ◁ C22 ◁ C23 | |
1567 conjugacy classes of subgroups; only the 47 normal subgroups are shown.
This group has 1567 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) = 96 |
48 conjugacy classes — view the full character table.
| maximal subgroup of | #72227785, #72227796, #72234234, … |
| maximal normal subgroup of | #72227796, #72234234 |
| maximal quotient of | C2×#72231199, #72260868, #72261345 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1090194);