Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| isoclinism | isoclinic class of S4 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C211 |
| direct product | C24×SL(2,ℤ/4) |
| derived | A4 ⋊ C24×C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C25 | S4 |
| commutator, hypocenter | A4 | C24×C4 |
| Frattini | C2 | C24×S4 |
| Fitting, socle | C27 | D3 |
| prime radical | C25×A4 | C2 |
| cosocle | C2×A4 | C25 |
| LCS | G ▷ A4 | |
| UCS | C1 ◁ C25 | |
13713 conjugacy classes of subgroups; only the 1429 normal subgroups are shown.
This group has 13713 conjugacy classes of subgroups — too many to draw. Only its 1429 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 192 |
160 conjugacy classes — view the full character table.
| maximal subgroup of | #72259081, #120×SL(2,ℤ/4), C24×GL(2,ℤ/4) |
| maximal normal subgroup of | #72259081, #120×SL(2,ℤ/4), C24×GL(2,ℤ/4) |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1090140);