Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C22×S4 |
| commutator | C22×A4 | C22×C4 |
| Fitting | C24×Q8 | D3 |
| socle | C25 | C22×D3 |
| prime radical, cosocle | C23×A4 | C23 |
| hypercenter | C22×Q8 | S4 |
| hypocenter | A4 | #391 |
| LCS | G ▷ C22×A4 ▷ A4 | |
| UCS | C1 ◁ C23 ◁ C22×Q8 | |
1277 conjugacy classes of subgroups; only the 129 normal subgroups are shown.
This group has 1277 conjugacy classes of subgroups — too many to draw. Only its 129 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 192 |
Polycyclic presentation:
2^6,3,2^2|a^2=f,b^2=e,c^2=e,c^a=cd,g^a=g^2,h^a=hi,c^b=ce,h^g=i,i^g=hi76 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1086091);