Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C8 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C42 | C2×S4 |
| commutator | C2×A4 | C4×C8 |
| Fitting | C22×C4×C8 | D3 |
| socle | C24 | C4×Dic3 |
| prime radical | C22×A4 | C42 |
| cosocle | C42×A4 | C22 |
| hypercenter | C4×C8 | S4 |
| hypocenter | A4 | #322 |
| LCS | G ▷ C2×A4 ▷ A4 | |
| UCS | C1 ◁ C42 ◁ C4×C8 | |
293 conjugacy classes of subgroups; only the 65 normal subgroups are shown.
This group has 293 conjugacy classes of subgroups — too many to draw. Only its 65 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=d,b^2=c,c^2=e,d^2=f,b^a=bf,g^a=g^2,h^a=hi,h^g=i,i^g=hi112 conjugacy classes — view the full character table.
| maximal subgroup of | #72342692, #72342696, S4×#322 |
| maximal normal subgroup of | #72342692, #72342696, S4×#322 |
| maximal quotient of | #72342751, #72342760 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1085366);