Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C16 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C16 | C2×S4 |
| commutator | C2×A4 | C2×C16 |
| Fitting | C23×C16 | D3 |
| socle | C23 | C2×C3⋊C16 |
| prime radical | C22×A4 | C16 |
| cosocle | A4×C16 | C22 |
| hypercenter | C2×C16 | S4 |
| hypocenter | A4 | M64 |
| LCS | G ▷ C2×A4 ▷ A4 | |
| UCS | C1 ◁ C16 ◁ C2×C16 | |
133 conjugacy classes of subgroups; only the 39 normal subgroups are shown.
This group has 133 conjugacy classes of subgroups — too many to draw. Only its 39 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 96 |
Polycyclic presentation:
2^6,3,2^2|a^2=c,c^2=d,d^2=e,e^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 | #72342686, #72342752, #72342754, … |
| maximal normal subgroup of | #72342686, #72342752, #72342754, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1086036);