Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C16 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Fitting | C22×C32 ⋊ D3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C16 | C2×S4 |
| commutator | C2×A4 | C2×C16 |
| Fitting | C22×C32 | D3 |
| socle | C23 | D3×C16 |
| prime radical | C2×S4 | C16 |
| cosocle | A4×C16 | C22 |
| hypercenter | C32 | S4 |
| hypocenter | A4 | M64 |
| LCS | G ▷ C2×A4 ▷ A4 | |
| UCS | C1 ◁ C16 ◁ C32 | |
118 conjugacy classes of subgroups; only the 27 normal subgroups are shown.
This group has 118 conjugacy classes of subgroups — too many to draw. Only its 27 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 96 |
Polycyclic presentation:
2^6,3,2^2|b^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 | S4×M64, #72342895, #72342897 |
| maximal normal subgroup of | S4×M64, #72342895, #72342897 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1086040);