Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2 | #73133 |
| commutator, cosocle | A4×C16 | C22 |
| Frattini | C16 | C2×S4 |
| Fitting | C22×Q32 | D3 |
| socle | C23 | #826 |
| prime radical | C2×A4 | D16 |
| hypercenter | Q32 | S4 |
| hypocenter | A4 | Q64 |
| LCS | G ▷ A4×C16 ▷ C8×A4 ▷ C4×A4 ▷ C2×A4 ▷ A4 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ C8 ◁ C16 ◁ Q32 | |
120 conjugacy classes of subgroups; only the 21 normal subgroups are shown.
This group has 120 conjugacy classes of subgroups — too many to draw. Only its 21 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=f,c^2=de,d^2=ef,e^2=f,b^a=bc,c^a=cd,d^a=de,e^a=ef,g^a=g^2,h^a=hi,c^b=cd,d^b=de,e^b=ef,h^g=i,i^g=hi49 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1086046);