Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | C3×SD128 ⋊ C2 |
| Sylow | C3 ⋊ #13741 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | D3×D32 |
| commutator, cosocle | C3×C32 | C23 |
| Frattini | C32 | C22×D3 |
| Fitting | C3×SD128 | C2 |
| socle | C2×C3 | C2×D32 |
| hypercenter | SD128 | D3 |
| hypocenter | C3 | #13741 |
| LCS | G ▷ C3×C32 ▷ C3×C16 ▷ C3×C8 ▷ C3×C4 ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ C8 ◁ C16 ◁ C32 ◁ SD128 | |
128 conjugacy classes of subgroups; only the 39 normal subgroups are shown.
This group has 128 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.
| minimal degree | μ(G) = 131 |
| transitive degree | μt(G) = 384 |
Polycyclic presentation:
2^8,3|c^2=h,d^2=ef,e^2=fg,f^2=gh,g^2=h,b^a=bd,d^a=de,e^a=ef,f^a=fg,g^a=gh,d^b=de,e^b=ef,f^b=fg,g^b=gh,i^b=i^2,i^c=i^2105 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 90621);