Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Fitting | C3×#2204 ⋊ C2 |
| Sylow | C3 ⋊ #13681 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | #70996 |
| commutator, cosocle | C3×C4⋊Q8 | C23 |
| Frattini | C4⋊Q8 | C22×D3 |
| Fitting | C3×#2204 | C2 |
| socle | C2×C3 | #2196 |
| hypercenter | #2204 | D3 |
| hypocenter | C3 | #13681 |
| LCS | G ▷ C3×C4⋊Q8 ▷ C2×C3×C4 ▷ C22×C3 ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C2 ◁ C22 ◁ C2×C4 ◁ C4⋊Q8 ◁ #2204 | |
398 conjugacy classes of subgroups; only the 43 normal subgroups are shown.
This group has 398 conjugacy classes of subgroups — too many to draw. Only its 43 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 48 |
Polycyclic presentation:
2^8,3|d^2=h,e^2=h,f^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fg,d^b=dh,e^b=efgh,g^b=gh,d^c=df,e^c=eh,f^c=fh,g^c=gh,i^c=i^2,e^d=egh,f^d=fh,f^e=fh45 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 90325);