Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| Fitting | C3×#3291 ⋊ C2 |
| Sylow | C3 ⋊ #61110 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | D3×D42 |
| commutator, cosocle | C2×C3×C4 | C25 |
| Frattini | C2×C4 | C24×D3 |
| Fitting | C3×#3291 | C2 |
| socle | C2×C3 | C2×D42 |
| hypercenter | #3291 | D3 |
| hypocenter | C3 | #61110 |
| LCS | G ▷ C2×C3×C4 ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ #3291 | |
2684 conjugacy classes of subgroups.
| transitive degree | μt(G) = 48 |
Polycyclic presentation:
2^8,3|a^2=h,f^2=h,g^2=h,b^a=bf,c^a=cg,e^a=eh,f^a=fh,g^a=gh,d^b=df,e^b=eh,f^b=fh,g^c=gh,i^c=i^2,f^d=fh,i^e=i^281 conjugacy classes — view the full character table.
| maximal subgroup of | #72214248, #72214940, D3×#61110, … |
| maximal normal subgroup of | D3×#61110, #72215210, #72215228 |
| maximal quotient of | #72214248, #72214940, #72226092, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1045872);