Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| isoclinism | isoclinic class of #1455⋊C3 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C4 · Schur multiplier (H2) = C8 |
| unambiguous semidirect product | #7846⋊C3 |
| Fitting, Sylow | #7846 ⋊ C3 |
| derived | C82 ⋊ C3×C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | #1455⋊C3 |
| commutator, hypocenter | C82 | C3×C4 |
| Frattini | C2×C42 | C2×A4 |
| Fitting | #7846 | C3 |
| socle | C23 | C2×C42⋊C3 |
| prime radical | C2×C82⋊C3 | C2 |
| cosocle | C2×C82 | C2×C3 |
| LCS | G ▷ C82 | |
| UCS | C1 ◁ C2 | |
117 conjugacy classes of subgroups.
| transitive degree | μt(G) = 24 |
Polycyclic presentation:
3,2^8|b^2=e,c^2=f,d^2=g,e^2=h,f^2=i,b^a=ceghi,c^a=bc,e^a=fh,f^a=ef,h^a=i,i^a=hi,d^b=dh,d^c=di48 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#7846⋊C3, #72211680, #72213074, … |
| maximal normal subgroup of | C2×#7846⋊C3, #72211680, #72213074 |
| maximal quotient of | #72210933, C2×#7846⋊C3 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1083491);