Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| isoclinism | isoclinic class of #857 |
| automorphism | ? |
| homology | Abelianization (H1) = C16 · Schur multiplier (H2) = C2 |
No aliases for this group.
| derived | C42⋊C3 ⋊ C16 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C8 | #857 |
| commutator, hypocenter | C42⋊C3 | C16 |
| Frattini | C22×C8 | S4 |
| Fitting | C42×C8 | D3 |
| socle | C23 | A4⋊C8 |
| prime radical | C2×C42⋊C3 | C8 |
| cosocle | C8×C42⋊C3 | C2 |
| LCS | G ▷ C42⋊C3 | |
| UCS | C1 ◁ C8 | |
102 conjugacy classes of subgroups.
| transitive degree | μt(G) = 48 |
Polycyclic presentation:
2^4,3,2^4|a^2=b,b^2=c,c^2=d,f^2=h,g^2=i,e^a=e^2,f^a=fg,g^a=gi,h^a=hi,f^e=ghi,g^e=fgh,h^e=i,i^e=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1085333);