Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
No aliases for this group.
| Fitting | C3×#3510 ⋊ C2 |
| Sylow | C3 ⋊ #62841 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C25×D3 |
| commutator, socle, cosocle | C22×C3 | C26 |
| Fitting | C3×#3510 | C2 |
| hypercenter | #3510 | D3 |
| hypocenter | C3 | #62841 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ #3510 | |
6624 conjugacy classes of subgroups.
| transitive degree | μt(G) = 192 |
Polycyclic presentation:
2^8,3|a^2=g,d^2=h,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=cg,e^b=eh,i^c=i^2,i^f=i^2120 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1078572);