Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C23×C42 |
No aliases for this group.
| Sylow | C3 ⋊ #13812 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C24 | C23×D3 |
| commutator | C22×C3 | C22×C42 |
| Fitting | C3×C4×#95 | C2 |
| socle, prime radical, cosocle | C24×C3 | C24 |
| hypercenter | C4×#95 | D3 |
| hypocenter | C3 | #13812 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C24 ◁ C4×#95 | |
1546 conjugacy classes of subgroups.
| transitive degree | μt(G) = 768 |
Polycyclic presentation:
2^8,3|a^2=g,b^2=h,c^2=ef,d^2=fg,b^a=be,c^a=cf,d^b=de,i^c=i^2180 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 91056);