Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C8 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Fitting | C3×#2168 ⋊ C2 |
| Sylow | C3 ⋊ #33432 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2×C8 | C23×D3 |
| commutator, socle | C22×C3 | C23×C8 |
| Fitting | C3×#2168 | C2 |
| prime radical | #5547 | C4 |
| cosocle | C2×C3×C8 | C24 |
| hypercenter | #2168 | D3 |
| hypocenter | C3 | #33432 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C2×C8 ◁ #2168 | |
710 conjugacy classes of subgroups.
| transitive degree | μt(G) = 192 |
Polycyclic presentation:
2^8,3|a^2=f,c^2=f,f^2=g,g^2=h,b^a=be,d^a=dh,i^c=i^2,i^d=i^2168 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 322736);