Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| Sylow | C3 ⋊ #34310 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C24×D3 |
| commutator | C22×C3 | C24×C4 |
| Fitting, prime radical | C2×C3×D42 | C2 |
| socle, cosocle | C23×C3 | C25 |
| hypercenter | C2×D42 | D3 |
| hypocenter | C3 | #34310 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ C2×D42 | |
3764 conjugacy classes of subgroups.
| transitive degree | μt(G) = 192 |
Polycyclic presentation:
2^8,3|a^2=h,c^2=g,d^2=f,e^2=h,b^a=bf,c^a=cg,e^a=ef,i^a=i^2,d^b=df,i^e=i^2138 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#34310, #72270700 |
| maximal normal subgroup of | D3×#34310, #72270700 |
| maximal quotient of | #72271183 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 331624);