Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| Fitting | C3×#2358 ⋊ C2 |
| Sylow | C3 ⋊ #34290 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C24×D3 |
| commutator | C22×C3 | C24×C4 |
| Fitting | C3×#2358 | C2 |
| socle, cosocle | C23×C3 | C25 |
| prime radical | C2×#5590 | C2 |
| hypercenter | #2358 | D3 |
| hypocenter | C3 | #34290 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ #2358 | |
3960 conjugacy classes of subgroups.
| transitive degree | μt(G) = 192 |
Polycyclic presentation:
2^8,3|a^2=h,d^2=f,e^2=h,b^a=bf,c^a=cg,e^a=ef,i^a=i^2,d^b=df,i^d=i^2,i^e=i^2138 conjugacy classes — view the full character table.
| maximal subgroup of | #72255283, D3×#34290, #72255332 |
| maximal normal subgroup of | #72255283, D3×#34290, #72255332 |
| maximal quotient of | #72270957 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 331172);