Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Sylow | C3 ⋊ #22853 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22×C4 | C23×D3 |
| commutator, socle | C23×C3 | C23×C4 |
| Fitting, prime radical | C3×#2388 | C2 |
| cosocle | C22×C3×C4 | C24 |
| hypercenter | #2388 | D3 |
| hypocenter | C3 | #22853 |
| LCS | G ▷ C23×C3 ▷ C3 | |
| UCS | C1 ◁ C22×C4 ◁ #2388 | |
1164 conjugacy classes of subgroups.
| transitive degree | μt(G) = 384 |
Polycyclic presentation:
2^8,3|a^2=g,b^2=g,g^2=h,b^a=be,c^a=cf,d^a=dh,i^a=i^2,c^b=ch,d^b=deh,i^b=i^2,d^c=dh120 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 189817);