Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
No aliases for this group.
| Fitting | C2×C3×C4×#41 ⋊ C2 |
| Sylow | C3 ⋊ #62686 |
| subgroup | G/H | |
|---|---|---|
| center | C42 | C23×D3 |
| commutator, socle, cosocle | C22×C3 | C26 |
| Frattini | C22 | C25×D3 |
| Fitting | C2×C3×C4×#41 | C2 |
| hypercenter | C2×C4×#41 | D3 |
| hypocenter | C3 | #62686 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C42 ◁ C2×C4×#41 | |
7226 conjugacy classes of subgroups.
| transitive degree | μt(G) = 192 |
Polycyclic presentation:
2^8,3|e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dh,c^b=ch,i^b=i^2168 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1076431);