Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C214 |
No aliases for this group.
| Fitting | C3×#3584 ⋊ C2 |
| Sylow | C3 ⋊ #63090 |
| subgroup | G/H | |
|---|---|---|
| center | C4 | C22×D3×D4 |
| commutator, cosocle | C3×C4 | C26 |
| Frattini | C4 | C25×D3 |
| Fitting | C3×#3584 | C2 |
| socle | C2×C3 | C24×D4 |
| hypercenter | #3584 | D3 |
| hypocenter | C3 | #63090 |
| LCS | G ▷ C3×C4 ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C4 ◁ C2×#41 ◁ #3584 | |
10836 conjugacy classes of subgroups.
| transitive degree | μt(G) = 96 |
Polycyclic presentation:
2^8,3|f^2=h,g^2=h,b^a=bg,e^a=eh,g^a=gh,g^b=gh,d^c=dh,i^e=i^2126 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1083420);