Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
No aliases for this group.
| Fitting | C3×#3503 ⋊ C2 |
| Sylow | C3 ⋊ Q8×H(5,𝔽2) |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C25×D3 |
| commutator, socle, cosocle | C22×C3 | C26 |
| Fitting | C3×#3503 | C2 |
| prime radical | C2×#1009 | C22 |
| hypercenter | #3503 | D3 |
| hypocenter | C3 | Q8×H(5,𝔽2) |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ #3503 | |
7406 conjugacy classes of subgroups.
| transitive degree | μt(G) = 192 |
Polycyclic presentation:
2^8,3|a^2=g,b^2=g,b^a=bg,c^a=ch,f^c=fh,e^d=eh,i^d=i^2,i^e=i^2,i^f=i^2129 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1081677);