Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29×C4 |
No aliases for this group.
| Fitting | C22×C3×H(5,𝔽2) ⋊ C2 |
| Sylow | C3 ⋊ #35527 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C24×D3 |
| commutator, socle, cosocle | C23×C3 | C25 |
| Fitting | C22×C3×H(5,𝔽2) | C2 |
| hypercenter | C22×H(5,𝔽2) | D3 |
| hypocenter | C3 | #35527 |
| LCS | G ▷ C23×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ C22×H(5,𝔽2) | |
6142 conjugacy classes of subgroups.
| transitive degree | μt(G) = 192 |
Polycyclic presentation:
2^8,3|a^2=g,d^2=h,b^a=bf,c^a=cg,i^a=i^2,c^b=ch,e^b=eh,d^c=dh132 conjugacy classes — view the full character table.
| maximal subgroup of | #72248002, #72248004, #72248010, … |
| maximal normal subgroup of | #72248002, #72248004, #72248010, … |
| maximal quotient of | #72253591, #72270846, #72270900, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 352823);