Finite Group DatabasebetaFinite Group Databasebeta| order | 704 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
| unambiguous semidirect product | C11⋊#564 |
| Sylow | C11 ⋊ #564 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C23×D11 |
| commutator, socle, prime radical, cosocle | C22×C11 | C24 |
| Fitting | C11×#125 | C2 |
| hypercenter | #125 | D11 |
| hypocenter | C11 | #564 |
| LCS | G ▷ C22×C11 ▷ C11 | |
| UCS | C1 ◁ C22 ◁ #125 | |
172 conjugacy classes of subgroups.
| transitive degree | μt(G) = 704 |
Polycyclic presentation:
2^6,11|a^2=f,b^2=e,c^2=e,d^2=f,c^a=cf,d^a=de,g^a=g^10,c^b=cef,d^b=df89 conjugacy classes — view the full character table.
| maximal subgroup of | #13067917, #13067919, #13067942, … |
| maximal normal subgroup of | #13067917, #13067919, #13067942, … |
| maximal quotient of | #13071798, #13071799, #13071800, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(704, 1136);