Finite Group DatabasebetaFinite Group Databasebeta| order | 704 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C1 |
No aliases for this group.
| Sylow | C11 ⋊ SD64 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | #66890 |
| commutator, cosocle | C11×C16 | C22 |
| Frattini | C16 | C2×D11 |
| Fitting, prime radical | C11×D16 | C2 |
| socle | C2×C11 | D16 |
| hypercenter | D16 | D11 |
| hypocenter | C11 | SD64 |
| LCS | G ▷ C11×C16 ▷ C8×C11 ▷ C4×C11 ▷ C2×C11 ▷ C11 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ C8 ◁ C16 ◁ D16 | |
32 conjugacy classes of subgroups.
| minimal degree | μ(G) = 43 |
| transitive degree | μt(G) = 352 |
Polycyclic presentation:
2^6,11|a^2=f,c^2=de,d^2=ef,e^2=f,b^a=bc,c^a=cd,d^a=de,e^a=ef,g^a=g^10,c^b=cd,d^b=de,e^b=ef74 conjugacy classes — view the full character table.
| maximal subgroup of | D11×SD64, C2×#10622158, #13070922, … |
| maximal normal subgroup of | D11×SD64, C2×#10622158, #13070922, … |
| maximal quotient of | #13066794, #13066796, #13066817, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(704, 77);