Finite Group DatabasebetaFinite Group Databasebeta| order | 704 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Sylow | C11 ⋊ #533 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C23×D11 |
| commutator, socle, cosocle | C22×C11 | C24 |
| Fitting | C11×C22⋊Q8 | C2 |
| prime radical | C23×C11 | C23 |
| hypercenter | C22⋊Q8 | D11 |
| hypocenter | C11 | #533 |
| LCS | G ▷ C22×C11 ▷ C11 | |
| UCS | C1 ◁ C22 ◁ C22⋊Q8 | |
222 conjugacy classes of subgroups; only the 109 normal subgroups are shown.
This group has 222 conjugacy classes of subgroups — too many to draw. Only its 109 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 352 |
Polycyclic presentation:
2^6,11|a^2=f,c^2=e,d^2=e,g^a=g^10,d^b=df,d^c=de98 conjugacy classes — view the full character table.
| maximal subgroup of | #13067113, #13067126, #13067165, … |
| maximal normal subgroup of | #13067113, #13067126, #13067165, … |
| maximal quotient of | #13071018, #13071087, #13071089, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(704, 1072);