Finite Group DatabasebetaFinite Group Databasebeta| order | 704 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Fitting | C11×C22⋊Q8 ⋊ C2 |
| Sylow | C11 ⋊ #556 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C23×D11 |
| commutator, socle, cosocle | C22×C11 | C24 |
| Fitting | C11×C22⋊Q8 | C2 |
| prime radical | C2×C22⋊D11 | C22 |
| hypercenter | C22⋊Q8 | D11 |
| hypocenter | C11 | #556 |
| LCS | G ▷ C22×C11 ▷ C11 | |
| UCS | C1 ◁ C22 ◁ C22⋊Q8 | |
212 conjugacy classes of subgroups; only the 97 normal subgroups are shown.
This group has 212 conjugacy classes of subgroups — too many to draw. Only its 97 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|b^2=e,c^2=e,c^a=cf,d^a=def,g^a=g^10,c^b=ce,d^c=df95 conjugacy classes — view the full character table.
| maximal subgroup of | #13080526, #13080528, #13080548, … |
| maximal normal subgroup of | #13080526, #13080528, #13080548, … |
| maximal quotient of | #13071508, #13071511, #13071560, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(704, 1091);