Finite Group DatabasebetaFinite Group Databasebeta| order | 704 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| Sylow | C11 ⋊ #390 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C22×D11 |
| commutator | C22×C11 | C22×C4 |
| Fitting, prime radical | C22×D4×C11 | C2 |
| socle, cosocle | C23×C11 | C23 |
| hypercenter | C22×D4 | D11 |
| hypocenter | C11 | #390 |
| LCS | G ▷ C22×C11 ▷ C11 | |
| UCS | C1 ◁ C23 ◁ C22×D4 | |
258 conjugacy classes of subgroups; only the 91 normal subgroups are shown.
This group has 258 conjugacy classes of subgroups — too many to draw. Only its 91 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=d,b^2=f,c^a=ce,g^a=g^10,c^b=cf128 conjugacy classes — view the full character table.
| maximal subgroup of | #13066740, #13068570, #13068572, … |
| maximal normal subgroup of | #13066740, #13068570, #13068572, … |
| maximal quotient of | #13066845, #13066853, #13066857, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(704, 753);