Finite Group DatabasebetaFinite Group Databasebeta| order | 704 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| Sylow | C11 ⋊ #391 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C22×D11 |
| commutator | C22×C11 | C22×C4 |
| Fitting | C22×Q8×C11 | C2 |
| socle, prime radical, cosocle | C23×C11 | C23 |
| hypercenter | C22×Q8 | D11 |
| hypocenter | C11 | #391 |
| LCS | G ▷ C22×C11 ▷ C11 | |
| UCS | C1 ◁ C23 ◁ C22×Q8 | |
186 conjugacy classes of subgroups.
| transitive degree | μt(G) = 704 |
Polycyclic presentation:
2^6,11|a^2=d,b^2=f,c^2=f,c^a=ce,g^a=g^10,c^b=cf128 conjugacy classes — view the full character table.
| maximal subgroup of | #13066742, #13068575, #13068577, … |
| maximal normal subgroup of | #13066742, #13068575, #13068577, … |
| maximal quotient of | #13066845, #13066848, #13066869, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(704, 762);