Finite Group DatabasebetaFinite Group Databasebeta| order | 704 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| Fitting | C11×H(5,𝔽2) ⋊ C2 |
| Sylow | C11 ⋊ #585 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C24×D11 |
| commutator, socle, cosocle | C2×C11 | C25 |
| Fitting | C11×H(5,𝔽2) | C2 |
| hypercenter | H(5,𝔽2) | D11 |
| hypocenter | C11 | #585 |
| LCS | G ▷ C2×C11 ▷ C11 | |
| UCS | C1 ◁ C2 ◁ H(5,𝔽2) | |
810 conjugacy classes of subgroups.
| transitive degree | μt(G) = 176 |
Polycyclic presentation:
2^6,11|c^2=f,e^a=ef,g^a=g^10,d^b=df,e^c=ef119 conjugacy classes — view the full character table.
| maximal subgroup of | #13077961, #13077966, #13077984, … |
| maximal normal subgroup of | #13077961, #13077966, #13077984, … |
| maximal quotient of | #13081814, #13081816, #13081820, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(704, 1374);