Finite Group DatabasebetaFinite Group Databasebeta| order | 704 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Fitting | C11×#137 ⋊ C2 |
| Sylow | C11 ⋊ #578 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | C2×D4×D11 |
| commutator, cosocle | C4×C11 | C24 |
| Frattini | C4 | C23×D11 |
| Fitting | C11×#137 | C2 |
| socle | C2×C11 | C22×D4 |
| hypercenter | #137 | D11 |
| hypocenter | C11 | #578 |
| LCS | G ▷ C4×C11 ▷ C2×C11 ▷ C11 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ #137 | |
248 conjugacy classes of subgroups; only the 99 normal subgroups are shown.
This group has 248 conjugacy classes of subgroups — too many to draw. Only its 99 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|c^2=f,d^2=f,e^2=f,b^a=bf,g^a=g^10,d^b=df,d^c=de,e^c=ef,e^d=ef77 conjugacy classes — view the full character table.
| maximal subgroup of | #13080386, #13080388, #13080390, … |
| maximal normal subgroup of | #13080386, #13080388, #13080390, … |
| maximal quotient of | #13077147, #13077148, #13077215, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(704, 1229);