Finite Group DatabasebetaFinite Group Databasebeta| order | 704 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | C11×#100 ⋊ C2 |
| Sylow | C11 ⋊ #472 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | #66937 |
| commutator, cosocle | C2×C4×C11 | C23 |
| Frattini | C2×C4 | C22×D11 |
| Fitting | C11×#100 | C2 |
| socle | C2×C11 | #121 |
| hypercenter | #100 | D11 |
| hypocenter | C11 | #472 |
| LCS | G ▷ C2×C4×C11 ▷ C2×C11 ▷ C11 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ #100 | |
108 conjugacy classes of subgroups; only the 35 normal subgroups are shown.
This group has 108 conjugacy classes of subgroups — too many to draw. Only its 35 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 176 |
Polycyclic presentation:
2^6,11|b^2=d,d^2=f,c^a=cde,e^a=ef,g^a=g^10,c^b=cef,e^b=ef,d^c=df71 conjugacy classes — view the full character table.
| maximal subgroup of | D11×#472, C2×#10622362, #13078331, … |
| maximal normal subgroup of | D11×#472, C2×#10622362, #13078331, … |
| maximal quotient of | #13067970, #13067975, #13068014, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(704, 281);