Finite Group DatabasebetaFinite Group Databasebeta| order | 704 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Fitting | C11×#131 ⋊ C2 |
| Sylow | C11 ⋊ #568 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C4 | C23×D11 |
| commutator, socle | C2×C11 | C23×C4 |
| Fitting | C11×#131 | C2 |
| prime radical | #41×D11 | C2 |
| cosocle | C4×C11 | C24 |
| hypercenter | #131 | D11 |
| hypocenter | C11 | #568 |
| LCS | G ▷ C2×C11 ▷ C11 | |
| UCS | C1 ◁ C4 ◁ #131 | |
258 conjugacy classes of subgroups; only the 147 normal subgroups are shown.
This group has 258 conjugacy classes of subgroups — too many to draw. Only its 147 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 176 |
Polycyclic presentation:
2^6,11|d^2=e,e^2=f,d^a=df,g^a=g^10,c^b=cf134 conjugacy classes — view the full character table.
| maximal subgroup of | #13077521, #13077524, #13077528, … |
| maximal normal subgroup of | #13077521, #13077524, #13077528, … |
| maximal quotient of | #13077032, #13077042, #13077043, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(704, 1200);