Finite Group DatabasebetaFinite Group Databasebeta| order | 704 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C11 ⋊ #400 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C22×D11 |
| commutator, socle, prime radical, cosocle | C23×C11 | C23 |
| Fitting | C11×#95 | C2 |
| hypercenter | #95 | D11 |
| hypocenter | C11 | #400 |
| LCS | G ▷ C23×C11 ▷ C11 | |
| UCS | C1 ◁ C23 ◁ #95 | |
134 conjugacy classes of subgroups; only the 55 normal subgroups are shown.
This group has 134 conjugacy classes of subgroups — too many to draw. Only its 55 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 704 |
Polycyclic presentation:
2^6,11|a^2=f,b^2=d,c^2=e,b^a=be,c^a=cde,g^a=g^10,c^b=cf122 conjugacy classes — view the full character table.
| maximal subgroup of | D11×#400, C2×#10622271, #13071874, … |
| maximal normal subgroup of | D11×#400, C2×#10622271, #13071874, … |
| maximal quotient of | #13066867, #13066871, #13066874, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(704, 190);