Finite Group DatabasebetaFinite Group Databasebeta| order | 32340 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | D55⋊(D49⋊C3) |
| unambiguous semidirect product | (C5⋊(D49⋊C3))⋊D11 |
| unambiguous semidirect product | (C11⋊(D49⋊C3))⋊D5 |
| unambiguous semidirect product | (D55⋊D49)⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, Fitting, cosocle, hypocenter | C5×C11×C49 | C22×C3 |
| Frattini | C7 | D55⋊AGL(1,𝔽7) |
| socle | C5×C7×C11 | C2×AGL(1,𝔽7) |
| LCS | G ▷ C5×C11×C49 | |
| UCS | C1 | |
120 conjugacy classes of subgroups; only the 30 normal subgroups are shown.
This group has 120 conjugacy classes of subgroups — too many to draw. Only its 30 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 2695 |
Polycyclic presentation:
2^2,3,5,7,11,7|e^7=g^6,e^a=e^6g,f^a=f^10,g^a=g^6,d^b=d^4,e^b=e^6g,g^b=g^6,e^c=e^2g^3,g^c=g^2316 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32340, 61);