Finite Group DatabasebetaFinite Group Databasebeta| order | 32340 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C5 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | #64422⋊AGL(1,𝔽11) |
| unambiguous semidirect product | (#64422⋊D11)⋊C5 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C72×C11 | C22×C3×C5 |
| LCS | G ▷ C72×C11 | |
| UCS | C1 | |
168 conjugacy classes of subgroups; only the 38 normal subgroups are shown.
This group has 168 conjugacy classes of subgroups — too many to draw. Only its 38 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 25 |
| transitive degree | μt(G) = 462 |
A minimal faithful permutation representation — 2 generators on 25 points:
(1,2,7,3,9)(4,11,10,5,8)(13,16,17,14,15,18)(19,20,25)(21,22,24);(2,3)(4,6)(5,8)(7,11)(9,10)(12,13,15,18,17,16,14)(20,21)(22,24)(23,25)Polycyclic presentation:
2^2,3,5,7^2,11|f^a=f^6,g^a=g^10,e^b=e^6,e^c=e^2,f^c=f^4,g^d=g^4112 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32340, 263);