Finite Group DatabasebetaFinite Group Databasebeta| order | 32340 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C5 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | #64424⋊AGL(1,𝔽11) |
| unambiguous semidirect product | (#64424⋊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 | |
208 conjugacy classes of subgroups; only the 37 normal subgroups are shown.
This group has 208 conjugacy classes of subgroups — too many to draw. Only its 37 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) = 154 |
A minimal faithful permutation representation — 2 generators on 25 points:
(1,2,7,3,9)(4,11,10,5,8)(12,13,14,18,15,17)(20,23,24,21,22,25);(2,3)(4,6)(5,8)(7,11)(9,10)(13,14)(15,16)(17,18)(19,20,22,25,24,23,21)Polycyclic presentation:
2^2,3,5,7^2,11|e^a=e^6,g^a=g^10,e^b=e^6,f^b=f^6,e^c=e^2,f^c=f^2,g^d=g^4109 conjugacy classes — view the full character table.
| maximal subgroup of | AGL(1,𝔽11)×#64423⋊C2 |
| maximal normal subgroup of | AGL(1,𝔽11)×#64423⋊C2 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32340, 260);