Finite Group DatabasebetaFinite Group Databasebeta| order | 32340 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C5 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | #20266354⋊C5 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | C11×#3862 | C22×C5 |
| Fitting, socle | C72×C11 | C2×C5×D3 |
| LCS | G ▷ C11×#3862 | |
| UCS | C1 | |
136 conjugacy classes of subgroups; only the 21 normal subgroups are shown.
This group has 136 conjugacy classes of subgroups — too many to draw. Only its 21 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:
(2,5,10,6,11,12)(3,13,9,7,8,14)(15,16,21,17,23)(18,25,24,19,22);(1,2)(3,6)(4,9)(5,7)(8,11)(10,14)(12,13)(16,17)(18,20)(19,22)(21,25)(23,24)Polycyclic presentation:
2^2,5,3,11,7^2|d^a=d^2,e^a=e^10,f^a=f^4g,g^a=f^6g^3,f^b=f^6,g^b=g^6,e^c=e^4,f^d=f^2g^6,g^d=g^4109 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32340, 248);