Finite Group DatabasebetaFinite Group Databasebeta| order | 32340 |
| automorphism | ? |
| homology | Abelianization (H1) = C4 · Schur multiplier (H2) = C1 |
| dicyclic | Dic8085 |
| unambiguous semidirect product | C3⋊Dic2695 |
| unambiguous semidirect product | C11⋊Dic735 |
| unambiguous semidirect product | C49⋊Dic165 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | D8085 |
| commutator, hypocenter | C3×C5×C11×C49 | C4 |
| Frattini | C2×C7 | D1155 |
| Fitting, cosocle | C2×C3×C5×C11×C49 | C2 |
| socle, prime radical | C2×C3×C5×C7×C11 | D7 |
| LCS | G ▷ C3×C5×C11×C49 | |
| UCS | C1 ◁ C2 | |
72 conjugacy classes of subgroups.
| minimal degree | μ(G) = 72 |
| transitive degree | μt(G) = 32340 |
A minimal faithful permutation representation — 2 generators on 72 points:
(1,2)(3,4)(5,6,7)(8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61)(62,63,64,65,66,67,68,69,70,71,72);(1,3,2,4)(6,7)(9,12)(10,11)(14,61)(15,60)(16,59)(17,58)(18,57)(19,56)(20,55)(21,54)(22,53)(23,52)(24,51)(25,50)(26,49)(27,48)(28,47)(29,46)(30,45)(31,44)(32,43)(33,42)(34,41)(35,40)(36,39)(37,38)(63,72)(64,71)(65,70)(66,69)(67,68)Polycyclic presentation:
2,3,5,7,11,2,7|a^2=f,d^7=g^6,b^a=b^2,c^a=c^4,d^a=d^6g,e^a=e^10,g^a=g^68088 conjugacy classes. The character table is too large to display.
| maximal subgroup of | Dic16170 |
| maximal normal subgroup of | Dic16170 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32340, 19);