Finite Group DatabasebetaFinite Group Databasebeta| order | 12210 |
| automorphism | Aut(G) order 17582400 |
| homology | Abelianization (H1) = C2 · Schur multiplier (H2) = C1 |
| dihedral | D6105 |
| unambiguous semidirect product | C3⋊D2035 |
| unambiguous semidirect product | C5⋊D1221 |
| unambiguous semidirect product | C11⋊D555 |
| unambiguous semidirect product | C37⋊D165 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C3×C5×C11×C37 | C2 |
| LCS | G ▷ C3×C5×C11×C37 | |
| UCS | C1 | |
32 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 6105 |
A minimal faithful permutation representation — 2 generators on 56 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);(2,3)(5,8)(6,7)(10,19)(11,18)(12,17)(13,16)(14,15)(21,56)(22,55)(23,54)(24,53)(25,52)(26,51)(27,50)(28,49)(29,48)(30,47)(31,46)(32,45)(33,44)(34,43)(35,42)(36,41)(37,40)(38,39)Polycyclic presentation:
2,3,5,11,37|b^a=b^2,c^a=c^4,d^a=d^10,e^a=e^363054 conjugacy classes. The character table is too large to display.
| maximal subgroup of | D37×D165, D11×D555, D15×D407, … |
| maximal normal subgroup of | D37×D165, D11×D555, D15×D407, … |
| maximal quotient of | Dic6105, C2×D6105, D18315, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(12210, 19);