Finite Group DatabasebetaFinite Group Databasebeta| order | 12012 |
| automorphism | Aut(G) order 8648640 |
| homology | Abelianization (H1) = C4 · Schur multiplier (H2) = C1 |
| dicyclic | Dic3003 |
| unambiguous semidirect product | C3⋊Dic1001 |
| unambiguous semidirect product | C7⋊Dic429 |
| unambiguous semidirect product | C11⋊Dic273 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | D3003 |
| commutator, hypocenter | C3×C7×C11×C13 | C4 |
| Fitting, socle, prime radical, cosocle | C2×C3×C7×C11×C13 | C2 |
| LCS | G ▷ C3×C7×C11×C13 | |
| UCS | C1 ◁ C2 | |
48 conjugacy classes of subgroups.
| minimal degree | μ(G) = 38 |
| transitive degree | μt(G) = 12012 |
A minimal faithful permutation representation — 2 generators on 38 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);(1,3,2,4)(6,7)(9,14)(10,13)(11,12)(16,25)(17,24)(18,23)(19,22)(20,21)(27,38)(28,37)(29,36)(30,35)(31,34)(32,33)Polycyclic presentation:
2,3,7,11,13,2|a^2=f,b^a=b^2,c^a=c^6,d^a=d^10,e^a=e^123006 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(12012, 20);