Finite Group DatabasebetaFinite Group Databasebeta| order | 1512 |
| automorphism | Aut(G) order 163296 |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C1 |
| dicyclic | Dic378 |
| Hall | C7×C27 ⋊ Q8 |
| Sylow | C27 ⋊ Dic14 |
| Sylow | C7 ⋊ Dic54 |
48 conjugacy classes of subgroups.
| minimal degree | μ(G) = 42 |
| transitive degree | μt(G) = 1512 |
A minimal faithful permutation representation — 2 generators on 42 points:
(1,2,3,4)(5,8,7,6)(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);(1,5,3,7)(2,6,4,8)(10,35)(11,34)(12,33)(13,32)(14,31)(15,30)(16,29)(17,28)(18,27)(19,26)(20,25)(21,24)(22,23)(37,42)(38,41)(39,40)Polycyclic presentation:
2^3,3,7,3^2|a^2=c,b^2=c,d^3=f,f^3=g,b^a=bc,d^a=d^2f^2g^2,e^a=e^6,f^a=f^2g^2,g^a=g^2381 conjugacy classes — view the full character table.
| maximal subgroup of | Dic756, Dic1134, Dic1890, … |
| maximal normal subgroup of | Dic756 |
| maximal quotient of | Dic1134, Dic1890, Dic2646, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1512, 34);