Finite Group DatabasebetaFinite Group Databasebeta| order | 5400 |
| automorphism | Aut(G) order 1944000 |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C1 |
| dicyclic | Dic1350 |
| Hall | C25×C27 ⋊ Q8 |
| Sylow | C27 ⋊ Dic50 |
| Sylow | C25 ⋊ Dic54 |
72 conjugacy classes of subgroups.
| minimal degree | μ(G) = 60 |
| transitive degree | μt(G) = 5400 |
A minimal faithful permutation representation — 2 generators on 60 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)(36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60);(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)(37,60)(38,59)(39,58)(40,57)(41,56)(42,55)(43,54)(44,53)(45,52)(46,51)(47,50)(48,49)Polycyclic presentation:
2^3,3,5,3,5,3|a^2=c,b^2=c,d^3=f,e^5=g,f^3=h,b^a=bc,d^a=d^2f^2h^2,e^a=e^4g^4,f^a=f^2h^2,g^a=g^4,h^a=h^21353 conjugacy classes. The character table is too large to display.
| maximal subgroup of | Dic2700, Dic4050, Dic6750, … |
| maximal normal subgroup of | Dic2700 |
| maximal quotient of | Dic4050, Dic6750, Dic9450, … |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:
gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f", "g", "h");;
gap> coll := SingleCollector(F, [2, 2, 2, 3, 5, 3, 5, 3]);;
gap> SetPower(coll, 1, F.3);;
gap> SetPower(coll, 2, F.3);;
gap> SetPower(coll, 4, F.6);;
gap> SetPower(coll, 5, F.7);;
gap> SetPower(coll, 6, F.8);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 4, 1, F.4^2*F.6^2*F.8^2);;
gap> SetConjugate(coll, 5, 1, F.5^4*F.7^4);;
gap> SetConjugate(coll, 6, 1, F.6^2*F.8^2);;
gap> SetConjugate(coll, 7, 1, F.7^4);;
gap> SetConjugate(coll, 8, 1, F.8^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);