Finite Group DatabasebetaFinite Group Databasebeta| order | 15288 |
| automorphism | Aut(G) order 15410304 |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2 |
| dihedral | D7644 |
| Fitting | C3×C4×C13×C49 ⋊ C2 |
| Hall | C3×C13×C49 ⋊ D4 |
| Hall | C13×C49 ⋊ D12 |
| Hall | C3×C49 ⋊ D52 |
| Sylow | C49 ⋊ D156 |
| Hall | C3×C13 ⋊ D196 |
| Sylow | C13 ⋊ D588 |
| Sylow | C3 ⋊ D2548 |
96 conjugacy classes of subgroups.
| minimal degree | μ(G) = 69 |
| transitive degree | μt(G) = 7644 |
A minimal faithful permutation representation — 2 generators on 69 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);(2,4)(6,7)(9,56)(10,55)(11,54)(12,53)(13,52)(14,51)(15,50)(16,49)(17,48)(18,47)(19,46)(20,45)(21,44)(22,43)(23,42)(24,41)(25,40)(26,39)(27,38)(28,37)(29,36)(30,35)(31,34)(32,33)(58,69)(59,68)(60,67)(61,66)(62,65)(63,64)Polycyclic presentation:
2^3,3,7,13,7|b^2=c,e^7=g,b^a=bc,d^a=d^2,e^a=e^6g^6,f^a=f^12,g^a=g^63825 conjugacy classes. The character table is too large to display.
| maximal subgroup of | D15288, D22932 |
| maximal normal subgroup of | D15288 |
| maximal quotient of | D15288, Dic7644, D22932 |
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");;
gap> coll := SingleCollector(F, [2, 2, 2, 3, 7, 13, 7]);;
gap> SetPower(coll, 2, F.3);;
gap> SetPower(coll, 5, F.7);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 4, 1, F.4^2);;
gap> SetConjugate(coll, 5, 1, F.5^6*F.7^6);;
gap> SetConjugate(coll, 6, 1, F.6^12);;
gap> SetConjugate(coll, 7, 1, F.7^6);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);