Finite Group DatabasebetaFinite Group Databasebeta| order | 21120 |
| automorphism | Aut(G) order 27033600 |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C1 |
| dicyclic | Dic5280 |
| Hall | C3×C5×C11 ⋊ Q128 |
| Hall | C5×C11 ⋊ Dic96 |
| Hall | C3×C11 ⋊ Dic160 |
| Hall | C3×C5 ⋊ Dic352 |
| Sylow | C11 ⋊ Dic480 |
| Sylow | C5 ⋊ Dic1056 |
| Sylow | C3 ⋊ Dic1760 |
144 conjugacy classes of subgroups; only the 59 normal subgroups are shown.
This group has 144 conjugacy classes of subgroups — too many to draw. Only its 59 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 147 |
| transitive degree | μt(G) = 21120 |
A minimal faithful permutation representation — 2 generators on 147 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)(65,128,127,126,125,124,123,122,121,120,119,118,117,116,115,114,113,112,111,110,109,108,107,106,105,104,103,102,101,100,99,98,97,96,95,94,93,92,91,90,89,88,87,86,85,84,83,82,81,80,79,78,77,76,75,74,73,72,71,70,69,68,67,66)(129,130,131)(132,133,134,135,136)(137,138,139,140,141,142,143,144,145,146,147);(1,65,33,97)(2,66,34,98)(3,67,35,99)(4,68,36,100)(5,69,37,101)(6,70,38,102)(7,71,39,103)(8,72,40,104)(9,73,41,105)(10,74,42,106)(11,75,43,107)(12,76,44,108)(13,77,45,109)(14,78,46,110)(15,79,47,111)(16,80,48,112)(17,81,49,113)(18,82,50,114)(19,83,51,115)(20,84,52,116)(21,85,53,117)(22,86,54,118)(23,87,55,119)(24,88,56,120)(25,89,57,121)(26,90,58,122)(27,91,59,123)(28,92,60,124)(29,93,61,125)(30,94,62,126)(31,95,63,127)(32,96,64,128)(130,131)(133,136)(134,135)(138,147)(139,146)(140,145)(141,144)(142,143)Polycyclic presentation:
2^7,3,5,11|a^2=g,b^2=cd,c^2=de,d^2=ef,e^2=fg,f^2=g,b^a=bc,c^a=cd,d^a=de,e^a=ef,f^a=fg,h^a=h^2,i^a=i^4,j^a=j^105283 conjugacy classes. The character table is too large to display.
| maximal subgroup of | Dic10560, Dic15840 |
| maximal normal subgroup of | Dic10560 |
| maximal quotient of | Dic15840 |
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", "i", "j");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 3, 5, 11]);;
gap> SetPower(coll, 1, F.7);;
gap> SetPower(coll, 2, F.3*F.4);;
gap> SetPower(coll, 3, F.4*F.5);;
gap> SetPower(coll, 4, F.5*F.6);;
gap> SetPower(coll, 5, F.6*F.7);;
gap> SetPower(coll, 6, F.7);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 3, 1, F.3*F.4);;
gap> SetConjugate(coll, 4, 1, F.4*F.5);;
gap> SetConjugate(coll, 5, 1, F.5*F.6);;
gap> SetConjugate(coll, 6, 1, F.6*F.7);;
gap> SetConjugate(coll, 8, 1, F.8^2);;
gap> SetConjugate(coll, 9, 1, F.9^4);;
gap> SetConjugate(coll, 10, 1, F.10^10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);