Finite Group DatabasebetaFinite Group Databasebeta| order | 28800 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2 |
| dihedral | D14400 |
| Fitting | C9×C25×C64 ⋊ C2 |
| Hall | C9×C25 ⋊ D64 |
| Sylow | C25 ⋊ D576 |
| Sylow | C9 ⋊ D1600 |
180 conjugacy classes of subgroups; only the 66 normal subgroups are shown.
This group has 180 conjugacy classes of subgroups — too many to draw. Only its 66 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 98 |
| transitive degree | μt(G) = 14400 |
A minimal faithful permutation representation — 2 generators on 98 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)(65,66,67,68,69,70,71,72,73)(74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98);(2,64)(3,63)(4,62)(5,61)(6,60)(7,59)(8,58)(9,57)(10,56)(11,55)(12,54)(13,53)(14,52)(15,51)(16,50)(17,49)(18,48)(19,47)(20,46)(21,45)(22,44)(23,43)(24,42)(25,41)(26,40)(27,39)(28,38)(29,37)(30,36)(31,35)(32,34)(66,73)(67,72)(68,71)(69,70)(75,98)(76,97)(77,96)(78,95)(79,94)(80,93)(81,92)(82,91)(83,90)(84,89)(85,88)(86,87)Polycyclic presentation:
2^7,3,5,3,5|b^2=cd,c^2=de,d^2=ef,e^2=fg,f^2=g,h^3=j,i^5=k,b^a=bc,c^a=cd,d^a=de,e^a=ef,f^a=fg,h^a=h^2j^2,i^a=i^4k^4,j^a=j^2,k^a=k^47203 conjugacy classes. The character table is too large to display.
| maximal subgroup of | D28800 |
| maximal normal subgroup of | D28800 |
| maximal quotient of | Dic14400, D28800 |
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", "k");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 3, 5, 3, 5]);;
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> SetPower(coll, 8, F.10);;
gap> SetPower(coll, 9, F.11);;
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*F.10^2);;
gap> SetConjugate(coll, 9, 1, F.9^4*F.11^4);;
gap> SetConjugate(coll, 10, 1, F.10^2);;
gap> SetConjugate(coll, 11, 1, F.11^4);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);