Finite Group DatabasebetaFinite Group Databasebeta| order | 8096 |
| automorphism | Aut(G) order 7124480 |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2 |
| dihedral | D4048 |
| Fitting | C11×C16×C23 ⋊ C2 |
| Hall | C11×C23 ⋊ D16 |
| Sylow | C23 ⋊ D176 |
| Sylow | C11 ⋊ D368 |
56 conjugacy classes of subgroups.
| minimal degree | μ(G) = 50 |
| transitive degree | μt(G) = 4048 |
A minimal faithful permutation representation — 2 generators on 50 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);(2,16)(3,15)(4,14)(5,13)(6,12)(7,11)(8,10)(18,27)(19,26)(20,25)(21,24)(22,23)(29,50)(30,49)(31,48)(32,47)(33,46)(34,45)(35,44)(36,43)(37,42)(38,41)(39,40)Polycyclic presentation:
2^5,11,23|b^2=cd,c^2=de,d^2=e,b^a=bc,c^a=cd,d^a=de,f^a=f^10,g^a=g^222027 conjugacy classes. The character table is too large to display.
| maximal subgroup of | D8096, D12144, D20240, … |
| maximal normal subgroup of | D8096 |
| maximal quotient of | D8096, Dic4048, D12144, … |
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, 2, 2, 11, 23]);;
gap> SetPower(coll, 2, F.3*F.4);;
gap> SetPower(coll, 3, F.4*F.5);;
gap> SetPower(coll, 4, F.5);;
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, 6, 1, F.6^10);;
gap> SetConjugate(coll, 7, 1, F.7^22);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);