Finite Group DatabasebetaFinite Group Databasebeta| order | 23664 |
| automorphism | Aut(G) order 42405888 |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2 |
| dihedral | D11832 |
| Fitting | C3×C8×C17×C29 ⋊ C2 |
| Hall | C3×C17×C29 ⋊ D8 |
| Hall | C17×C29 ⋊ D24 |
| Hall | C3×C29 ⋊ D136 |
| Hall | C3×C17 ⋊ D232 |
| Sylow | C29 ⋊ D408 |
| Sylow | C17 ⋊ D696 |
| Sylow | C3 ⋊ D3944 |
88 conjugacy classes of subgroups.
| minimal degree | μ(G) = 57 |
| transitive degree | μt(G) = 11832 |
A minimal faithful permutation representation — 2 generators on 57 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);(2,8)(3,7)(4,6)(10,11)(13,28)(14,27)(15,26)(16,25)(17,24)(18,23)(19,22)(20,21)(30,57)(31,56)(32,55)(33,54)(34,53)(35,52)(36,51)(37,50)(38,49)(39,48)(40,47)(41,46)(42,45)(43,44)Polycyclic presentation:
2^4,3,17,29|b^2=cd,c^2=d,b^a=bc,c^a=cd,e^a=e^2,f^a=f^16,g^a=g^285919 conjugacy classes. The character table is too large to display.
| maximal subgroup of | D23664 |
| maximal normal subgroup of | D23664 |
| maximal quotient of | Dic11832, D23664 |
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, 3, 17, 29]);;
gap> SetPower(coll, 2, F.3*F.4);;
gap> SetPower(coll, 3, F.4);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 3, 1, F.3*F.4);;
gap> SetConjugate(coll, 5, 1, F.5^2);;
gap> SetConjugate(coll, 6, 1, F.6^16);;
gap> SetConjugate(coll, 7, 1, F.7^28);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);