Finite Group DatabasebetaFinite Group Databasebeta| order | 1536 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
1324 conjugacy classes of subgroups; only the 66 normal subgroups are shown.
This group has 1324 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) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 48 points:
(1,4,5,12)(2,7,8,15)(3,9)(6,13)(10,11)(14,16)(17,31,33,20,22,34,21,32)(18,41,27,37,25,48,40,35)(19,46,45,29,28,47,30,44)(23,43,36,26,24,42,38,39);(1,3,5,10)(2,6,8,14)(4,11,12,9)(7,13,15,16)(17,18,22,25)(19,23,28,24)(20,27,31,39,21,43,32,40,34,26,33,42)(29,36,46,37,30,48,44,38,47,35,45,41);(1,2)(3,6)(4,7)(5,8)(9,16)(10,14)(11,13)(12,15)(19,28)(23,24)(29,44)(30,45)(35,37)(36,38)(41,48)(46,47)Polycyclic presentation:
2^6,3,2^3|b^2=f,c^2=efj,d^2=j,h^2=j,i^2=j,c^a=ce,c^b=cd,d^a=df,d^b=dj,d^c=dfj,e^b=ef,g^c=g^2,h^c=hi,h^g=i,i^c=ij,i^g=hij,i^h=ij61 conjugacy classes — view the full character table.
| maximal subgroup of | #72268143 |
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, 3, 2, 2, 2]);;
gap> SetPower(coll, 2, F.6);;
gap> SetPower(coll, 3, F.5*F.6*F.10);;
gap> SetPower(coll, 4, F.10);;
gap> SetPower(coll, 8, F.10);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 3, 1, F.3*F.5);;
gap> SetConjugate(coll, 3, 2, F.3*F.4);;
gap> SetConjugate(coll, 4, 1, F.4*F.6);;
gap> SetConjugate(coll, 4, 2, F.4*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.6*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.6);;
gap> SetConjugate(coll, 7, 3, F.7^2);;
gap> SetConjugate(coll, 8, 3, F.8*F.9);;
gap> SetConjugate(coll, 8, 7, F.9);;
gap> SetConjugate(coll, 9, 3, F.9*F.10);;
gap> SetConjugate(coll, 9, 7, F.8*F.9*F.10);;
gap> SetConjugate(coll, 9, 8, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);