Finite Group DatabasebetaFinite Group Databasebeta| order | 1536 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
675 conjugacy classes of subgroups; only the 34 normal subgroups are shown.
This group has 675 conjugacy classes of subgroups — too many to draw. Only its 34 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) = 96 |
A minimal faithful permutation representation — 2 generators on 48 points:
(1,14,11,9,13,8,3,7)(2,5,16,12,15,4,6,10)(17,31,33,20,22,34,21,32)(18,39,43,35,27,48,26,37)(19,46,45,44,28,47,30,29)(23,40,38,25,24,42,36,41);(2,3,8)(4,13,5)(6,15,10)(7,11,16)(9,12,14)(17,18)(19,23)(20,26,34,25,21,42)(22,27)(24,28)(29,36,46,35,30,48)(31,41,33,40,32,43)(37,45,39,44,38,47)Polycyclic presentation:
2^4,3,2^5|b^2=c,d^2=j,f^2=j,g^2=j,b^a=bd,d^a=dj,d^b=dj,e^b=e^2,f^b=fgh,f^c=fh,f^e=g,g^b=gij,g^c=ghi,g^e=fgj,g^f=gj,h^b=i,h^e=hi,i^b=h,i^e=h42 conjugacy classes — view the full character table.
| maximal quotient of | #72268906 |
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, 3, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 2, F.3);;
gap> SetPower(coll, 4, F.10);;
gap> SetPower(coll, 6, F.10);;
gap> SetPower(coll, 7, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.4);;
gap> SetConjugate(coll, 4, 1, F.4*F.10);;
gap> SetConjugate(coll, 4, 2, F.4*F.10);;
gap> SetConjugate(coll, 5, 2, F.5^2);;
gap> SetConjugate(coll, 6, 2, F.6*F.7*F.8);;
gap> SetConjugate(coll, 6, 3, F.6*F.8);;
gap> SetConjugate(coll, 6, 5, F.7);;
gap> SetConjugate(coll, 7, 2, F.7*F.9*F.10);;
gap> SetConjugate(coll, 7, 3, F.7*F.8*F.9);;
gap> SetConjugate(coll, 7, 5, F.6*F.7*F.10);;
gap> SetConjugate(coll, 7, 6, F.7*F.10);;
gap> SetConjugate(coll, 8, 2, F.9);;
gap> SetConjugate(coll, 8, 5, F.8*F.9);;
gap> SetConjugate(coll, 9, 2, F.8);;
gap> SetConjugate(coll, 9, 5, F.8);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);