Finite Group DatabasebetaFinite Group Databasebeta| order | 1536 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
665 conjugacy classes of subgroups; only the 32 normal subgroups are shown.
This group has 665 conjugacy classes of subgroups — too many to draw. Only its 32 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) = 48 |
A minimal faithful permutation representation — 2 generators on 48 points:
(1,4,41,6,22,48)(2,26,36,12,24,40)(3,13,44,7,19,38)(5,43,37,23,17,16)(8,28,47,9,10,46)(11,31,32,20,45,15)(14,42,39,21,18,33)(25,29,35,27,30,34);(1,2)(3,12,6,8)(4,15)(5,20,21,25)(7,9)(10,16)(11,14)(13,32,22,34)(17,40)(18,36,43,46)(19,35)(23,27)(24,33,28,37)(26,39)(29,44,45,48)(30,38)(31,41)(42,47)Polycyclic presentation:
2^4,3,2^5|a^2=c,f^2=h,g^2=i,b^a=bd,c^b=cj,d^a=dj,e^a=e^2,e^b=ehj,f^a=fg,f^c=fhj,f^e=ghj,g^a=ghij,g^c=gij,g^e=fgj,g^f=gj,h^a=hij,h^e=i,i^e=hij36 conjugacy classes — view the full character table.
| maximal subgroup of | #72268833 |
| maximal normal subgroup of | #72268833 |
| maximal quotient of | #72268873 |
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, 1, F.3);;
gap> SetPower(coll, 6, F.8);;
gap> SetPower(coll, 7, F.9);;
gap> SetConjugate(coll, 2, 1, F.2*F.4);;
gap> SetConjugate(coll, 3, 2, F.3*F.10);;
gap> SetConjugate(coll, 4, 1, F.4*F.10);;
gap> SetConjugate(coll, 5, 1, F.5^2);;
gap> SetConjugate(coll, 5, 2, F.5*F.8*F.10);;
gap> SetConjugate(coll, 6, 1, F.6*F.7);;
gap> SetConjugate(coll, 6, 3, F.6*F.8*F.10);;
gap> SetConjugate(coll, 6, 5, F.7*F.8*F.10);;
gap> SetConjugate(coll, 7, 1, F.7*F.8*F.9*F.10);;
gap> SetConjugate(coll, 7, 3, F.7*F.9*F.10);;
gap> SetConjugate(coll, 7, 5, F.6*F.7*F.10);;
gap> SetConjugate(coll, 7, 6, F.7*F.10);;
gap> SetConjugate(coll, 8, 1, F.8*F.9*F.10);;
gap> SetConjugate(coll, 8, 5, F.9);;
gap> SetConjugate(coll, 9, 5, F.8*F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);