Finite Group DatabasebetaFinite Group Databasebeta| order | 1536 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Fitting | #63076 ⋊ D3 |
1974 conjugacy classes of subgroups; only the 42 normal subgroups are shown.
This group has 1974 conjugacy classes of subgroups — too many to draw. Only its 42 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 2 generators on 32 points:
(1,13,4,9,5,21,11,26)(2,7,14,16,10,18,3,28)(6,30,27,8,17,12,15,24)(19,32,25,20,31,29,23,22);(1,2,4,3,5,10,11,14)(6,9,29,12,18,23,27,21,22,8,16,31,17,26,32,30,7,25,15,13,20,24,28,19)Polycyclic presentation:
2^4,3,2^5|a^2=c,b^2=cd,c^2=j,f^2=j,g^2=j,b^a=bd,c^b=cj,d^a=dj,d^b=dj,e^b=e^2,f^b=fg,f^e=g,g^b=gj,g^e=fgj,g^f=gj,h^b=i,h^e=i,h^f=hj,h^g=hj,i^b=h,i^e=hi,i^g=ij59 conjugacy classes — view the full character table.
| maximal subgroup of | #72247755, #72252443, #72252444 |
| maximal normal subgroup of | #72247755, #72252443, #72252444 |
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, 2, F.3*F.4);;
gap> SetPower(coll, 3, 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, 3, 2, F.3*F.10);;
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);;
gap> SetConjugate(coll, 6, 5, F.7);;
gap> SetConjugate(coll, 7, 2, F.7*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, 2, F.9);;
gap> SetConjugate(coll, 8, 5, F.9);;
gap> SetConjugate(coll, 8, 6, F.8*F.10);;
gap> SetConjugate(coll, 8, 7, F.8*F.10);;
gap> SetConjugate(coll, 9, 2, F.8);;
gap> SetConjugate(coll, 9, 5, F.8*F.9);;
gap> SetConjugate(coll, 9, 7, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);