Finite Group DatabasebetaFinite Group Databasebeta| order | 1536 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| Fitting | #20610 ⋊ D3 |
1240 conjugacy classes of subgroups; only the 31 normal subgroups are shown.
This group has 1240 conjugacy classes of subgroups — too many to draw. Only its 31 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 2 generators on 36 points:
(5,20,9,8)(6,29,14,25)(7,34,18,32)(10,21,24,22)(11,31,26,13)(12,30,28,19)(15,36,23,27)(16,35,33,17);(1,2,3,4)(5,6)(7,11)(8,23,20,19,9,30)(10,15)(12,16)(13,24,31,22,14,21)(17,28,35,27,18,36)(25,33,29,32,26,34)Polycyclic presentation:
2^4,3,2^5|a^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^a=fhij,f^b=fgij,f^e=g,g^a=ghj,g^b=gh,g^e=fgj,g^f=gj,h^e=i,h^f=hj,i^b=hi,i^e=hi,i^f=ij,i^g=ij50 conjugacy classes — view the full character table.
| maximal subgroup of | #72341722, #72341747, #72341764, … |
| maximal normal subgroup of | #72341722, #72341747, #72341764, … |
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, 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, 1, F.6*F.8*F.9*F.10);;
gap> SetConjugate(coll, 6, 2, F.6*F.7*F.9*F.10);;
gap> SetConjugate(coll, 6, 5, F.7);;
gap> SetConjugate(coll, 7, 1, F.7*F.8*F.10);;
gap> SetConjugate(coll, 7, 2, F.7*F.8);;
gap> SetConjugate(coll, 7, 5, F.6*F.7*F.10);;
gap> SetConjugate(coll, 7, 6, F.7*F.10);;
gap> SetConjugate(coll, 8, 5, F.9);;
gap> SetConjugate(coll, 8, 6, F.8*F.10);;
gap> SetConjugate(coll, 9, 2, F.8*F.9);;
gap> SetConjugate(coll, 9, 5, F.8*F.9);;
gap> SetConjugate(coll, 9, 6, F.9*F.10);;
gap> SetConjugate(coll, 9, 7, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);