Finite Group DatabasebetaFinite Group Databasebeta| order | 1536 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C2 |
No aliases for this group.
269 conjugacy classes of subgroups; only the 60 normal subgroups are shown.
This group has 269 conjugacy classes of subgroups — too many to draw. Only its 60 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) = 96 |
A minimal faithful permutation representation — 2 generators on 36 points:
(1,4,3)(5,7,9,20,11,22,26,19,12,23,25,34,10,21,8,24)(6,13,15,18,16,29,27,33,17,30,14,31,28,36,32,35);(2,4)(5,6,8,14,10,16,25,32,12,17,26,15,11,28,9,27)(7,18,19,29,21,33,20,30,23,31,24,36,22,35,34,13)Polycyclic presentation:
2^7,3,2^2|a^2=c,b^2=d,c^2=e,d^2=f,e^2=g,f^2=g,b^a=bd,c^b=cg,d^a=df,f^a=fg,h^a=h^2,i^a=ij,i^h=j,j^h=ij104 conjugacy classes — view the full character table.
| maximal subgroup of | #72342835, #72342844, #72342860, … |
| maximal normal subgroup of | #72342835, #72342844, #72342860, … |
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, 2, 3, 2, 2]);;
gap> SetPower(coll, 1, F.3);;
gap> SetPower(coll, 2, F.4);;
gap> SetPower(coll, 3, F.5);;
gap> SetPower(coll, 4, F.6);;
gap> SetPower(coll, 5, F.7);;
gap> SetPower(coll, 6, F.7);;
gap> SetConjugate(coll, 2, 1, F.2*F.4);;
gap> SetConjugate(coll, 3, 2, F.3*F.7);;
gap> SetConjugate(coll, 4, 1, F.4*F.6);;
gap> SetConjugate(coll, 6, 1, F.6*F.7);;
gap> SetConjugate(coll, 8, 1, F.8^2);;
gap> SetConjugate(coll, 9, 1, F.9*F.10);;
gap> SetConjugate(coll, 9, 8, F.10);;
gap> SetConjugate(coll, 10, 8, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);