Finite Group DatabasebetaFinite Group Databasebeta| order | 1536 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C42 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| Fitting, Sylow | #10574037 ⋊ C3 |
4325 conjugacy classes of subgroups; only the 123 normal subgroups are shown.
This group has 4325 conjugacy classes of subgroups — too many to draw. Only its 123 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) = 128 |
A minimal faithful permutation representation — 3 generators on 36 points:
(5,6,7,12,10,11,17,21)(8,26,16,24,30,23,28,36,9,13,27,25,19,33,29,31,18,14,15,34,20,22,35,32);(1,2,3,4)(6,11)(8,19)(12,21)(14,23)(15,28)(18,30)(25,32)(27,35);(5,18)(6,26)(7,27)(8,9)(10,30)(11,33)(12,34)(13,14)(15,16)(17,35)(19,20)(21,36)(22,23)(24,25)(28,29)(31,32)Polycyclic presentation:
2^2,3,2^7|a^2=d,b^2=e,e^2=j,b^a=bj,f^c=g,g^c=fg,h^c=i,h^f=hj,i^c=hi,i^f=ij,i^g=ij140 conjugacy classes — view the full character table.
| maximal subgroup of | #72227323, #72253041, #72341690, … |
| maximal normal subgroup of | #72227323, #72253041, #72341690, … |
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, 3, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 1, F.4);;
gap> SetPower(coll, 2, F.5);;
gap> SetPower(coll, 5, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.10);;
gap> SetConjugate(coll, 6, 3, F.7);;
gap> SetConjugate(coll, 7, 3, F.6*F.7);;
gap> SetConjugate(coll, 8, 3, F.9);;
gap> SetConjugate(coll, 8, 6, F.8*F.10);;
gap> SetConjugate(coll, 9, 3, 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);