Finite Group DatabasebetaFinite Group Databasebeta| order | 2592 |
| automorphism | ? |
| homology | Abelianization (H1) = C32×C4 · Schur multiplier (H2) = C1 |
No aliases for this group.
303 conjugacy classes of subgroups; only the 56 normal subgroups are shown.
This group has 303 conjugacy classes of subgroups — too many to draw. Only its 56 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 144 |
A minimal faithful permutation representation — 2 generators on 44 points:
(1,2,3,7,6,10,13,20)(4,21,12,18,16,24,26,31,5,8,25,19,15,23,28,30,14,9,11,29,17,22,27,32)(34,35)(36,42,43,38,37,44,41,39,40);(1,14,6,16)(2,21,10,23)(3,25,13,27)(4,17,15,5)(7,29,20,31)(8,24,22,9)(11,28,26,12)(18,32,30,19)(33,34,35)(37,39,42)(40,44,43)Polycyclic presentation:
2,3^2,2,3,2^2,3,2|a^2=d,c^3=e,d^2=i,f^2=i,g^2=i,c^b=ce,f^c=g,g^c=fgi,g^f=gi,h^a=h^2186 conjugacy classes — view the full character table.
| maximal subgroup of | #72253003, #72253017 |
| maximal normal subgroup of | #72253003, #72253017 |
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");;
gap> coll := SingleCollector(F, [2, 3, 3, 2, 3, 2, 2, 3, 2]);;
gap> SetPower(coll, 1, F.4);;
gap> SetPower(coll, 3, F.5);;
gap> SetPower(coll, 4, F.9);;
gap> SetPower(coll, 6, F.9);;
gap> SetPower(coll, 7, F.9);;
gap> SetConjugate(coll, 3, 2, F.3*F.5);;
gap> SetConjugate(coll, 6, 3, F.7);;
gap> SetConjugate(coll, 7, 3, F.6*F.7*F.9);;
gap> SetConjugate(coll, 7, 6, F.7*F.9);;
gap> SetConjugate(coll, 8, 1, F.8^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);