Finite Group DatabasebetaFinite Group Databasebeta| order | 1536 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Fitting | C2×#3573 ⋊ D3 |
2809 conjugacy classes of subgroups; only the 68 normal subgroups are shown.
This group has 2809 conjugacy classes of subgroups — too many to draw. Only its 68 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 64 points:
(1,4,8,5)(2,10,13,20)(3,15,19,16)(6,24,50,23,27,48,26,22,25,21,7,49)(9,28,14,36)(11,47,55,46,35,54,34,32,33,31,12,62)(17,42,61,41,45,59,44,40,43,39,18,60)(29,57,58,56,38,63,37,52,53,51,30,64);(1,17,5,41,8,44,4,39)(2,29,10,51,13,37,20,56)(3,23,16,26,19,21,15,6)(7,43,24,60,27,61,22,59)(9,31,28,34,14,46,36,11)(12,53,32,63,35,58,47,64)(18,49,42,50,45,48,40,25)(30,54,52,55,38,62,57,33);(1,2,5,10,8,13,4,20)(3,14,16,36,19,9,15,28)(6,12,23,32,26,35,21,47)(7,11,24,31,27,34,22,46)(17,38,41,57,44,30,39,52)(18,37,42,56,45,29,40,51)(25,33,49,54,50,55,48,62)(43,58,60,64,61,53,59,63)Polycyclic presentation:
2^4,3,2^5|a^2=d,b^2=d,c^2=j,d^2=j,b^a=bj,c^a=cj,e^a=e^2,f^a=g,f^e=g,g^a=f,g^e=fg,h^a=hi,h^e=i,h^f=hj,i^e=hi,i^f=ij,i^g=ij92 conjugacy classes — view the full character table.
| maximal subgroup of | #72341899, #72342749, #72342787, … |
| maximal normal subgroup of | #72341899, #72342749, #72342787, … |
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.4);;
gap> SetPower(coll, 2, F.4);;
gap> SetPower(coll, 3, F.10);;
gap> SetPower(coll, 4, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.10);;
gap> SetConjugate(coll, 3, 1, F.3*F.10);;
gap> SetConjugate(coll, 5, 1, F.5^2);;
gap> SetConjugate(coll, 6, 1, F.7);;
gap> SetConjugate(coll, 6, 5, F.7);;
gap> SetConjugate(coll, 7, 1, F.6);;
gap> SetConjugate(coll, 7, 5, F.6*F.7);;
gap> SetConjugate(coll, 8, 1, F.8*F.9);;
gap> SetConjugate(coll, 8, 5, F.9);;
gap> SetConjugate(coll, 8, 6, F.8*F.10);;
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);