Finite Group DatabasebetaFinite Group Databasebeta| order | 4608 |
| automorphism | ? |
| homology | Abelianization (H1) = C32×C42 · Schur multiplier (H2) = C2×C3×C4 |
No aliases for this group.
| Fitting, Sylow | #10574037 ⋊ C32 |
2335 conjugacy classes of subgroups; only the 168 normal subgroups are shown.
This group has 2335 conjugacy classes of subgroups — too many to draw. Only its 168 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 100 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 2 generators on 100 points:
(5,46,64,33,30,35,26,81,31,12,61,43,11,45,25,14,28,16,8,82,10,6,63,68)(7,69,54,66,57,40,56,80,21,32,60,75,23,44,53,37,27,39,18,93,20,13,88,76)(9,34,24,67,29,15,62,42)(17,79,99,91,59,92,86,96,87,65,85,95,51,73,84,70,50,72,47,98,49,36,90,100)(19,41,52,77,22,38,55,74)(48,78,83,97,58,71,89,94);(1,2,3,4)(5,22,51,10,7,59,11,19,17,31,23,50)(6,38,65,35,13,73,12,41,36,16,32,79)(8,55,86,25,18,90,26,52,47,64,56,85)(9,20,87,30,27,48,29,21,49,28,57,58)(14,74,91,68,37,96,33,77,70,43,66,98)(15,40,92,45,44,78,34,39,72,46,69,71)(24,53,99,63,60,83,62,54,84,61,88,89)(42,76,100,81,80,97,67,75,95,82,93,94)Polycyclic presentation:
2^2,3^2,2^7|a^2=e,b^2=f,f^2=k,g^2=k,h^2=k,b^a=bk,g^c=hk,g^d=ghk,h^c=gh,h^d=gk,h^g=hk,i^c=gi,i^d=j,i^g=ik,j^c=ghjk,j^d=ij,j^h=jk292 conjugacy classes — view the full character table.
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", "k");;
gap> coll := SingleCollector(F, [2, 2, 3, 3, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 1, F.5);;
gap> SetPower(coll, 2, F.6);;
gap> SetPower(coll, 6, F.11);;
gap> SetPower(coll, 7, F.11);;
gap> SetPower(coll, 8, F.11);;
gap> SetConjugate(coll, 2, 1, F.2*F.11);;
gap> SetConjugate(coll, 7, 3, F.8*F.11);;
gap> SetConjugate(coll, 7, 4, F.7*F.8*F.11);;
gap> SetConjugate(coll, 8, 3, F.7*F.8);;
gap> SetConjugate(coll, 8, 4, F.7*F.11);;
gap> SetConjugate(coll, 8, 7, F.8*F.11);;
gap> SetConjugate(coll, 9, 3, F.7*F.9);;
gap> SetConjugate(coll, 9, 4, F.10);;
gap> SetConjugate(coll, 9, 7, F.9*F.11);;
gap> SetConjugate(coll, 10, 3, F.7*F.8*F.10*F.11);;
gap> SetConjugate(coll, 10, 4, F.9*F.10);;
gap> SetConjugate(coll, 10, 8, F.10*F.11);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);