Finite Group DatabasebetaFinite Group Databasebeta| order | 3072 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Fitting, Sylow | #72241337 ⋊ C3 |
4722 conjugacy classes of subgroups; only the 44 normal subgroups are shown.
This group has 4722 conjugacy classes of subgroups — too many to draw. Only its 44 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 2 generators on 32 points:
(1,2,3,11,4,8,14,7)(5,24,13,23,18,22,27,31,6,9,28,25,15,26,29,19,17,10,12,32,16,21,30,20);(1,18)(2,32)(3,30)(4,17)(5,16,15,6)(7,26)(8,31)(9,25,21,20)(10,19,22,23)(11,24)(12,29,27,13)(14,28)Polycyclic presentation:
2^2,3,2^8|b^2=de,g^2=k,h^2=k,b^a=bd,d^b=df,e^a=ef,e^b=ef,e^d=ek,f^b=fk,g^a=gj,g^c=h,h^a=hi,h^c=ghk,h^g=hk,i^c=ijk,i^g=ik,j^c=i,j^h=jk70 conjugacy classes — view the full character table.
| maximal subgroup of | #72252563, #72267052, #72267085 |
| maximal normal subgroup of | #72252563, #72267052, #72267085 |
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, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 2, F.4*F.5);;
gap> SetPower(coll, 7, F.11);;
gap> SetPower(coll, 8, F.11);;
gap> SetConjugate(coll, 2, 1, F.2*F.4);;
gap> SetConjugate(coll, 4, 2, F.4*F.6);;
gap> SetConjugate(coll, 5, 1, F.5*F.6);;
gap> SetConjugate(coll, 5, 2, F.5*F.6);;
gap> SetConjugate(coll, 5, 4, F.5*F.11);;
gap> SetConjugate(coll, 6, 2, F.6*F.11);;
gap> SetConjugate(coll, 7, 1, F.7*F.10);;
gap> SetConjugate(coll, 7, 3, F.8);;
gap> SetConjugate(coll, 8, 1, F.8*F.9);;
gap> SetConjugate(coll, 8, 3, F.7*F.8*F.11);;
gap> SetConjugate(coll, 8, 7, F.8*F.11);;
gap> SetConjugate(coll, 9, 3, F.9*F.10*F.11);;
gap> SetConjugate(coll, 9, 7, F.9*F.11);;
gap> SetConjugate(coll, 10, 3, F.9);;
gap> SetConjugate(coll, 10, 8, F.10*F.11);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);