Finite Group DatabasebetaFinite Group Databasebeta| order | 4096 |
| nilpotency class | 5 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C8 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C8 | #129088 |
| commutator | C4×C16 | C2×C4×C8 |
| Frattini, cosocle | C8×#346 | C23 |
| socle | C2 | #72227185 |
| prime radical | C23 | #129159 |
| LCS | G ▷ C4×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8 ◁ C2×M64 ◁ C4×#1398 ◁ C8×#1398 ◁ G | |
| LECS | G ▷ C8×#346 ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
782 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 64 points:
(1,2,5,15,8,17,6,39,11,19,30,58,45,60,41,52,12,21,7,42,10,46,38,16,36,20,40,59,9,18,37,57)(3,22,4,29,25,54,31,64,27,13,33,47,26,49,32,61,28,50,34,55,43,14,44,48,63,51,24,62,35,53,23,56);(1,3,6,23,9,26,5,33,36,63,37,44,10,25,40,4,12,28,38,32,45,35,7,24,11,27,41,31,8,43,30,34)(2,13,16,48,18,14,42,29,20,22,52,61,46,49,58,62,21,51,39,64,60,54,15,55,19,50,57,56,17,53,59,47);(1,4,30,25,8,44,41,63,11,33,7,26,45,23,38,3,12,34,40,43,10,31,37,27,36,24,5,35,9,32,6,28)(2,14,52,62,60,50,59,48,20,49,39,55,17,13,42,61,21,54,57,47,18,22,58,64,19,53,16,29,46,51,15,56)Polycyclic presentation:
2^12|a^2=d,b^2=e,c^2=fjl,d^2=g,e^2=h,f^2=i,g^2=j,h^2=jl,i^2=k,j^2=l,k^2=l,b^a=bhijk,c^a=cf,d^b=dj,d^c=dkl,e^a=ejkl,e^d=el,f^a=fi,f^d=fl,g^b=gl,g^c=gl,i^a=ik,k^a=kl304 conjugacy classes — view the full character table.
| maximal subgroup of | #72227218, #72227244, #72227257, … |
| maximal normal subgroup of | #72227218, #72227244, #72227257, … |
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", "l");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 1, F.4);;
gap> SetPower(coll, 2, F.5);;
gap> SetPower(coll, 3, F.6*F.10*F.12);;
gap> SetPower(coll, 4, F.7);;
gap> SetPower(coll, 5, F.8);;
gap> SetPower(coll, 6, F.9);;
gap> SetPower(coll, 7, F.10);;
gap> SetPower(coll, 8, F.10*F.12);;
gap> SetPower(coll, 9, F.11);;
gap> SetPower(coll, 10, F.12);;
gap> SetPower(coll, 11, F.12);;
gap> SetConjugate(coll, 2, 1, F.2*F.8*F.9*F.10*F.11);;
gap> SetConjugate(coll, 3, 1, F.3*F.6);;
gap> SetConjugate(coll, 4, 2, F.4*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.11*F.12);;
gap> SetConjugate(coll, 5, 1, F.5*F.10*F.11*F.12);;
gap> SetConjugate(coll, 5, 4, F.5*F.12);;
gap> SetConjugate(coll, 6, 1, F.6*F.9);;
gap> SetConjugate(coll, 6, 4, F.6*F.12);;
gap> SetConjugate(coll, 7, 2, F.7*F.12);;
gap> SetConjugate(coll, 7, 3, F.7*F.12);;
gap> SetConjugate(coll, 9, 1, F.9*F.11);;
gap> SetConjugate(coll, 11, 1, F.11*F.12);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);