Finite Group DatabasebetaFinite Group Databasebeta| order | 4096 |
| nilpotency class | 6 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #72227271 |
| commutator | C162 | C2×C8 |
| Frattini, cosocle | #72216220 | C22 |
| prime radical | C2×#123403 | C4 |
| LCS | G ▷ C162 ▷ C8×C16 ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×C4×C8 ◁ C2×C8×C16 ◁ #72216220 ◁ G | |
| LECS | G ▷ #72216220 ▷ C2×C8×C16 ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
668 conjugacy classes of subgroups; only the 87 normal subgroups are shown.
This group has 668 conjugacy classes of subgroups — too many to draw. Only its 87 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 — 2 generators on 64 points:
(1,2,3,13,4,15,21,46,29,17,54,48,5,33,34,47,9,18,25,49,31,14,22,51,8,39,24,59,30,53,57,64,11,19,27,50,32,42,55,61,38,52,58,63,35,12,20,41,10,40,26,60,6,36,37,45,7,16,23,44,28,43,56,62);(2,12)(3,20)(4,28)(5,6)(7,29)(8,38)(9,10)(13,44)(14,39)(15,52)(16,36)(17,42)(18,53)(19,33)(21,58)(22,24)(23,37)(25,57)(26,56)(27,34)(30,32)(31,35)(40,43)(41,47)(45,46)(48,60)(49,63)(50,59)(51,61)(54,55)Polycyclic presentation:
2^12|a^2=c,c^2=e,d^2=fghik,e^2=hj,f^2=hl,g^2=ij,h^2=jl,i^2=k,j^2=l,k^2=l,b^a=bd,c^b=cf,d^a=dg,d^b=dfg,d^c=djk,e^b=eh,f^a=fjk,f^b=fhjl,f^c=fl,g^a=gi,g^b=gijk,g^c=gl,h^b=hj,i^a=ik,i^b=ikl,j^b=jl,k^a=kl,k^b=kl166 conjugacy classes — view the full character table.
| maximal subgroup of | #72227285 |
| maximal normal subgroup of | #72227285 |
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.3);;
gap> SetPower(coll, 3, F.5);;
gap> SetPower(coll, 4, F.6*F.7*F.8*F.9*F.11);;
gap> SetPower(coll, 5, F.8*F.10);;
gap> SetPower(coll, 6, F.8*F.12);;
gap> SetPower(coll, 7, F.9*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.4);;
gap> SetConjugate(coll, 3, 2, F.3*F.6);;
gap> SetConjugate(coll, 4, 1, F.4*F.7);;
gap> SetConjugate(coll, 4, 2, F.4*F.6*F.7);;
gap> SetConjugate(coll, 4, 3, F.4*F.10*F.11);;
gap> SetConjugate(coll, 5, 2, F.5*F.8);;
gap> SetConjugate(coll, 6, 1, F.6*F.10*F.11);;
gap> SetConjugate(coll, 6, 2, F.6*F.8*F.10*F.12);;
gap> SetConjugate(coll, 6, 3, F.6*F.12);;
gap> SetConjugate(coll, 7, 1, F.7*F.9);;
gap> SetConjugate(coll, 7, 2, F.7*F.9*F.10*F.11);;
gap> SetConjugate(coll, 7, 3, F.7*F.12);;
gap> SetConjugate(coll, 8, 2, F.8*F.10);;
gap> SetConjugate(coll, 9, 1, F.9*F.11);;
gap> SetConjugate(coll, 9, 2, F.9*F.11*F.12);;
gap> SetConjugate(coll, 10, 2, F.10*F.12);;
gap> SetConjugate(coll, 11, 1, F.11*F.12);;
gap> SetConjugate(coll, 11, 2, F.11*F.12);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);