Finite Group DatabasebetaFinite Group Databasebeta| order | 3072 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
6464 conjugacy classes of subgroups; only the 55 normal subgroups are shown.
This group has 6464 conjugacy classes of subgroups — too many to draw. Only its 55 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) = 128 |
A minimal faithful permutation representation — 2 generators on 64 points:
(1,2,24,22,6,7,30,31)(3,8,18,21,11,20,15,9)(4,25,26,29,16,14,13,23)(5,10,27,28,19,17,12,32)(33,49,37,36,38,47,50,48)(34,64,56,53,41,57,43,51)(35,61,45,44,46,59,62,60)(39,63,54,42,40,58,52,55);(2,7)(3,11)(4,5,18)(9,17,25,21,10,14)(12,26,30,27,13,24)(15,16,19)(22,23,32)(28,31,29)(33,34)(35,39)(36,43,49,42,37,58)(38,41)(40,46)(44,52,61,51,45,64)(47,55,50,63,48,56)(53,62,57,60,54,59)Polycyclic presentation:
2^4,3,2^6|b^2=d,c^2=k,d^2=j,f^2=j,g^2=j,b^a=bc,c^a=ck,c^b=cj,d^a=djk,e^b=e^2,f^b=fg,f^e=g,g^b=gj,g^e=fgj,g^f=gj,h^b=fghijk,h^e=i,h^f=hj,h^g=hjk,i^b=gik,i^e=hi,i^f=ik,i^g=ij71 conjugacy classes — view the full character table.
| maximal subgroup of | #72269004, #72272066 |
| maximal normal subgroup of | #72269004, #72272066 |
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, 2, 2, 3, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 2, F.4);;
gap> SetPower(coll, 3, F.11);;
gap> SetPower(coll, 4, F.10);;
gap> SetPower(coll, 6, F.10);;
gap> SetPower(coll, 7, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 3, 1, F.3*F.11);;
gap> SetConjugate(coll, 3, 2, F.3*F.10);;
gap> SetConjugate(coll, 4, 1, F.4*F.10*F.11);;
gap> SetConjugate(coll, 5, 2, F.5^2);;
gap> SetConjugate(coll, 6, 2, F.6*F.7);;
gap> SetConjugate(coll, 6, 5, F.7);;
gap> SetConjugate(coll, 7, 2, F.7*F.10);;
gap> SetConjugate(coll, 7, 5, F.6*F.7*F.10);;
gap> SetConjugate(coll, 7, 6, F.7*F.10);;
gap> SetConjugate(coll, 8, 2, F.6*F.7*F.8*F.9*F.10*F.11);;
gap> SetConjugate(coll, 8, 5, F.9);;
gap> SetConjugate(coll, 8, 6, F.8*F.10);;
gap> SetConjugate(coll, 8, 7, F.8*F.10*F.11);;
gap> SetConjugate(coll, 9, 2, F.7*F.9*F.11);;
gap> SetConjugate(coll, 9, 5, F.8*F.9);;
gap> SetConjugate(coll, 9, 6, F.9*F.11);;
gap> SetConjugate(coll, 9, 7, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);