Finite Group DatabasebetaFinite Group Databasebeta| order | 3072 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
3390 conjugacy classes of subgroups; only the 36 normal subgroups are shown.
This group has 3390 conjugacy classes of subgroups — too many to draw. Only its 36 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 2 generators on 48 points:
(1,14,11,9,13,8,3,7)(2,5,16,12,15,4,6,10)(17,30,35,25,22,40,21,39)(18,31,33,20,24,34,26,32)(19,43,29,47,27,48,42,37)(23,46,38,28,36,44,45,41);(2,3,8)(4,13,5)(6,15,10)(7,11,16)(9,12,14)(18,23)(19,27)(20,21,34)(24,36)(25,38,40,37,26,48)(28,42,46,41,29,44)(30,47,33,43,39,45)(31,32,35)Polycyclic presentation:
2^5,3,2^5|b^2=c,d^2=k,g^2=k,h^2=k,b^a=bd,c^a=ce,d^a=dk,d^b=dek,d^c=dk,e^b=ek,f^b=f^2,g^b=ghi,g^c=gi,g^f=h,h^b=hjk,h^c=hij,h^f=ghk,h^g=hk,i^b=j,i^f=ij,j^b=i,j^f=i48 conjugacy classes — view the full character table.
| maximal quotient of | #72268929 |
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, 2, 3, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 2, F.3);;
gap> SetPower(coll, 4, F.11);;
gap> SetPower(coll, 7, F.11);;
gap> SetPower(coll, 8, F.11);;
gap> SetConjugate(coll, 2, 1, F.2*F.4);;
gap> SetConjugate(coll, 3, 1, F.3*F.5);;
gap> SetConjugate(coll, 4, 1, F.4*F.11);;
gap> SetConjugate(coll, 4, 2, F.4*F.5*F.11);;
gap> SetConjugate(coll, 4, 3, F.4*F.11);;
gap> SetConjugate(coll, 5, 2, F.5*F.11);;
gap> SetConjugate(coll, 6, 2, F.6^2);;
gap> SetConjugate(coll, 7, 2, F.7*F.8*F.9);;
gap> SetConjugate(coll, 7, 3, F.7*F.9);;
gap> SetConjugate(coll, 7, 6, F.8);;
gap> SetConjugate(coll, 8, 2, F.8*F.10*F.11);;
gap> SetConjugate(coll, 8, 3, F.8*F.9*F.10);;
gap> SetConjugate(coll, 8, 6, F.7*F.8*F.11);;
gap> SetConjugate(coll, 8, 7, F.8*F.11);;
gap> SetConjugate(coll, 9, 2, F.10);;
gap> SetConjugate(coll, 9, 6, F.9*F.10);;
gap> SetConjugate(coll, 10, 2, F.9);;
gap> SetConjugate(coll, 10, 6, F.9);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);