Finite Group DatabasebetaFinite Group Databasebeta| order | 3072 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Fitting | C22×#1313 ⋊ D3 |
991 conjugacy classes of subgroups; only the 99 normal subgroups are shown.
This group has 991 conjugacy classes of subgroups — too many to draw. Only its 99 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,4,3)(5,8,9,16,11,25,26,17,12,14,27,24,22,15,10,6)(7,13,28,33,20,31,35,29,21,32,34,36,19,30,18,23);(2,4)(5,7)(6,13)(8,23)(9,18)(10,28)(11,19)(12,21)(14,29)(15,33)(16,30)(17,32)(20,22)(24,31)(25,36)(26,34)(27,35);(5,6,9,14,11,16,26,15,12,17,27,8,22,24,10,25)(7,13,18,29,19,30,34,33,21,32,35,23,20,31,28,36)Polycyclic presentation:
2^8,3,2^2|a^2=d,c^2=eg,d^2=f,e^2=gh,f^2=h,g^2=h,c^a=cgh,c^b=ce,d^c=dh,e^a=eh,e^b=eg,g^b=gh,i^b=i^2,j^b=jk,j^i=k,k^i=jk136 conjugacy classes — view the full character table.
| maximal subgroup of | #72342940, #72343039, S4×#12068 |
| maximal normal subgroup of | #72342940, #72343039, S4×#12068 |
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, 2, 2, 2, 3, 2, 2]);;
gap> SetPower(coll, 1, F.4);;
gap> SetPower(coll, 3, F.5*F.7);;
gap> SetPower(coll, 4, F.6);;
gap> SetPower(coll, 5, F.7*F.8);;
gap> SetPower(coll, 6, F.8);;
gap> SetPower(coll, 7, F.8);;
gap> SetConjugate(coll, 3, 1, F.3*F.7*F.8);;
gap> SetConjugate(coll, 3, 2, F.3*F.5);;
gap> SetConjugate(coll, 4, 3, F.4*F.8);;
gap> SetConjugate(coll, 5, 1, F.5*F.8);;
gap> SetConjugate(coll, 5, 2, F.5*F.7);;
gap> SetConjugate(coll, 7, 2, F.7*F.8);;
gap> SetConjugate(coll, 9, 2, F.9^2);;
gap> SetConjugate(coll, 10, 2, F.10*F.11);;
gap> SetConjugate(coll, 10, 9, F.11);;
gap> SetConjugate(coll, 11, 9, F.10*F.11);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);