Finite Group DatabasebetaFinite Group Databasebeta| order | 3072 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| Fitting, Sylow | #72281182 ⋊ C3 |
3655 conjugacy classes of subgroups; only the 112 normal subgroups are shown.
This group has 3655 conjugacy classes of subgroups — too many to draw. Only its 112 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) = 128 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,3,4,2)(5,7,8,9)(6,13,14,12)(10,33,20,21,24,18,19,34,11,17,26,22)(15,36,25,27,31,30,23,35,16,29,32,28);(5,26)(6,36)(7,33)(8,24)(9,34)(10,20,19,11)(12,32)(13,31)(14,35)(15,30,23,28)(16,27,25,29)(17,18,21,22);(1,2,4,3)(5,6,8,14)(7,13,9,12)(10,23,19,15)(11,25,20,16)(17,27,21,29)(18,28,22,30)(24,32,26,31)(33,35,34,36)Polycyclic presentation:
2^3,3,2^7|a^2=e,c^2=e,f^2=k,g^2=k,h^2=k,b^a=bf,c^b=ck,f^a=fk,f^b=fk,g^a=gij,g^b=gij,g^d=h,h^a=hi,h^b=hi,h^d=ghk,h^g=hk,i^d=jk,i^g=ik,j^d=ijk,j^g=jk,j^h=jk118 conjugacy classes — view the full character table.
| maximal subgroup of | #72341808 |
| maximal normal subgroup of | #72341808 |
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, 3, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 1, F.5);;
gap> SetPower(coll, 3, F.5);;
gap> SetPower(coll, 6, F.11);;
gap> SetPower(coll, 7, F.11);;
gap> SetPower(coll, 8, F.11);;
gap> SetConjugate(coll, 2, 1, F.2*F.6);;
gap> SetConjugate(coll, 3, 2, F.3*F.11);;
gap> SetConjugate(coll, 6, 1, F.6*F.11);;
gap> SetConjugate(coll, 6, 2, F.6*F.11);;
gap> SetConjugate(coll, 7, 1, F.7*F.9*F.10);;
gap> SetConjugate(coll, 7, 2, F.7*F.9*F.10);;
gap> SetConjugate(coll, 7, 4, F.8);;
gap> SetConjugate(coll, 8, 1, F.8*F.9);;
gap> SetConjugate(coll, 8, 2, F.8*F.9);;
gap> SetConjugate(coll, 8, 4, F.7*F.8*F.11);;
gap> SetConjugate(coll, 8, 7, F.8*F.11);;
gap> SetConjugate(coll, 9, 4, F.10*F.11);;
gap> SetConjugate(coll, 9, 7, F.9*F.11);;
gap> SetConjugate(coll, 10, 4, F.9*F.10*F.11);;
gap> SetConjugate(coll, 10, 7, F.10*F.11);;
gap> SetConjugate(coll, 10, 8, F.10*F.11);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);