Finite Group DatabasebetaFinite Group Databasebeta| order | 3072 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Fitting | #392665 ⋊ D3 |
3779 conjugacy classes of subgroups; only the 69 normal subgroups are shown.
This group has 3779 conjugacy classes of subgroups — too many to draw. Only its 69 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) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(2,4)(5,33,30,27,10,20,15,35)(6,29,34,22,11,28,26,9)(7,36,8,25,17,31,19,14)(12,21,13,32,23,18,24,16);(1,2)(3,4)(8,19)(9,22)(13,24)(14,25)(15,30)(16,32)(18,21)(20,33)(26,34)(27,35)(28,29)(31,36);(6,11)(8,18)(9,22)(12,23)(13,20)(15,29)(16,32)(19,21)(24,33)(26,31)(28,30)(34,36)Polycyclic presentation:
2^5,3,2^5|c^2=e,e^2=k,c^a=ck,c^b=cd,f^a=f^2,f^c=f^2,g^a=ghijk,g^b=gik,g^c=gh,g^f=h,h^a=hjk,h^b=hjk,h^f=gh,i^a=ij,i^c=ij,i^f=j,i^h=ik,j^f=ij,j^g=jk94 conjugacy classes — view the full character table.
| maximal subgroup of | #72343956, #72343985 |
| maximal normal subgroup of | #72343956, #72343985 |
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, 3, F.5);;
gap> SetPower(coll, 5, F.11);;
gap> SetConjugate(coll, 3, 1, F.3*F.11);;
gap> SetConjugate(coll, 3, 2, F.3*F.4);;
gap> SetConjugate(coll, 6, 1, F.6^2);;
gap> SetConjugate(coll, 6, 3, F.6^2);;
gap> SetConjugate(coll, 7, 1, F.7*F.8*F.9*F.10*F.11);;
gap> SetConjugate(coll, 7, 2, F.7*F.9*F.11);;
gap> SetConjugate(coll, 7, 3, F.7*F.8);;
gap> SetConjugate(coll, 7, 6, F.8);;
gap> SetConjugate(coll, 8, 1, F.8*F.10*F.11);;
gap> SetConjugate(coll, 8, 2, F.8*F.10*F.11);;
gap> SetConjugate(coll, 8, 6, F.7*F.8);;
gap> SetConjugate(coll, 9, 1, F.9*F.10);;
gap> SetConjugate(coll, 9, 3, F.9*F.10);;
gap> SetConjugate(coll, 9, 6, F.10);;
gap> SetConjugate(coll, 9, 8, F.9*F.11);;
gap> SetConjugate(coll, 10, 6, F.9*F.10);;
gap> SetConjugate(coll, 10, 7, F.10*F.11);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);