Finite Group DatabasebetaFinite Group Databasebeta| order | 3072 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C23 |
No aliases for this group.
1156 conjugacy classes of subgroups; only the 127 normal subgroups are shown.
This group has 1156 conjugacy classes of subgroups — too many to draw. Only its 127 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)(5,8,10,25,11,26,33,20,12,27,9,28,29,34,30,13)(6,14,16,23,18,21,31,24,19,35,36,22,17,32,15,7);(5,7)(6,13)(8,15)(9,21)(10,32)(11,22)(12,24)(14,30)(16,34)(17,25)(18,28)(19,20)(23,29)(26,36)(27,31)(33,35);(2,4)(5,6,9,15,11,17,30,36,12,19,10,31,29,18,33,16)(7,20,21,27,22,28,14,34,24,13,32,8,23,25,35,26)Polycyclic presentation:
2^8,3,2^2|a^2=d,c^2=e,d^2=f,e^2=g,f^2=h,g^2=h,b^a=bh,c^a=ce,c^b=cegh,d^c=dh,e^a=eg,e^b=egh,g^a=gh,g^b=gh,i^a=i^2,j^a=jk,j^i=k,k^i=jk124 conjugacy classes — view the full character table.
| maximal subgroup of | #72342952, #72342975, S4×#12068 |
| maximal normal subgroup of | #72342952, #72342975, 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);;
gap> SetPower(coll, 4, F.6);;
gap> SetPower(coll, 5, F.7);;
gap> SetPower(coll, 6, F.8);;
gap> SetPower(coll, 7, F.8);;
gap> SetConjugate(coll, 2, 1, F.2*F.8);;
gap> SetConjugate(coll, 3, 1, F.3*F.5);;
gap> SetConjugate(coll, 3, 2, F.3*F.5*F.7*F.8);;
gap> SetConjugate(coll, 4, 3, F.4*F.8);;
gap> SetConjugate(coll, 5, 1, F.5*F.7);;
gap> SetConjugate(coll, 5, 2, F.5*F.7*F.8);;
gap> SetConjugate(coll, 7, 1, F.7*F.8);;
gap> SetConjugate(coll, 7, 2, F.7*F.8);;
gap> SetConjugate(coll, 9, 1, F.9^2);;
gap> SetConjugate(coll, 10, 1, 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);