Finite Group DatabasebetaFinite Group Databasebeta| order | 4608 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C22×C3 |
No aliases for this group.
1482 conjugacy classes of subgroups; only the 102 normal subgroups are shown.
This group has 1482 conjugacy classes of subgroups — too many to draw. Only its 102 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 144 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,4)(2,3)(5,6,8)(10,14)(15,20)(16,21)(22,24);(2,4)(6,7)(9,10,11,15,12,16,17,22,13,14,18,20,19,21,23,24);(2,3,4)(6,7,8)Polycyclic presentation:
2^5,3^2,2^4|a^2=c,c^2=d,d^2=e,b^a=be,f^a=f^2,g^a=g^2,h^a=i,h^f=i,h^g=i,i^a=h,i^f=hi,i^g=hi,j^a=hjk,j^f=k,j^g=hij,k^a=hik,k^f=jk,k^g=hk176 conjugacy classes — view the full character table.
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, 3, 2, 2, 2, 2]);;
gap> SetPower(coll, 1, F.3);;
gap> SetPower(coll, 3, F.4);;
gap> SetPower(coll, 4, F.5);;
gap> SetConjugate(coll, 2, 1, F.2*F.5);;
gap> SetConjugate(coll, 6, 1, F.6^2);;
gap> SetConjugate(coll, 7, 1, F.7^2);;
gap> SetConjugate(coll, 8, 1, F.9);;
gap> SetConjugate(coll, 8, 6, F.9);;
gap> SetConjugate(coll, 8, 7, F.9);;
gap> SetConjugate(coll, 9, 1, F.8);;
gap> SetConjugate(coll, 9, 6, F.8*F.9);;
gap> SetConjugate(coll, 9, 7, F.8*F.9);;
gap> SetConjugate(coll, 10, 1, F.8*F.10*F.11);;
gap> SetConjugate(coll, 10, 6, F.11);;
gap> SetConjugate(coll, 10, 7, F.8*F.9*F.10);;
gap> SetConjugate(coll, 11, 1, F.8*F.9*F.11);;
gap> SetConjugate(coll, 11, 6, F.10*F.11);;
gap> SetConjugate(coll, 11, 7, F.8*F.11);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);