Finite Group DatabasebetaFinite Group Databasebeta| order | 7488 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C13 ⋊ C3×SL(2,ℤ/4)⋊C4 |
468 conjugacy classes of subgroups; only the 53 normal subgroups are shown.
This group has 468 conjugacy classes of subgroups — too many to draw. Only its 53 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 21 |
| transitive degree | μt(G) = 156 |
A minimal faithful permutation representation — 2 generators on 21 points:
(2,4)(5,7,6)(10,15,19,16,12,21,13,14,17,11,18,20);(1,2,3,4)(7,8)(9,10,14,12,19,16,20,21,11,17,18,15,13)Polycyclic presentation:
2^2,3,2^2,3,13,2^2|a^2=e,b^2=d,b^a=be,f^a=f^2,g^b=g^8,g^c=g^3,g^d=g^12,h^f=i,i^a=hi,i^f=hi94 conjugacy classes — view the full character table.
| maximal subgroup of | GL(2,ℤ/4)×AGL(1,𝔽13) |
| maximal normal subgroup of | GL(2,ℤ/4)×AGL(1,𝔽13) |
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");;
gap> coll := SingleCollector(F, [2, 2, 3, 2, 2, 3, 13, 2, 2]);;
gap> SetPower(coll, 1, F.5);;
gap> SetPower(coll, 2, F.4);;
gap> SetConjugate(coll, 2, 1, F.2*F.5);;
gap> SetConjugate(coll, 6, 1, F.6^2);;
gap> SetConjugate(coll, 7, 2, F.7^8);;
gap> SetConjugate(coll, 7, 3, F.7^3);;
gap> SetConjugate(coll, 7, 4, F.7^12);;
gap> SetConjugate(coll, 8, 6, F.9);;
gap> SetConjugate(coll, 9, 1, F.8*F.9);;
gap> SetConjugate(coll, 9, 6, F.8*F.9);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);