Finite Group DatabasebetaFinite Group Databasebeta| order | 9600 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Fitting | C24×C52 ⋊ C4×D3 |
| Sylow | C52 ⋊ C22⋊C4×S4 |
2974 conjugacy classes of subgroups; only the 97 normal subgroups are shown.
This group has 2974 conjugacy classes of subgroups — too many to draw. Only its 97 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 120 |
A minimal faithful permutation representation — 3 generators on 18 points:
(1,3,4)(6,8)(9,10,13,11,12)(14,15,18,17,16);(2,3)(5,6)(7,8)(10,11,12,13)(15,17,16,18);(2,3)(15,16)(17,18)Polycyclic presentation:
2^5,3,5^2,2^2|b^2=d,c^b=ce,f^a=f^2,f^b=f^2,g^a=g^4,g^b=g^3,g^d=g^4,h^a=g^2h,h^b=h^3,h^d=h^4,i^a=j,i^b=j,i^f=ij,j^a=i,j^b=i,j^f=i122 conjugacy classes — view the full character table.
| maximal subgroup of | #72260373 |
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");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 3, 5, 5, 2, 2]);;
gap> SetPower(coll, 2, F.4);;
gap> SetConjugate(coll, 3, 2, F.3*F.5);;
gap> SetConjugate(coll, 6, 1, F.6^2);;
gap> SetConjugate(coll, 6, 2, F.6^2);;
gap> SetConjugate(coll, 7, 1, F.7^4);;
gap> SetConjugate(coll, 7, 2, F.7^3);;
gap> SetConjugate(coll, 7, 4, F.7^4);;
gap> SetConjugate(coll, 8, 1, F.7^2*F.8);;
gap> SetConjugate(coll, 8, 2, F.8^3);;
gap> SetConjugate(coll, 8, 4, F.8^4);;
gap> SetConjugate(coll, 9, 1, F.10);;
gap> SetConjugate(coll, 9, 2, F.10);;
gap> SetConjugate(coll, 9, 6, F.9*F.10);;
gap> SetConjugate(coll, 10, 1, F.9);;
gap> SetConjugate(coll, 10, 2, F.9);;
gap> SetConjugate(coll, 10, 6, F.9);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);