Finite Group DatabasebetaFinite Group Databasebeta| order | 1536 |
| automorphism | ? |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C24 |
No aliases for this group.
3015 conjugacy classes of subgroups; only the 73 normal subgroups are shown.
This group has 3015 conjugacy classes of subgroups — too many to draw. Only its 73 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 2 generators on 20 points:
(1,4,3,2)(5,7,8,6)(13,14,15,18,16,17,19,20);(2,3,4)(6,7,8)(9,10,11,12)(14,17)(18,20)Polycyclic presentation:
2^5,3,2^4|a^2=c,b^2=d,d^2=e,b^a=be,f^b=f^2,g^b=h,g^f=h,h^b=g,h^f=gh,i^b=gij,i^f=j,j^b=ghj,j^f=ij112 conjugacy classes — view the full character table.
| maximal subgroup of | #97×#1020 |
| maximal normal subgroup of | #97×#1020 |
| maximal quotient of | #72341750 |
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, 2, 2, 2, 2]);;
gap> SetPower(coll, 1, F.3);;
gap> SetPower(coll, 2, F.4);;
gap> SetPower(coll, 4, F.5);;
gap> SetConjugate(coll, 2, 1, F.2*F.5);;
gap> SetConjugate(coll, 6, 2, F.6^2);;
gap> SetConjugate(coll, 7, 2, F.8);;
gap> SetConjugate(coll, 7, 6, F.8);;
gap> SetConjugate(coll, 8, 2, F.7);;
gap> SetConjugate(coll, 8, 6, F.7*F.8);;
gap> SetConjugate(coll, 9, 2, F.7*F.9*F.10);;
gap> SetConjugate(coll, 9, 6, F.10);;
gap> SetConjugate(coll, 10, 2, F.7*F.8*F.10);;
gap> SetConjugate(coll, 10, 6, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);