Finite Group DatabasebetaFinite Group Databasebeta| order | 1536 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Fitting | #20612 ⋊ D3 |
1480 conjugacy classes of subgroups; only the 49 normal subgroups are shown.
This group has 1480 conjugacy classes of subgroups — too many to draw. Only its 49 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,26,25,8,16,10,13,31,19,2,27,30,6,23,12,20,14,18,3,32,5,7,29,22)(4,15,11,28,17,9,24,21);(2,7)(8,20)(9,15)(10,18)(21,28)(22,30)(23,26)(31,32);(2,7)(4,14)(8,20)(9,23)(10,18)(11,27)(15,26)(16,17)(21,31)(22,30)(24,29)(28,32)Polycyclic presentation:
2^4,3,2^5|c^2=d,d^2=j,f^2=j,g^2=j,c^a=cj,c^b=cj,e^a=e^2,f^a=fgij,f^c=fhij,f^e=g,g^a=gh,g^c=ghj,g^e=fgj,g^f=gj,h^e=i,h^f=hj,i^a=hi,i^e=hi,i^f=ij,i^g=ij62 conjugacy classes — view the full character table.
| maximal subgroup of | #72236967, #72240998, #72241001, … |
| maximal normal subgroup of | #72236967, #72240998, #72241001, … |
| maximal quotient of | #72342063, #72343269, C2×#72236889 |
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, 3, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 3, F.4);;
gap> SetPower(coll, 4, F.10);;
gap> SetPower(coll, 6, F.10);;
gap> SetPower(coll, 7, F.10);;
gap> SetConjugate(coll, 3, 1, F.3*F.10);;
gap> SetConjugate(coll, 3, 2, F.3*F.10);;
gap> SetConjugate(coll, 5, 1, F.5^2);;
gap> SetConjugate(coll, 6, 1, F.6*F.7*F.9*F.10);;
gap> SetConjugate(coll, 6, 3, F.6*F.8*F.9*F.10);;
gap> SetConjugate(coll, 6, 5, F.7);;
gap> SetConjugate(coll, 7, 1, F.7*F.8);;
gap> SetConjugate(coll, 7, 3, F.7*F.8*F.10);;
gap> SetConjugate(coll, 7, 5, F.6*F.7*F.10);;
gap> SetConjugate(coll, 7, 6, F.7*F.10);;
gap> SetConjugate(coll, 8, 5, F.9);;
gap> SetConjugate(coll, 8, 6, F.8*F.10);;
gap> SetConjugate(coll, 9, 1, F.8*F.9);;
gap> SetConjugate(coll, 9, 5, F.8*F.9);;
gap> SetConjugate(coll, 9, 6, F.9*F.10);;
gap> SetConjugate(coll, 9, 7, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);