Finite Group DatabasebetaFinite Group Databasebeta| order | 1536 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
5994 conjugacy classes of subgroups; only the 52 normal subgroups are shown.
This group has 5994 conjugacy classes of subgroups — too many to draw. Only its 52 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 — 2 generators on 32 points:
(1,2)(3,11)(4,30,17,9,28,21)(5,20,15,10,16,22)(6,8)(7,14)(12,29,27,23,32,19)(13,18,25,24,26,31);(1,4,15,5)(2,18,30,24)(3,12,25,13)(6,16,28,17)(7,20,31,10)(8,23,22,19)(9,29,21,11)(14,26,32,27)Polycyclic presentation:
2^4,3,2^5|a^2=d,c^2=j,b^a=bc,c^b=cj,d^b=dj,e^a=e^2,f^a=g,f^e=fg,g^a=f,g^e=f,h^a=hi,h^e=i,h^g=hj,i^e=hi,i^f=ij,i^g=ij62 conjugacy classes — view the full character table.
| maximal subgroup of | #72341458, #72341463, #72341539 |
| maximal normal subgroup of | #72341458, #72341463, #72341539 |
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, 1, F.4);;
gap> SetPower(coll, 3, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 3, 2, F.3*F.10);;
gap> SetConjugate(coll, 4, 2, F.4*F.10);;
gap> SetConjugate(coll, 5, 1, F.5^2);;
gap> SetConjugate(coll, 6, 1, F.7);;
gap> SetConjugate(coll, 6, 5, F.6*F.7);;
gap> SetConjugate(coll, 7, 1, F.6);;
gap> SetConjugate(coll, 7, 5, F.6);;
gap> SetConjugate(coll, 8, 1, F.8*F.9);;
gap> SetConjugate(coll, 8, 5, F.9);;
gap> SetConjugate(coll, 8, 7, F.8*F.10);;
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);