Finite Group DatabasebetaFinite Group Databasebeta| order | 1536 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Fitting | C4×#585 ⋊ D3 |
2006 conjugacy classes of subgroups; only the 42 normal subgroups are shown.
This group has 2006 conjugacy classes of subgroups — too many to draw. Only its 42 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:
(2,8,12,9)(5,19,17)(6,29,18)(10,23,32,22,26,30,25,21,24,20,11,31)(13,16,27)(14,28,15);(1,10,27,23,7,25,28,21)(2,13,31,19,12,15,30,6)(3,22,29,26,4,20,17,11)(5,24,14,9,18,32,16,8)Polycyclic presentation:
2^4,3,2^5|a^2=c,b^2=cd,c^2=j,d^2=j,b^a=bd,d^a=dj,e^a=e^2,f^a=g,f^e=g,g^a=f,g^e=fg,h^a=hi,h^e=i,h^f=hj,i^e=hi,i^f=ij,i^g=ij68 conjugacy classes — view the full character table.
| maximal subgroup of | #72341899 |
| maximal normal subgroup of | #72341899 |
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.3);;
gap> SetPower(coll, 2, F.3*F.4);;
gap> SetPower(coll, 3, F.10);;
gap> SetPower(coll, 4, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.4);;
gap> SetConjugate(coll, 4, 1, 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.7);;
gap> SetConjugate(coll, 7, 1, F.6);;
gap> SetConjugate(coll, 7, 5, F.6*F.7);;
gap> SetConjugate(coll, 8, 1, F.8*F.9);;
gap> SetConjugate(coll, 8, 5, F.9);;
gap> SetConjugate(coll, 8, 6, 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);