Finite Group DatabasebetaFinite Group Databasebeta| order | 1536 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Fitting | C22×H(3,𝔽4) ⋊ D3 |
1820 conjugacy classes of subgroups; only the 59 normal subgroups are shown.
This group has 1820 conjugacy classes of subgroups — too many to draw. Only its 59 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) = 48 |
A minimal faithful permutation representation — 3 generators on 20 points:
(2,4)(5,7,12)(8,15,20)(9,13,19)(10,11,17)(14,18,16);(6,10)(7,13)(11,17)(12,18)(14,15)(19,20);(1,2)(3,4)(6,7)(8,9)(10,15)(11,14)(13,17)(18,19)Polycyclic presentation:
2^5,3,2^4|b^a=bd,c^a=ce,f^a=f^2,g^a=h,g^b=gi,g^d=gj,g^f=gh,h^a=g,h^b=hij,h^d=hi,h^f=g,i^a=j,i^f=j,j^a=i,j^f=ij49 conjugacy classes — view the full character table.
| maximal subgroup of | #72258621, C2×#72258617, #72340897, … |
| maximal normal subgroup of | C2×#72258617 |
| maximal quotient of | C2×#72258617 |
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> SetConjugate(coll, 2, 1, F.2*F.4);;
gap> SetConjugate(coll, 3, 1, F.3*F.5);;
gap> SetConjugate(coll, 6, 1, F.6^2);;
gap> SetConjugate(coll, 7, 1, F.8);;
gap> SetConjugate(coll, 7, 2, F.7*F.9);;
gap> SetConjugate(coll, 7, 4, F.7*F.10);;
gap> SetConjugate(coll, 7, 6, F.7*F.8);;
gap> SetConjugate(coll, 8, 1, F.7);;
gap> SetConjugate(coll, 8, 2, F.8*F.9*F.10);;
gap> SetConjugate(coll, 8, 4, F.8*F.9);;
gap> SetConjugate(coll, 8, 6, F.7);;
gap> SetConjugate(coll, 9, 1, F.10);;
gap> SetConjugate(coll, 9, 6, F.10);;
gap> SetConjugate(coll, 10, 1, F.9);;
gap> SetConjugate(coll, 10, 6, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);