Finite Group DatabasebetaFinite Group Databasebeta| order | 1536 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| Sylow | C3 ⋊ #494600 |
4346 conjugacy classes of subgroups.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 4 generators on 23 points:
(6,11)(10,18)(12,19)(13,20)(21,22,23);(1,2,3,4)(10,18)(14,16)(22,23);(5,7)(6,10)(8,16)(9,15)(11,18)(12,20)(13,19)(14,17);(5,6)(7,13,15,20)(8,12)(9,11)(10,14,18,16)(17,19)Polycyclic presentation:
2^9,3|a^2=hi,c^2=ef,g^2=i,b^a=bg,c^b=ce,d^a=di,e^a=eh,f^a=fh,g^b=gi,g^c=ghi,g^e=gi,g^f=gi,h^b=hi,j^c=j^2126 conjugacy classes — view the full character table.
| maximal subgroup of | #72248347, D3×#494600, #72248603 |
| maximal normal subgroup of | #72248347, D3×#494600, #72248603 |
| maximal quotient of | #72270085 |
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, 2, 2, 2, 2, 3]);;
gap> SetPower(coll, 1, F.8*F.9);;
gap> SetPower(coll, 3, F.5*F.6);;
gap> SetPower(coll, 7, F.9);;
gap> SetConjugate(coll, 2, 1, F.2*F.7);;
gap> SetConjugate(coll, 3, 2, F.3*F.5);;
gap> SetConjugate(coll, 4, 1, F.4*F.9);;
gap> SetConjugate(coll, 5, 1, F.5*F.8);;
gap> SetConjugate(coll, 6, 1, F.6*F.8);;
gap> SetConjugate(coll, 7, 2, F.7*F.9);;
gap> SetConjugate(coll, 7, 3, F.7*F.8*F.9);;
gap> SetConjugate(coll, 7, 5, F.7*F.9);;
gap> SetConjugate(coll, 7, 6, F.7*F.9);;
gap> SetConjugate(coll, 8, 2, F.8*F.9);;
gap> SetConjugate(coll, 10, 3, F.10^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);