Finite Group DatabasebetaFinite Group Databasebeta| order | 1536 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| Fitting | C3×#20609 ⋊ C2 |
| 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)(7,15)(9,16)(18,19)(21,22,23);(1,3,4,2)(7,14,15,20)(9,16)(10,18,13,19)(12,17)(22,23);(5,7,8,15)(6,10,11,13)(9,14,16,20)(12,18,17,19);(1,2,4,3)(5,6,8,11)(7,13,15,10)(9,17,16,12)(14,19,20,18)Polycyclic presentation:
2^9,3|a^2=e,b^2=i,c^2=ehi,g^2=i,b^a=bef,c^b=cefgi,d^a=di,d^b=di,d^c=dh,f^c=fh,g^a=ghi,g^c=ghi,g^d=gi,g^f=gi,h^b=hi,j^c=j^2126 conjugacy classes — view the full character table.
| maximal subgroup of | #72248482, D3×#494600, #72248627 |
| maximal normal subgroup of | #72248482, D3×#494600, #72248627 |
| maximal quotient of | #72270346 |
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.5);;
gap> SetPower(coll, 2, F.9);;
gap> SetPower(coll, 3, F.5*F.8*F.9);;
gap> SetPower(coll, 7, F.9);;
gap> SetConjugate(coll, 2, 1, F.2*F.5*F.6);;
gap> SetConjugate(coll, 3, 2, F.3*F.5*F.6*F.7*F.9);;
gap> SetConjugate(coll, 4, 1, F.4*F.9);;
gap> SetConjugate(coll, 4, 2, F.4*F.9);;
gap> SetConjugate(coll, 4, 3, F.4*F.8);;
gap> SetConjugate(coll, 6, 3, F.6*F.8);;
gap> SetConjugate(coll, 7, 1, F.7*F.8*F.9);;
gap> SetConjugate(coll, 7, 3, F.7*F.8*F.9);;
gap> SetConjugate(coll, 7, 4, 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);