Finite Group DatabasebetaFinite Group Databasebeta| order | 1536 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C26 |
No aliases for this group.
| Sylow | C3 ⋊ #146654 |
2434 conjugacy classes of subgroups.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 3 generators on 23 points:
(1,2,3,4)(10,17)(14,20)(21,22,23);(5,7)(6,10,11,17)(8,14)(9,15)(12,13,18,19)(16,20)(22,23);(5,6)(7,13,15,19)(8,12)(9,11)(10,14,17,20)(16,18)Polycyclic presentation:
2^9,3|a^2=hi,b^2=e,c^2=d,g^2=i,b^a=bg,c^b=cf,e^a=ei,f^a=fh,g^c=ghi,g^f=gi,h^b=hi,j^b=j^2144 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#146654, #72248361, #72248578 |
| maximal normal subgroup of | D3×#146654, #72248361, #72248578 |
| maximal quotient of | #72253520 |
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, 2, F.5);;
gap> SetPower(coll, 3, F.4);;
gap> SetPower(coll, 7, F.9);;
gap> SetConjugate(coll, 2, 1, F.2*F.7);;
gap> SetConjugate(coll, 3, 2, F.3*F.6);;
gap> SetConjugate(coll, 5, 1, F.5*F.9);;
gap> SetConjugate(coll, 6, 1, F.6*F.8);;
gap> SetConjugate(coll, 7, 3, F.7*F.8*F.9);;
gap> SetConjugate(coll, 7, 6, F.7*F.9);;
gap> SetConjugate(coll, 8, 2, F.8*F.9);;
gap> SetConjugate(coll, 10, 2, F.10^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);