Finite Group DatabasebetaFinite Group Databasebeta| order | 1536 |
| automorphism | ? |
| homology | Abelianization (H1) = C3 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| Fitting, derived, Sylow | #93144 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, Fitting, cosocle, hypocenter | #93144 | C3 |
| Frattini | C22×#120 | A4 |
| socle | C22 | #1304⋊C3 |
| LCS | G ▷ #93144 | |
| UCS | C1 | |
313 conjugacy classes of subgroups; only the 8 normal subgroups are shown.
This group has 313 conjugacy classes of subgroups — too many to draw. Only its 8 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:
(2,8,23)(3,14,13)(5,20,19)(6,22,7)(9,10,28)(11,16,24)(12,17,26)(15,31,30)(18,27,21)(25,29,32);(1,9,20,26,7,25,4,24)(2,28,10,16,12,18,30,8)(3,11,32,21,17,13,15,29)(5,14,22,31,19,23,6,27)Polycyclic presentation:
3,2^9|b^2=e,c^2=f,e^2=i,f^2=ij,b^a=bc,c^a=bdfg,c^b=cd,d^a=dgi,d^b=dg,d^c=dh,e^a=defhj,e^c=eg,e^d=ei,f^a=eghi,f^b=fh,f^d=fij,f^e=fj,g^a=ghi,g^b=gi,g^c=gj,h^a=gj,h^b=hj,h^c=hij,i^a=j,j^a=ij16 conjugacy classes — view the full character table.
| maximal subgroup of | #72235030, #72235536, #72235539 |
| maximal normal subgroup of | #72235030, #72235536, #72235539 |
| maximal quotient of | #72236028 |
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, [3, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 2, F.5);;
gap> SetPower(coll, 3, F.6);;
gap> SetPower(coll, 5, F.9);;
gap> SetPower(coll, 6, F.9*F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 3, 1, F.2*F.4*F.6*F.7);;
gap> SetConjugate(coll, 3, 2, F.3*F.4);;
gap> SetConjugate(coll, 4, 1, F.4*F.7*F.9);;
gap> SetConjugate(coll, 4, 2, F.4*F.7);;
gap> SetConjugate(coll, 4, 3, F.4*F.8);;
gap> SetConjugate(coll, 5, 1, F.4*F.5*F.6*F.8*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.7);;
gap> SetConjugate(coll, 5, 4, F.5*F.9);;
gap> SetConjugate(coll, 6, 1, F.5*F.7*F.8*F.9);;
gap> SetConjugate(coll, 6, 2, F.6*F.8);;
gap> SetConjugate(coll, 6, 4, F.6*F.9*F.10);;
gap> SetConjugate(coll, 6, 5, F.6*F.10);;
gap> SetConjugate(coll, 7, 1, F.7*F.8*F.9);;
gap> SetConjugate(coll, 7, 2, F.7*F.9);;
gap> SetConjugate(coll, 7, 3, F.7*F.10);;
gap> SetConjugate(coll, 8, 1, F.7*F.10);;
gap> SetConjugate(coll, 8, 2, F.8*F.10);;
gap> SetConjugate(coll, 8, 3, F.8*F.9*F.10);;
gap> SetConjugate(coll, 9, 1, F.10);;
gap> SetConjugate(coll, 10, 1, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);