Finite Group DatabasebetaFinite Group Databasebeta| order | 1536 |
| automorphism | ? |
| homology | Abelianization (H1) = C3 · Schur multiplier (H2) = C2×C42 |
No aliases for this group.
| Fitting, derived, Sylow | #346281 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, Fitting, cosocle, hypocenter | #346281 | C3 |
| Frattini | C25 | C22⋊A4 |
| socle | C22 | #73292 |
| LCS | G ▷ #346281 | |
| UCS | C1 | |
225 conjugacy classes of subgroups; only the 13 normal subgroups are shown.
This group has 225 conjugacy classes of subgroups — too many to draw. Only its 13 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 64 points:
(2,9,29)(3,10,15)(4,19,49)(5,21,24)(6,7,8)(11,34,59)(12,40,32)(13,18,44)(14,28,30)(16,41,62)(17,47,38)(20,52,61)(22,27,51)(23,26,56)(25,45,53)(31,58,39)(33,55,46)(35,42,63)(36,54,48)(37,50,64)(43,60,57);(1,56,6,21)(2,62,13,36)(3,60,18,45)(4,26,24,19)(5,22,27,23)(7,49,8,51)(9,63,28,40)(10,58,29,47)(11,42,39,25)(12,37,43,38)(14,55,15,61)(16,31,46,50)(17,54,34,20)(30,59,44,64)(32,52,53,41)(33,35,48,57);(1,29,7,30)(2,28,13,9)(3,14,18,15)(4,57,24,35)(5,48,26,41)(6,10,8,44)(11,54,37,16)(12,34)(17,43)(19,52,27,33)(20,38,46,39)(21,64)(22,53,23,32)(25,31)(36,60,61,63)(40,62,45,55)(42,50)(47,49)(51,58)(56,59)Polycyclic presentation:
3,2^9|b^2=f,c^2=fg,d^2=j,e^2=i,b^a=bc,c^a=bghi,c^b=cfh,d^a=de,d^b=dgij,d^c=dghij,e^a=di,e^b=eghi,e^c=ehij,f^a=fghij,f^c=fi,f^d=fij,f^e=fj,g^a=ghi,g^b=gij,g^c=gi,h^a=gj,h^b=hi,h^c=hj,i^a=j,j^a=ij16 conjugacy classes — view the full character table.
| maximal subgroup of | #72235274, #72235867 |
| maximal normal subgroup of | #72235274, #72235867 |
| maximal quotient of | #72235873 |
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.6);;
gap> SetPower(coll, 3, F.6*F.7);;
gap> SetPower(coll, 4, F.10);;
gap> SetPower(coll, 5, F.9);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 3, 1, F.2*F.7*F.8*F.9);;
gap> SetConjugate(coll, 3, 2, F.3*F.6*F.8);;
gap> SetConjugate(coll, 4, 1, F.4*F.5);;
gap> SetConjugate(coll, 4, 2, F.4*F.7*F.9*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.7*F.8*F.9*F.10);;
gap> SetConjugate(coll, 5, 1, F.4*F.9);;
gap> SetConjugate(coll, 5, 2, F.5*F.7*F.8*F.9);;
gap> SetConjugate(coll, 5, 3, F.5*F.8*F.9*F.10);;
gap> SetConjugate(coll, 6, 1, F.6*F.7*F.8*F.9*F.10);;
gap> SetConjugate(coll, 6, 3, F.6*F.9);;
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*F.10);;
gap> SetConjugate(coll, 7, 3, F.7*F.9);;
gap> SetConjugate(coll, 8, 1, F.7*F.10);;
gap> SetConjugate(coll, 8, 2, F.8*F.9);;
gap> SetConjugate(coll, 8, 3, F.8*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);