Finite Group DatabasebetaFinite Group Databasebeta| order | 1536 |
| automorphism | ? |
| homology | Abelianization (H1) = C3 · Schur multiplier (H2) = C4 |
No aliases for this group.
| Fitting, derived, Sylow | #346293 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, Fitting, cosocle, hypocenter | #346293 | C3 |
| Frattini | C25 | C22⋊A4 |
| socle | C22 | #73293 |
| LCS | G ▷ #346293 | |
| UCS | C1 | |
374 conjugacy classes of subgroups; only the 25 normal subgroups are shown.
This group has 374 conjugacy classes of subgroups — too many to draw. Only its 25 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 3 generators on 48 points:
(1,2,7)(3,13,21)(4,8,24)(5,10,23)(6,12,22)(9,20,14)(11,31,19)(15,36,44)(16,33,43)(17,25,46)(18,27,45)(26,47,39)(28,41,34)(29,48,38)(30,40,35)(32,42,37);(1,19,6,5)(3,15,16,37)(4,39,18,17)(7,23,22,31)(13,33)(14,38,35,34)(20,41,40,48)(21,44,43,42)(24,46,45,47)(32,36);(1,14)(2,13)(3,4)(5,38,19,34)(6,35)(7,24)(8,9)(10,32,11,36)(12,33)(15,17,37,39)(16,18)(20,21)(22,45)(23,47,31,46)(25,28,26,29)(27,30)(40,43)(41,44,48,42)Polycyclic presentation:
3,2^9|b^2=f,c^2=f,d^2=i,e^2=j,b^a=c,c^a=bcf,c^b=cf,d^a=de,d^b=dfg,d^c=dfgh,e^a=dj,e^b=efghij,e^c=eh,f^d=fi,f^e=fj,g^b=gi,g^c=gij,g^d=gi,g^e=gj,h^a=hj,h^b=hij,h^c=hj,i^a=ij,j^a=i32 conjugacy classes — view the full character table.
| derived subgroup of | #72340910, #72341073, #72341186 |
| maximal subgroup of | #72340910, #72340981, #72341029, … |
| maximal normal subgroup of | #72340910, #72341073, #72341183 |
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);;
gap> SetPower(coll, 4, F.9);;
gap> SetPower(coll, 5, F.10);;
gap> SetConjugate(coll, 2, 1, F.3);;
gap> SetConjugate(coll, 3, 1, F.2*F.3*F.6);;
gap> SetConjugate(coll, 3, 2, F.3*F.6);;
gap> SetConjugate(coll, 4, 1, F.4*F.5);;
gap> SetConjugate(coll, 4, 2, F.4*F.6*F.7);;
gap> SetConjugate(coll, 4, 3, F.4*F.6*F.7*F.8);;
gap> SetConjugate(coll, 5, 1, F.4*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.6*F.7*F.8*F.9*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.8);;
gap> SetConjugate(coll, 6, 4, F.6*F.9);;
gap> SetConjugate(coll, 6, 5, F.6*F.10);;
gap> SetConjugate(coll, 7, 2, F.7*F.9);;
gap> SetConjugate(coll, 7, 3, F.7*F.9*F.10);;
gap> SetConjugate(coll, 7, 4, F.7*F.9);;
gap> SetConjugate(coll, 7, 5, F.7*F.10);;
gap> SetConjugate(coll, 8, 1, F.8*F.10);;
gap> SetConjugate(coll, 8, 2, F.8*F.9*F.10);;
gap> SetConjugate(coll, 8, 3, F.8*F.10);;
gap> SetConjugate(coll, 9, 1, F.9*F.10);;
gap> SetConjugate(coll, 10, 1, F.9);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);