Finite Group DatabasebetaFinite Group Databasebeta| order | 2048 |
| nilpotency class | 5 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#2177 |
| commutator | C2×C16 | C24×C4 |
| Frattini, cosocle | C4×C16 | C25 |
| prime radical | C2×#33982 | C4 |
| LCS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×#41 ◁ #17335 ◁ #479917 ◁ G | |
| LECS | G ▷ C4×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
3306 conjugacy classes of subgroups.
| minimal degree | μ(G) = 72 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 72 points:
(11,24)(12,29)(13,16)(14,27)(17,38)(19,44)(20,23)(21,46)(22,40)(25,54)(26,28)(30,60)(31,63)(32,59)(33,58)(34,53)(35,36)(37,57)(39,65)(41,43)(42,68)(45,70)(47,72)(48,71)(49,69)(50,67)(51,64)(52,55)(56,62)(61,66);(1,2,5,4)(3,7,8,6)(9,10,15,18)(11,22,28,17)(12,19,32,42)(13,20,16,23)(14,21,36,43)(24,40,26,38)(25,48,56,39)(27,46,35,41)(29,44,59,68)(30,61,52,45)(31,47,33,49)(34,50,37,51)(53,67,57,64)(54,71,62,65)(55,70,60,66)(58,69,63,72);(1,3,4,7,5,8,2,6)(9,11,13,26,15,28,16,24)(10,17,20,40,18,22,23,38)(12,30,31,25,32,52,33,56)(14,35,34,53,36,27,37,57)(19,45,47,39,42,61,49,48)(21,46,50,64,43,41,51,67)(29,54,58,60,59,62,63,55)(44,65,69,70,68,71,72,66);(10,18)(17,22)(19,42)(20,23)(21,43)(38,40)(39,48)(41,46)(44,68)(45,61)(47,49)(50,51)(64,67)(65,71)(66,70)(69,72);(9,12,27,55,11,25,34,63,13,33,57,62,26,30,36,59,15,32,35,60,28,56,37,58,16,31,53,54,24,52,14,29)(10,19,46,70,22,48,50,72,20,49,64,65,38,61,43,68,18,42,41,66,17,39,51,69,23,47,67,71,40,45,21,44)Polycyclic presentation:
2^11|b^2=h,c^2=f,e^2=gi,f^2=h,g^2=ij,i^2=jk,j^2=k,c^a=cj,c^b=ch,d^b=dk,e^a=eg,e^c=ejk,f^a=fk,f^e=fk,g^a=gi,g^c=gk,i^a=ij,j^a=jk176 conjugacy classes — view the full character table.
| maximal subgroup of | #72322589, #72323055, #72323074, … |
| maximal normal subgroup of | #72322589, #72323055, #72323074, … |
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", "k");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 2, F.8);;
gap> SetPower(coll, 3, F.6);;
gap> SetPower(coll, 5, F.7*F.9);;
gap> SetPower(coll, 6, F.8);;
gap> SetPower(coll, 7, F.9*F.10);;
gap> SetPower(coll, 9, F.10*F.11);;
gap> SetPower(coll, 10, F.11);;
gap> SetConjugate(coll, 3, 1, F.3*F.10);;
gap> SetConjugate(coll, 3, 2, F.3*F.8);;
gap> SetConjugate(coll, 4, 2, F.4*F.11);;
gap> SetConjugate(coll, 5, 1, F.5*F.7);;
gap> SetConjugate(coll, 5, 3, F.5*F.10*F.11);;
gap> SetConjugate(coll, 6, 1, F.6*F.11);;
gap> SetConjugate(coll, 6, 5, F.6*F.11);;
gap> SetConjugate(coll, 7, 1, F.7*F.9);;
gap> SetConjugate(coll, 7, 3, F.7*F.11);;
gap> SetConjugate(coll, 9, 1, F.9*F.10);;
gap> SetConjugate(coll, 10, 1, F.10*F.11);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);