Finite Group DatabasebetaFinite Group Databasebeta| order | 2048 |
| nilpotency class | 5 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C8 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#1368 |
| commutator | C2×C16 | C23×C8 |
| Frattini, cosocle | #1311 | C24 |
| prime radical | C2×#41 | #334 |
| LCS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×#41 ◁ C8×#131 ◁ #479899 ◁ G | |
| LECS | G ▷ #1311 ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
812 conjugacy classes of subgroups.
| minimal degree | μ(G) = 96 |
| transitive degree | μt(G) = 1024 |
A minimal faithful permutation representation — 4 generators on 96 points:
(1,2,4,9,6,10,15,22,5,8,14,20,16,21,26,30)(3,7,12,17,13,19,25,28,11,18,23,27,24,29,31,32)(35,45,47,65,48,66,71,68)(36,50,46,69)(37,52)(38,54)(40,55,58,79,57,78,84,81)(42,62,56,82)(43,64)(44,61)(49,74,72,67,73,92,91,93)(51,76,70,75)(53,77)(59,87,86,80,85,96,95,94)(60,88)(63,90,83,89);(1,3)(2,7)(4,12)(5,11)(6,13)(8,18)(9,17)(10,19)(14,23)(15,25)(16,24)(20,27)(21,29)(22,28)(26,31)(30,32)(33,34)(35,40)(36,42)(37,43)(38,44)(39,41)(45,55)(46,56)(47,58)(48,57)(49,59)(50,62)(51,63)(52,64)(53,60)(54,61)(65,79)(66,78)(67,80)(68,81)(69,82)(70,83)(71,84)(72,86)(73,85)(74,87)(75,89)(76,90)(77,88)(91,95)(92,96)(93,94);(2,8)(3,11)(9,20)(10,21)(12,23)(13,24)(22,30)(25,31)(33,35,46,66,37,49,69,93,38,47,70,65,52,72,76,92,39,48,36,45,53,73,50,67,54,71,51,68,77,91,75,74)(34,40,56,78,43,59,82,94,44,58,83,79,64,86,90,96,41,57,42,55,60,85,62,80,61,84,63,81,88,95,89,87);(34,41)(40,57)(42,56)(43,60)(44,61)(55,78)(58,84)(59,85)(62,82)(63,83)(64,88)(79,81)(80,94)(86,95)(87,96)(89,90)Polycyclic presentation:
2^11|a^2=e,c^2=fik,e^2=g,f^2=h,g^2=i,h^2=j,j^2=k,c^a=cf,c^b=ci,d^b=dk,e^c=ejk,f^a=fh,f^e=fk,g^c=gk,h^a=hj,j^a=jk188 conjugacy classes — view the full character table.
| maximal subgroup of | #72231993, #72234318, #72323133, … |
| maximal normal subgroup of | #72231993, #72234318, #72323133, … |
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, 1, F.5);;
gap> SetPower(coll, 3, F.6*F.9*F.11);;
gap> SetPower(coll, 5, F.7);;
gap> SetPower(coll, 6, F.8);;
gap> SetPower(coll, 7, F.9);;
gap> SetPower(coll, 8, F.10);;
gap> SetPower(coll, 10, F.11);;
gap> SetConjugate(coll, 3, 1, F.3*F.6);;
gap> SetConjugate(coll, 3, 2, F.3*F.9);;
gap> SetConjugate(coll, 4, 2, F.4*F.11);;
gap> SetConjugate(coll, 5, 3, F.5*F.10*F.11);;
gap> SetConjugate(coll, 6, 1, F.6*F.8);;
gap> SetConjugate(coll, 6, 5, F.6*F.11);;
gap> SetConjugate(coll, 7, 3, F.7*F.11);;
gap> SetConjugate(coll, 8, 1, F.8*F.10);;
gap> SetConjugate(coll, 10, 1, F.10*F.11);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);