Finite Group DatabasebetaFinite Group Databasebeta| order | 1536 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
1629 conjugacy classes of subgroups; only the 52 normal subgroups are shown.
This group has 1629 conjugacy classes of subgroups — too many to draw. Only its 52 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 — 2 generators on 64 points:
(1,3,5,16,8,19,4,15)(2,9,10,14,13,30,22,38)(6,45,49,41,7,61,20,42,25,17,21,60,26,18,46,39,27,43,23,40,50,44,24,59)(11,53,54,36,12,63,47,37,33,28,48,56,34,29,62,57,35,51,31,58,55,52,32,64);(1,11,46,32,8,34,49,48)(2,20,54,27,13,23,62,7)(3,36,59,53,19,57,60,29)(4,31,50,35,5,47,25,12)(6,33,21,10,26,55,24,22)(9,17,56,40,30,44,64,42)(14,39,63,45,38,41,51,18)(15,52,61,58,16,28,43,37)Polycyclic presentation:
2^4,3,2^5|a^2=c,b^2=d,c^2=j,d^2=j,b^a=bd,d^a=dj,e^a=e^2,f^a=g,f^e=g,g^a=f,g^e=fg,h^a=hi,h^e=i,h^f=hj,i^e=hi,i^f=ij,i^g=ij68 conjugacy classes — view the full character table.
| maximal subgroup of | #72341787, #72341825, #72341869, … |
| maximal normal subgroup of | #72341787, #72341825, #72341869, … |
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, [2, 2, 2, 2, 3, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 1, F.3);;
gap> SetPower(coll, 2, F.4);;
gap> SetPower(coll, 3, F.10);;
gap> SetPower(coll, 4, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.4);;
gap> SetConjugate(coll, 4, 1, F.4*F.10);;
gap> SetConjugate(coll, 5, 1, F.5^2);;
gap> SetConjugate(coll, 6, 1, F.7);;
gap> SetConjugate(coll, 6, 5, F.7);;
gap> SetConjugate(coll, 7, 1, F.6);;
gap> SetConjugate(coll, 7, 5, F.6*F.7);;
gap> SetConjugate(coll, 8, 1, F.8*F.9);;
gap> SetConjugate(coll, 8, 5, F.9);;
gap> SetConjugate(coll, 8, 6, F.8*F.10);;
gap> SetConjugate(coll, 9, 5, F.8*F.9);;
gap> SetConjugate(coll, 9, 6, F.9*F.10);;
gap> SetConjugate(coll, 9, 7, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);