Finite Group DatabasebetaFinite Group Databasebeta| order | 1536 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
994 conjugacy classes of subgroups; only the 35 normal subgroups are shown.
This group has 994 conjugacy classes of subgroups — too many to draw. Only its 35 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 2 generators on 36 points:
(2,4)(5,19,16,22,10,31,30,33)(6,26,23,18,14,35,34,8)(7,29,21,25,17,36,9,12)(11,32,27,28,24,20,13,15);(1,2)(3,4)(8,21,20,18,9,32)(12,27,19,25,13,31)(15,30,26,28,16,35)(22,34,29,33,23,36)Polycyclic presentation:
2^4,3,2^5|b^2=d,d^2=j,b^a=bc,e^b=e^2,f^a=fhj,f^b=fg,f^e=g,g^a=gij,g^e=fg,h^b=hi,h^e=i,h^g=hj,i^e=hi,i^f=ij62 conjugacy classes — view the full character table.
| maximal subgroup of | #72343900, #72343905, #72343908, … |
| maximal normal subgroup of | #72343900, #72343905, #72343908, … |
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, 2, F.4);;
gap> SetPower(coll, 4, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 5, 2, F.5^2);;
gap> SetConjugate(coll, 6, 1, F.6*F.8*F.10);;
gap> SetConjugate(coll, 6, 2, F.6*F.7);;
gap> SetConjugate(coll, 6, 5, F.7);;
gap> SetConjugate(coll, 7, 1, F.7*F.9*F.10);;
gap> SetConjugate(coll, 7, 5, F.6*F.7);;
gap> SetConjugate(coll, 8, 2, F.8*F.9);;
gap> SetConjugate(coll, 8, 5, F.9);;
gap> SetConjugate(coll, 8, 7, F.8*F.10);;
gap> SetConjugate(coll, 9, 5, F.8*F.9);;
gap> SetConjugate(coll, 9, 6, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);