Finite Group DatabasebetaFinite Group Databasebeta| order | 1536 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C2 |
No aliases for this group.
783 conjugacy classes of subgroups; only the 41 normal subgroups are shown.
This group has 783 conjugacy classes of subgroups — too many to draw. Only its 41 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 2 generators on 32 points:
(1,3,5,4)(2,6,7,8)(9,18,31,13,11,30)(10,23,32)(12,20,19,22,21,25)(14,15,29)(16,28,17)(24,27,26);(1,2,4,6,5,7,3,8)(9,10)(11,17,18,26)(12,16)(13,14)(15,25)(19,23)(20,29,21,32)(22,24)(27,31)(28,30)Polycyclic presentation:
2^4,3,2^5|a^2=c,b^2=d,c^2=d,f^2=h,g^2=i,b^a=bdj,e^a=e^2hj,e^b=ehj,e^c=ehj,f^a=fgij,f^e=ghj,g^a=gi,g^e=fgj,g^f=gj,h^a=hij,h^e=i,i^e=hij84 conjugacy classes — view the full character table.
| maximal subgroup of | #72227454, #72227455, #72228212, … |
| maximal normal subgroup of | #72227454, #72227455, #72228212, … |
| maximal quotient of | C2×#72227449 |
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.4);;
gap> SetPower(coll, 6, F.8);;
gap> SetPower(coll, 7, F.9);;
gap> SetConjugate(coll, 2, 1, F.2*F.4*F.10);;
gap> SetConjugate(coll, 5, 1, F.5^2*F.8*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.8*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.8*F.10);;
gap> SetConjugate(coll, 6, 1, F.6*F.7*F.9*F.10);;
gap> SetConjugate(coll, 6, 5, F.7*F.8*F.10);;
gap> SetConjugate(coll, 7, 1, F.7*F.9);;
gap> SetConjugate(coll, 7, 5, F.6*F.7*F.10);;
gap> SetConjugate(coll, 7, 6, F.7*F.10);;
gap> SetConjugate(coll, 8, 1, F.8*F.9*F.10);;
gap> SetConjugate(coll, 8, 5, F.9);;
gap> SetConjugate(coll, 9, 5, F.8*F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);