Finite Group DatabasebetaFinite Group Databasebeta| order | 1536 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| derived | Q8⋊SL(2,𝔽3) ⋊ C2×C4 |
1263 conjugacy classes of subgroups; only the 47 normal subgroups are shown.
This group has 1263 conjugacy classes of subgroups — too many to draw. Only its 47 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 2 generators on 40 points:
(1,16,11,6,9,8,5,15,4,2,12,14)(3,7,10,13)(17,34,29,20,27,38,24,21,40,19,28,36)(18,25,39,23,33,37,26,32,30,22,31,35);(2,6)(3,9)(4,11)(7,15)(10,12)(13,16)(18,22,26,23)(21,35,32,29)(25,40,34,37)(27,30,33,36)(28,39,31,38)Polycyclic presentation:
2^3,3,2^6|a^2=c,b^2=i,c^2=j,e^2=j,f^2=j,g^2=i,h^2=i,b^a=bij,d^a=d^2,e^a=ef,e^d=f,f^a=fj,f^d=efj,f^e=fj,g^a=gh,g^d=h,g^f=gj,h^a=hi,h^d=ghi,h^e=hj,h^g=hi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72268245, #72268251, #72268255, … |
| maximal normal subgroup of | #72268245, #72268251, #72268255, … |
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, 3, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 1, F.3);;
gap> SetPower(coll, 2, F.9);;
gap> SetPower(coll, 3, F.10);;
gap> SetPower(coll, 5, F.10);;
gap> SetPower(coll, 6, F.10);;
gap> SetPower(coll, 7, F.9);;
gap> SetPower(coll, 8, F.9);;
gap> SetConjugate(coll, 2, 1, F.2*F.9*F.10);;
gap> SetConjugate(coll, 4, 1, F.4^2);;
gap> SetConjugate(coll, 5, 1, F.5*F.6);;
gap> SetConjugate(coll, 5, 4, F.6);;
gap> SetConjugate(coll, 6, 1, F.6*F.10);;
gap> SetConjugate(coll, 6, 4, F.5*F.6*F.10);;
gap> SetConjugate(coll, 6, 5, F.6*F.10);;
gap> SetConjugate(coll, 7, 1, F.7*F.8);;
gap> SetConjugate(coll, 7, 4, F.8);;
gap> SetConjugate(coll, 7, 6, F.7*F.10);;
gap> SetConjugate(coll, 8, 1, F.8*F.9);;
gap> SetConjugate(coll, 8, 4, F.7*F.8*F.9);;
gap> SetConjugate(coll, 8, 5, F.8*F.10);;
gap> SetConjugate(coll, 8, 7, F.8*F.9);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);