Finite Group DatabasebetaFinite Group Databasebeta| order | 1536 |
| automorphism | ? |
| homology | Abelianization (H1) = C2 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Fitting | #62817 ⋊ D3 |
| derived | #11710730 ⋊ C2 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #11715850 |
| commutator, cosocle, hypocenter | #11710730 | C2 |
| Frattini, hypercenter | C22 | #87588 |
| Fitting | #62817 | D3 |
| LCS | G ▷ #11710730 | |
| UCS | C1 ◁ C2 ◁ C22 | |
1587 conjugacy classes of subgroups; only the 21 normal subgroups are shown.
This group has 1587 conjugacy classes of subgroups — too many to draw. Only its 21 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
Polycyclic presentation:
2,3,2^8|a^2=j,c^2=i,d^2=i,g^2=ij,h^2=ij,b^a=b^2,c^a=gij,c^b=d,d^a=gh,d^b=cdi,d^c=di,e^a=f,e^b=f,e^c=ei,f^a=e,f^b=ef,f^c=fi,f^d=fi,g^a=ci,g^b=h,g^c=gj,g^e=gij,g^f=gij,h^a=cd,h^b=ghij,h^c=hj,h^d=hj,h^f=hij,h^g=hij,i^a=ij33 conjugacy classes — view the full character table.
| maximal subgroup of | #72209832, #72209845, #72210418, … |
| maximal normal subgroup of | #72209832, #72209845, #72210418, … |
| maximal quotient of | #72209773, #72209791, C2×#72209772, … |
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, 3, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 1, F.10);;
gap> SetPower(coll, 3, F.9);;
gap> SetPower(coll, 4, F.9);;
gap> SetPower(coll, 7, F.9*F.10);;
gap> SetPower(coll, 8, F.9*F.10);;
gap> SetConjugate(coll, 2, 1, F.2^2);;
gap> SetConjugate(coll, 3, 1, F.7*F.9*F.10);;
gap> SetConjugate(coll, 3, 2, F.4);;
gap> SetConjugate(coll, 4, 1, F.7*F.8);;
gap> SetConjugate(coll, 4, 2, F.3*F.4*F.9);;
gap> SetConjugate(coll, 4, 3, F.4*F.9);;
gap> SetConjugate(coll, 5, 1, F.6);;
gap> SetConjugate(coll, 5, 2, F.6);;
gap> SetConjugate(coll, 5, 3, F.5*F.9);;
gap> SetConjugate(coll, 6, 1, F.5);;
gap> SetConjugate(coll, 6, 2, F.5*F.6);;
gap> SetConjugate(coll, 6, 3, F.6*F.9);;
gap> SetConjugate(coll, 6, 4, F.6*F.9);;
gap> SetConjugate(coll, 7, 1, F.3*F.9);;
gap> SetConjugate(coll, 7, 2, F.8);;
gap> SetConjugate(coll, 7, 3, F.7*F.10);;
gap> SetConjugate(coll, 7, 5, F.7*F.9*F.10);;
gap> SetConjugate(coll, 7, 6, F.7*F.9*F.10);;
gap> SetConjugate(coll, 8, 1, F.3*F.4);;
gap> SetConjugate(coll, 8, 2, F.7*F.8*F.9*F.10);;
gap> SetConjugate(coll, 8, 3, F.8*F.10);;
gap> SetConjugate(coll, 8, 4, F.8*F.10);;
gap> SetConjugate(coll, 8, 6, F.8*F.9*F.10);;
gap> SetConjugate(coll, 8, 7, F.8*F.9*F.10);;
gap> SetConjugate(coll, 9, 1, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);