Finite Group DatabasebetaFinite Group Databasebeta| order | 1536 |
| automorphism | ? |
| homology | Abelianization (H1) = C2 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| Fitting | #13827 ⋊ D3 |
| derived | #11709219 ⋊ C2 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | #11709219 | C2 |
| Frattini, socle | C24 | #1020 |
| Fitting | #13827 | D3 |
| LCS | G ▷ #11709219 | |
| UCS | C1 | |
689 conjugacy classes of subgroups; only the 13 normal subgroups are shown.
This group has 689 conjugacy classes of subgroups — too many to draw. Only its 13 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 2 generators on 28 points:
(1,3,4,8)(2,9,12,10,7,11,6,5)(13,19,26,24,18,22,27,20)(14,25,21,16,17,15,28,23);(1,2,5)(3,6,11)(4,7,10)(8,12,9)(13,17,14)(15,24,26)(16,27,19)(20,21,22)(23,28,25)Polycyclic presentation:
2,3,2^8|a^2=ij,c^2=g,d^2=h,b^a=b^2i,c^a=dg,c^b=dg,d^a=ch,d^b=cd,e^b=f,e^c=ei,e^d=ej,f^a=ef,f^b=ef,f^c=fj,f^d=fij,g^a=h,g^b=h,h^a=g,h^b=gh,i^a=j,i^b=ij,j^a=i,j^b=i29 conjugacy classes — view the full character table.
| maximal subgroup of | #72237384, #72237386 |
| maximal normal subgroup of | #72237384, #72237386 |
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.9*F.10);;
gap> SetPower(coll, 3, F.7);;
gap> SetPower(coll, 4, F.8);;
gap> SetConjugate(coll, 2, 1, F.2^2*F.9);;
gap> SetConjugate(coll, 3, 1, F.4*F.7);;
gap> SetConjugate(coll, 3, 2, F.4*F.7);;
gap> SetConjugate(coll, 4, 1, F.3*F.8);;
gap> SetConjugate(coll, 4, 2, F.3*F.4);;
gap> SetConjugate(coll, 5, 2, F.6);;
gap> SetConjugate(coll, 5, 3, F.5*F.9);;
gap> SetConjugate(coll, 5, 4, F.5*F.10);;
gap> SetConjugate(coll, 6, 1, F.5*F.6);;
gap> SetConjugate(coll, 6, 2, F.5*F.6);;
gap> SetConjugate(coll, 6, 3, F.6*F.10);;
gap> SetConjugate(coll, 6, 4, F.6*F.9*F.10);;
gap> SetConjugate(coll, 7, 1, F.8);;
gap> SetConjugate(coll, 7, 2, F.8);;
gap> SetConjugate(coll, 8, 1, F.7);;
gap> SetConjugate(coll, 8, 2, F.7*F.8);;
gap> SetConjugate(coll, 9, 1, F.10);;
gap> SetConjugate(coll, 9, 2, F.9*F.10);;
gap> SetConjugate(coll, 10, 1, F.9);;
gap> SetConjugate(coll, 10, 2, F.9);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);