Finite Group DatabasebetaFinite Group Databasebeta| order | 1536 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Fitting | #52301 ⋊ D3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C4 | #85558 |
| commutator, cosocle, hypocenter | #2679⋊C3 | C22 |
| Frattini, socle | C23 | C2×#1020 |
| Fitting | #52301 | D3 |
| prime radical | #11714247 | C2 |
| LCS | G ▷ #2679⋊C3 | |
| UCS | C1 ◁ C4 | |
632 conjugacy classes of subgroups; only the 19 normal subgroups are shown.
This group has 632 conjugacy classes of subgroups — too many to draw. Only its 19 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) = 48 |
A minimal faithful permutation representation — 2 generators on 40 points:
(2,24,9,22,11,16,7,18)(3,12,15,4)(6,10,19,8,20,23,17,21)(13,14)(25,33,36,35,29,28,34,27)(26,37,40,39,32,31,38,30);(1,2,6)(3,8,22)(4,10,24)(5,11,20)(7,19,14)(9,17,13)(12,23,16)(15,21,18)(25,26,29,32)(27,31,34,39,28,40,35,37,36,30,33,38)Polycyclic presentation:
2^2,3,2^7|a^2=i,d^2=h,e^2=ij,f^2=hi,g^2=j,c^b=c^2,d^b=ehj,d^c=ehi,e^b=dhj,e^c=dei,e^d=ei,f^b=ghij,f^c=ghi,f^d=fhi,f^e=fhj,g^b=fhij,g^c=fg,g^d=ghj,g^e=gj,h^b=ij,h^c=ij,j^b=hi,j^c=hij40 conjugacy classes — view the full character table.
| maximal subgroup of | #72340826, #72340836, #72340842 |
| maximal normal subgroup of | #72340826, #72340836, #72340842 |
| maximal quotient of | #72266667 |
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, 3, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 1, F.9);;
gap> SetPower(coll, 4, F.8);;
gap> SetPower(coll, 5, F.9*F.10);;
gap> SetPower(coll, 6, F.8*F.9);;
gap> SetPower(coll, 7, F.10);;
gap> SetConjugate(coll, 3, 2, F.3^2);;
gap> SetConjugate(coll, 4, 2, F.5*F.8*F.10);;
gap> SetConjugate(coll, 4, 3, F.5*F.8*F.9);;
gap> SetConjugate(coll, 5, 2, F.4*F.8*F.10);;
gap> SetConjugate(coll, 5, 3, F.4*F.5*F.9);;
gap> SetConjugate(coll, 5, 4, F.5*F.9);;
gap> SetConjugate(coll, 6, 2, F.7*F.8*F.9*F.10);;
gap> SetConjugate(coll, 6, 3, F.7*F.8*F.9);;
gap> SetConjugate(coll, 6, 4, F.6*F.8*F.9);;
gap> SetConjugate(coll, 6, 5, F.6*F.8*F.10);;
gap> SetConjugate(coll, 7, 2, F.6*F.8*F.9*F.10);;
gap> SetConjugate(coll, 7, 3, F.6*F.7);;
gap> SetConjugate(coll, 7, 4, F.7*F.8*F.10);;
gap> SetConjugate(coll, 7, 5, F.7*F.10);;
gap> SetConjugate(coll, 8, 2, F.9*F.10);;
gap> SetConjugate(coll, 8, 3, F.9*F.10);;
gap> SetConjugate(coll, 10, 2, F.8*F.9);;
gap> SetConjugate(coll, 10, 3, F.8*F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);