Finite Group DatabasebetaFinite Group Databasebeta| order | 1536 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting, Sylow | #123675 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22 | #1437⋊C3 |
| commutator, cosocle, hypocenter | #13794 | C2×C3 |
| Frattini | C26 | C2×A4 |
| Fitting | #123675 | C3 |
| socle | C24 | #120⋊C3 |
| prime radical | #13794⋊C3 | C2 |
| LCS | G ▷ #13794 | |
| UCS | C1 ◁ C22 | |
1216 conjugacy classes of subgroups; only the 26 normal subgroups are shown.
This group has 1216 conjugacy classes of subgroups — too many to draw. Only its 26 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 2 generators on 20 points:
(1,6,11,4,3,10,5,8,7,2,9,12)(14,19,20,17,15,18);(1,2)(3,9)(4,5)(7,12,11,10)(13,18,16,20)(14,19,17,15)Polycyclic presentation:
2,3,2^8|a^2=j,c^2=h,d^2=g,c^a=cf,c^b=cd,d^a=de,d^b=cghi,d^c=dhi,e^b=f,e^c=ehij,f^b=efhij,f^d=fhij,g^b=h,h^b=gi,i^b=gh56 conjugacy classes — view the full character table.
| maximal subgroup of | #72267501 |
| maximal normal subgroup of | #72267501 |
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.8);;
gap> SetPower(coll, 4, F.7);;
gap> SetConjugate(coll, 3, 1, F.3*F.6);;
gap> SetConjugate(coll, 3, 2, F.3*F.4);;
gap> SetConjugate(coll, 4, 1, F.4*F.5);;
gap> SetConjugate(coll, 4, 2, F.3*F.7*F.8*F.9);;
gap> SetConjugate(coll, 4, 3, F.4*F.8*F.9);;
gap> SetConjugate(coll, 5, 2, F.6);;
gap> SetConjugate(coll, 5, 3, F.5*F.8*F.9*F.10);;
gap> SetConjugate(coll, 6, 2, F.5*F.6*F.8*F.9*F.10);;
gap> SetConjugate(coll, 6, 4, F.6*F.8*F.9*F.10);;
gap> SetConjugate(coll, 7, 2, F.8);;
gap> SetConjugate(coll, 8, 2, F.7*F.9);;
gap> SetConjugate(coll, 9, 2, F.7*F.8);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);