Finite Group DatabasebetaFinite Group Databasebeta| order | 1536 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Fitting | C22×#458 ⋊ D3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | #87588⋊C2 |
| commutator, cosocle, hypocenter | C22⋊#997 | C22 |
| Frattini | C23 | C2×#1020 |
| Fitting | C22×#458 | D3 |
| socle | C23 | #1020⋊C2 |
| prime radical | #11715850 | C2 |
| LCS | G ▷ C22⋊#997 | |
| UCS | C1 ◁ C2 | |
1382 conjugacy classes of subgroups; only the 20 normal subgroups are shown.
This group has 1382 conjugacy classes of subgroups — too many to draw. Only its 20 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) = 48 |
A minimal faithful permutation representation — 2 generators on 20 points:
(1,3,2)(5,6,9,12)(7,16,20,10,8,19,15,11,14,13,17,18);(3,4)(5,7,14,8)(6,10,18,11)(9,15,20,17)(12,13,19,16)Polycyclic presentation:
2^2,3,2^7|b^2=j,c^a=c^2,d^a=e,d^b=dhj,d^c=e,e^a=d,e^b=eij,e^c=de,f^c=g,g^a=fg,g^c=fg,h^a=i,h^c=i,h^e=hj,i^a=h,i^c=hi,i^d=ij40 conjugacy classes — view the full character table.
| maximal subgroup of | #72258360, #72311983 |
| maximal normal subgroup of | #72258360, #72311983 |
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, 2, F.10);;
gap> SetConjugate(coll, 3, 1, F.3^2);;
gap> SetConjugate(coll, 4, 1, F.5);;
gap> SetConjugate(coll, 4, 2, F.4*F.8*F.10);;
gap> SetConjugate(coll, 4, 3, F.5);;
gap> SetConjugate(coll, 5, 1, F.4);;
gap> SetConjugate(coll, 5, 2, F.5*F.9*F.10);;
gap> SetConjugate(coll, 5, 3, F.4*F.5);;
gap> SetConjugate(coll, 6, 3, F.7);;
gap> SetConjugate(coll, 7, 1, F.6*F.7);;
gap> SetConjugate(coll, 7, 3, F.6*F.7);;
gap> SetConjugate(coll, 8, 1, F.9);;
gap> SetConjugate(coll, 8, 3, F.9);;
gap> SetConjugate(coll, 8, 5, F.8*F.10);;
gap> SetConjugate(coll, 9, 1, F.8);;
gap> SetConjugate(coll, 9, 3, F.8*F.9);;
gap> SetConjugate(coll, 9, 4, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);