Finite Group DatabasebetaFinite Group Databasebeta| order | 3072 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Fitting | C22×#1580 ⋊ D3 |
1384 conjugacy classes of subgroups; only the 105 normal subgroups are shown.
This group has 1384 conjugacy classes of subgroups — too many to draw. Only its 105 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) = 384 |
A minimal faithful permutation representation — 3 generators on 28 points:
(1,2,4)(5,7,12,11,9,10,8,6)(13,15,25,24,18,17,16,26)(14,19,23,21,22,27,20,28);(2,3)(14,19)(15,24)(16,25)(17,26)(20,21)(22,27)(23,28);(5,6,8,10,9,11,12,7)(13,14,16,20,18,22,25,23)(15,19,26,28,17,27,24,21)Polycyclic presentation:
2^8,3,2^2|a^2=d,c^2=df,d^2=f,e^2=g,g^2=h,b^a=be,c^b=cgh,d^b=dh,e^a=eg,e^b=egh,e^c=eh,g^a=gh,g^b=gh,i^b=i^2,j^b=k,j^i=k,k^b=j,k^i=jk136 conjugacy classes — view the full character table.
| maximal subgroup of | S4×#12067 |
| maximal normal subgroup of | S4×#12067 |
| maximal quotient of | #72342940, #72342996 |
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", "k");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2]);;
gap> SetPower(coll, 1, F.4);;
gap> SetPower(coll, 3, F.4*F.6);;
gap> SetPower(coll, 4, F.6);;
gap> SetPower(coll, 5, F.7);;
gap> SetPower(coll, 7, F.8);;
gap> SetConjugate(coll, 2, 1, F.2*F.5);;
gap> SetConjugate(coll, 3, 2, F.3*F.7*F.8);;
gap> SetConjugate(coll, 4, 2, F.4*F.8);;
gap> SetConjugate(coll, 5, 1, F.5*F.7);;
gap> SetConjugate(coll, 5, 2, F.5*F.7*F.8);;
gap> SetConjugate(coll, 5, 3, F.5*F.8);;
gap> SetConjugate(coll, 7, 1, F.7*F.8);;
gap> SetConjugate(coll, 7, 2, F.7*F.8);;
gap> SetConjugate(coll, 9, 2, F.9^2);;
gap> SetConjugate(coll, 10, 2, F.11);;
gap> SetConjugate(coll, 10, 9, F.11);;
gap> SetConjugate(coll, 11, 2, F.10);;
gap> SetConjugate(coll, 11, 9, F.10*F.11);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);