Finite Group DatabasebetaFinite Group Databasebeta| order | 3072 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C2 |
No aliases for this group.
344 conjugacy classes of subgroups; only the 69 normal subgroups are shown.
This group has 344 conjugacy classes of subgroups — too many to draw. Only its 69 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 2 generators on 36 points:
(1,4,3)(5,6,7,11,8,14,15,23,9,12,17,24,18,26,30,28,10,13,16,22,19,27,29,36,20,25,31,21,33,34,32,35);(2,4)(6,11,12,21,13,22,25,24)(7,15,16,29)(8,18)(9,20)(14,28,26,23,27,35,34,36)(17,32,31,30)(19,33)Polycyclic presentation:
2^8,3,2^2|a^2=c,b^2=d,c^2=e,d^2=f,f^2=g,g^2=h,b^a=bd,c^b=cgh,d^a=df,d^c=dh,e^b=eh,f^a=fg,g^a=gh,i^a=i^2,j^a=jk,j^i=k,k^i=jk112 conjugacy classes — view the full character table.
| maximal subgroup of | S4×#7457, #72342975, #72343039 |
| maximal normal subgroup of | S4×#7457, #72342975, #72343039 |
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.3);;
gap> SetPower(coll, 2, F.4);;
gap> SetPower(coll, 3, F.5);;
gap> SetPower(coll, 4, F.6);;
gap> SetPower(coll, 6, F.7);;
gap> SetPower(coll, 7, F.8);;
gap> SetConjugate(coll, 2, 1, F.2*F.4);;
gap> SetConjugate(coll, 3, 2, F.3*F.7*F.8);;
gap> SetConjugate(coll, 4, 1, F.4*F.6);;
gap> SetConjugate(coll, 4, 3, F.4*F.8);;
gap> SetConjugate(coll, 5, 2, F.5*F.8);;
gap> SetConjugate(coll, 6, 1, F.6*F.7);;
gap> SetConjugate(coll, 7, 1, F.7*F.8);;
gap> SetConjugate(coll, 9, 1, F.9^2);;
gap> SetConjugate(coll, 10, 1, F.10*F.11);;
gap> SetConjugate(coll, 10, 9, F.11);;
gap> SetConjugate(coll, 11, 9, F.10*F.11);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);