Finite Group DatabasebetaFinite Group Databasebeta| order | 3072 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C22 |
No aliases for this group.
658 conjugacy classes of subgroups; only the 69 normal subgroups are shown.
This group has 658 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)(5,7)(6,11)(8,21)(9,25)(10,15)(12,20)(13,29)(14,23)(16,26)(17,31)(18,33)(19,27)(22,32)(24,36)(28,30)(34,35);(2,4)(5,6,8,13,9,15,22,31,11,7,17,14,27,19,33,21,12,16,24,18,10,25,23,35,26,20,34,32,28,30,29,36)Polycyclic presentation:
2^8,3,2^2|a^2=c,c^2=e,d^2=f,e^2=g,f^2=g,g^2=h,b^a=bd,c^b=cf,d^a=dgh,d^b=dfgh,d^c=dh,e^b=eg,f^a=fh,f^b=fgh,g^b=gh,i^a=i^2,j^a=jk,j^i=k,k^i=jk100 conjugacy classes — view the full character table.
| maximal subgroup of | S4×#7389, #72342975, #72342996 |
| maximal normal subgroup of | S4×#7389, #72342975, #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.3);;
gap> SetPower(coll, 3, F.5);;
gap> SetPower(coll, 4, F.6);;
gap> SetPower(coll, 5, F.7);;
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.6);;
gap> SetConjugate(coll, 4, 1, F.4*F.7*F.8);;
gap> SetConjugate(coll, 4, 2, F.4*F.6*F.7*F.8);;
gap> SetConjugate(coll, 4, 3, F.4*F.8);;
gap> SetConjugate(coll, 5, 2, F.5*F.7);;
gap> SetConjugate(coll, 6, 1, F.6*F.8);;
gap> SetConjugate(coll, 6, 2, F.6*F.7*F.8);;
gap> SetConjugate(coll, 7, 2, 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);