Finite Group DatabasebetaFinite Group Databasebeta| order | 1536 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Fitting | C22×#346 ⋊ D3 |
503 conjugacy classes of subgroups; only the 58 normal subgroups are shown.
This group has 503 conjugacy classes of subgroups — too many to draw. Only its 58 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) = 192 |
A minimal faithful permutation representation — 2 generators on 28 points:
(1,2,4,3)(5,7,9,8)(6,10,12,11)(14,20)(15,25)(16,26)(17,18)(21,22)(23,27)(24,28);(2,3,4)(5,6,8,11,9,12,7,10)(13,14,15,20,17,22,16,27,19,24,26,28,18,23,25,21)Polycyclic presentation:
2^7,3,2^2|a^2=c,b^2=ef,c^2=e,d^2=f,e^2=g,f^2=g,b^a=bd,c^b=cf,d^a=dg,d^b=dfg,e^b=eg,f^b=fg,h^b=h^2,i^b=j,i^h=j,j^b=i,j^h=ij104 conjugacy classes — view the full character table.
| maximal subgroup of | S4×#1333, #72342753, #72342810 |
| maximal normal subgroup of | S4×#1333, #72342753, #72342810 |
| maximal quotient of | #72342851, #72342860 |
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, 2, 2, 2, 2, 2, 3, 2, 2]);;
gap> SetPower(coll, 1, F.3);;
gap> SetPower(coll, 2, F.5*F.6);;
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> 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);;
gap> SetConjugate(coll, 4, 2, F.4*F.6*F.7);;
gap> SetConjugate(coll, 5, 2, F.5*F.7);;
gap> SetConjugate(coll, 6, 2, F.6*F.7);;
gap> SetConjugate(coll, 8, 2, F.8^2);;
gap> SetConjugate(coll, 9, 2, F.10);;
gap> SetConjugate(coll, 9, 8, F.10);;
gap> SetConjugate(coll, 10, 2, F.9);;
gap> SetConjugate(coll, 10, 8, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);