Finite Group DatabasebetaFinite Group Databasebeta| order | 1536 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | C22×M64 ⋊ D3 |
329 conjugacy classes of subgroups; only the 44 normal subgroups are shown.
This group has 329 conjugacy classes of subgroups — too many to draw. Only its 44 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:
(1,4,3,2)(5,6)(7,15)(8,20)(9,24)(10,12)(11,17)(13,22)(14,21)(16,28)(18,36)(19,29)(23,25)(26,35)(27,32)(30,34)(31,33);(2,3,4)(5,6,7,11,16,25,32,36,9,14,18,27,8,13,17,15,10,12,19,26,22,20,34,31,21,24,33,30,23,28,35,29)Polycyclic presentation:
2^7,3,2^2|a^2=c,c^2=efg,e^2=fg,f^2=g,b^a=bcd,c^b=ce,d^a=dg,d^b=dg,e^b=ef,f^b=fg,h^b=h^2,i^b=ij,i^h=j,j^h=ij92 conjugacy classes — view the full character table.
| maximal subgroup of | #72342827, #72342903, S4×#1418 |
| maximal normal subgroup of | #72342827, #72342903, S4×#1418 |
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, 3, F.5*F.6*F.7);;
gap> SetPower(coll, 5, F.6*F.7);;
gap> SetPower(coll, 6, F.7);;
gap> SetConjugate(coll, 2, 1, F.2*F.3*F.4);;
gap> SetConjugate(coll, 3, 2, F.3*F.5);;
gap> SetConjugate(coll, 4, 1, F.4*F.7);;
gap> SetConjugate(coll, 4, 2, F.4*F.7);;
gap> SetConjugate(coll, 5, 2, F.5*F.6);;
gap> SetConjugate(coll, 6, 2, F.6*F.7);;
gap> SetConjugate(coll, 8, 2, F.8^2);;
gap> SetConjugate(coll, 9, 2, F.9*F.10);;
gap> SetConjugate(coll, 9, 8, F.10);;
gap> SetConjugate(coll, 10, 8, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);