Finite Group DatabasebetaFinite Group Databasebeta| order | 1536 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| Fitting | C23×H(5,𝔽2) ⋊ D3 |
8437 conjugacy classes of subgroups; only the 220 normal subgroups are shown.
This group has 8437 conjugacy classes of subgroups — too many to draw. Only its 220 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 52 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 4 generators on 52 points:
(5,41,52,20,37,47,10,22,35,7,26,25)(6,14,39,28,49,34,11,29,44,12,30,45)(8,27,50,17,18,40,23,9,46,19,36,51)(13,38,24,15,33,48,21,31,42,16,32,43);(6,11)(12,28)(13,21)(14,29)(15,16)(24,42)(30,49)(31,38)(32,33)(34,45)(39,44)(43,48);(2,4);(1,2,3,4)(5,6)(7,16)(8,21)(9,24)(10,11)(12,19)(13,23)(14,25)(15,20)(17,28)(18,34)(22,39)(26,48)(27,42)(29,47)(30,50)(31,51)(32,52)(33,35)(36,45)(37,43)(38,40)(41,44)(46,49)Polycyclic presentation:
2^6,3,2^3|a^2=e,d^2=j,f^2=j,h^2=j,i^2=j,b^a=be,c^a=cj,d^a=df,f^a=fj,f^d=fj,g^a=g^2,h^a=ij,h^g=i,i^a=hj,i^g=hij,i^h=ij100 conjugacy classes — view the full character table.
| maximal subgroup of | #72228357, #72258787 |
| maximal normal subgroup of | #72258787 |
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, 3, 2, 2, 2]);;
gap> SetPower(coll, 1, F.5);;
gap> SetPower(coll, 4, F.10);;
gap> SetPower(coll, 6, F.10);;
gap> SetPower(coll, 8, F.10);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.5);;
gap> SetConjugate(coll, 3, 1, F.3*F.10);;
gap> SetConjugate(coll, 4, 1, F.4*F.6);;
gap> SetConjugate(coll, 6, 1, F.6*F.10);;
gap> SetConjugate(coll, 6, 4, F.6*F.10);;
gap> SetConjugate(coll, 7, 1, F.7^2);;
gap> SetConjugate(coll, 8, 1, F.9*F.10);;
gap> SetConjugate(coll, 8, 7, F.9);;
gap> SetConjugate(coll, 9, 1, F.8*F.10);;
gap> SetConjugate(coll, 9, 7, F.8*F.9*F.10);;
gap> SetConjugate(coll, 9, 8, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);