Finite Group DatabasebetaFinite Group Databasebeta| order | 1536 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Fitting, Sylow | #6430992 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C4 | #3549⋊C3 |
| commutator, hypocenter | #2679 | C3×C4 |
| Frattini | C22×C4 | C2×C22⋊A4 |
| Fitting | #6430992 | C3 |
| socle | C23 | C4×C22⋊A4 |
| prime radical | #2679⋊C3 | C4 |
| cosocle | #52301 | C2×C3 |
| LCS | G ▷ #2679 | |
| UCS | C1 ◁ C4 | |
629 conjugacy classes of subgroups; only the 28 normal subgroups are shown.
This group has 629 conjugacy classes of subgroups — too many to draw. Only its 28 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 3 generators on 48 points:
(1,9,33,10,19,46,5,31,15,2,16,42,8,28,41,11,3,24,4,25,37,14,22,48)(6,27,35,30,18,43,21,38,34,13,40,47,7,32,39,29,23,45,20,26,36,12,17,44);(1,7)(2,13)(4,21)(5,20)(6,8)(10,30)(11,29)(12,14)(15,36)(24,45)(33,39)(34,37)(35,41)(42,47)(43,46)(44,48);(1,20)(2,29)(3,23)(4,7)(5,6)(8,21)(9,32)(10,13)(11,12)(14,30)(15,41,37,33)(16,40)(17,22)(18,19)(24,48,46,42)(25,38)(26,31)(27,28)(34,35,36,39)(43,44,45,47)Polycyclic presentation:
2,3,2^8|a^2=c,c^2=ij,d^2=i,e^2=h,d^a=dj,d^b=de,e^a=ehij,e^b=dhij,e^d=eij,f^a=fhij,f^b=g,f^d=fhij,f^e=fi,g^a=gj,g^b=fg,g^d=ghj,g^e=gh,h^b=i,i^b=hj,j^b=hi64 conjugacy classes — view the full character table.
| maximal subgroup of | #72340588, #72340601, #72340711, … |
| maximal normal subgroup of | #72340588, #72340601, #72340711, … |
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, 3, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 1, F.3);;
gap> SetPower(coll, 3, F.9*F.10);;
gap> SetPower(coll, 4, F.9);;
gap> SetPower(coll, 5, F.8);;
gap> SetConjugate(coll, 4, 1, F.4*F.10);;
gap> SetConjugate(coll, 4, 2, F.4*F.5);;
gap> SetConjugate(coll, 5, 1, F.5*F.8*F.9*F.10);;
gap> SetConjugate(coll, 5, 2, F.4*F.8*F.9*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.9*F.10);;
gap> SetConjugate(coll, 6, 1, F.6*F.8*F.9*F.10);;
gap> SetConjugate(coll, 6, 2, F.7);;
gap> SetConjugate(coll, 6, 4, F.6*F.8*F.9*F.10);;
gap> SetConjugate(coll, 6, 5, F.6*F.9);;
gap> SetConjugate(coll, 7, 1, F.7*F.10);;
gap> SetConjugate(coll, 7, 2, F.6*F.7);;
gap> SetConjugate(coll, 7, 4, F.7*F.8*F.10);;
gap> SetConjugate(coll, 7, 5, F.7*F.8);;
gap> SetConjugate(coll, 8, 2, F.9);;
gap> SetConjugate(coll, 9, 2, F.8*F.10);;
gap> SetConjugate(coll, 10, 2, F.8*F.9);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);