Finite Group DatabasebetaFinite Group Databasebeta| order | 1536 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| Fitting, Sylow | #6407968 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C8 | #5464 |
| commutator, hypocenter | #2679 | C3×C4 |
| Frattini | C22×C4 | C2×C22⋊A4 |
| Fitting | #6407968 | C3 |
| socle | C23 | C4×C22⋊A4 |
| prime radical | #2679⋊C3 | C4 |
| cosocle | #52301 | C2×C3 |
| LCS | G ▷ #2679 | |
| UCS | C1 ◁ C8 | |
701 conjugacy classes of subgroups; only the 29 normal subgroups are shown.
This group has 701 conjugacy classes of subgroups — too many to draw. Only its 29 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:
(1,9,33,10,19,45,5,31,15,2,16,41,8,28,40,11,3,24,4,25,37,14,22,48)(6,32,35,29,18,44,21,26,34,12,39,43,7,27,38,30,23,42,20,47,36,13,17,46);(1,7)(2,13)(4,21)(5,20)(6,8)(10,30)(11,29)(12,14)(15,36)(24,44)(33,38)(34,37)(35,40)(41,46)(42,45)(43,48);(1,20)(2,29)(3,23)(4,7)(5,6)(8,21)(9,32)(10,13)(11,12)(14,30)(15,40,37,33)(16,39)(17,22)(18,19)(24,48,45,41)(25,47)(26,31)(27,28)(34,35,36,38)(42,43,44,46)Polycyclic presentation:
2,3,2^8|a^2=c,c^2=ij,d^2=i,e^2=h,d^b=de,e^b=dhij,e^d=eij,f^b=g,f^d=fhij,f^e=fi,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, #72340598, #72340711, … |
| maximal normal subgroup of | #72340588, #72340598, #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, 2, F.4*F.5);;
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, 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, 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);