Finite Group DatabasebetaFinite Group Databasebeta| order | 1536 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Fitting, Sylow | #6508708 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center, socle, hypercenter | C22 | #85645 |
| commutator, cosocle, hypocenter | #62871 | C2×C3 |
| Frattini | C24 | C2×C22⋊A4 |
| Fitting | #6508708 | C3 |
| prime radical | #62871⋊C3 | C2 |
| LCS | G ▷ #62871 | |
| UCS | C1 ◁ C22 | |
1106 conjugacy classes of subgroups; only the 26 normal subgroups are shown.
This group has 1106 conjugacy classes of subgroups — too many to draw. Only its 26 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) = 128 |
A minimal faithful permutation representation — 3 generators on 48 points:
(1,2,5,8)(3,7,9,14,4,15,10,16,11,6,12,13)(18,27,24,23,19,34)(20,21,32)(25,43,45,35,29,48)(26,44,37,36,40,46)(28,47,41,42,38,39)(30,31,33);(3,10)(6,14)(9,11)(13,15)(17,30,22,32)(18,39,23,41)(19,44,27,40)(20,33,31,21)(24,48,34,45)(25,36,35,26)(28,43,42,29)(37,47,46,38);(1,9,5,11)(2,13,8,15)(3,12,10,4)(6,16,14,7)(17,24)(18,19)(20,37)(21,38)(22,34)(23,27)(25,28)(26,29)(30,48)(31,46)(32,45)(33,47)(35,42)(36,43)(39,44)(40,41)Polycyclic presentation:
2,3,2^8|a^2=i,c^2=i,d^2=i,e^2=j,f^2=j,c^a=cg,c^b=cd,d^a=dh,d^b=c,d^c=di,e^b=ef,e^c=ei,f^b=e,f^c=fi,f^d=fi,f^e=fj,g^b=ghj,g^d=gj,h^b=g,h^c=hj44 conjugacy classes — view the full character table.
| maximal subgroup of | #72266650 |
| maximal normal subgroup of | #72266650 |
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.9);;
gap> SetPower(coll, 3, F.9);;
gap> SetPower(coll, 4, F.9);;
gap> SetPower(coll, 5, F.10);;
gap> SetPower(coll, 6, F.10);;
gap> SetConjugate(coll, 3, 1, F.3*F.7);;
gap> SetConjugate(coll, 3, 2, F.3*F.4);;
gap> SetConjugate(coll, 4, 1, F.4*F.8);;
gap> SetConjugate(coll, 4, 2, F.3);;
gap> SetConjugate(coll, 4, 3, F.4*F.9);;
gap> SetConjugate(coll, 5, 2, F.5*F.6);;
gap> SetConjugate(coll, 5, 3, F.5*F.9);;
gap> SetConjugate(coll, 6, 2, F.5);;
gap> SetConjugate(coll, 6, 3, F.6*F.9);;
gap> SetConjugate(coll, 6, 4, F.6*F.9);;
gap> SetConjugate(coll, 6, 5, F.6*F.10);;
gap> SetConjugate(coll, 7, 2, F.7*F.8*F.10);;
gap> SetConjugate(coll, 7, 4, F.7*F.10);;
gap> SetConjugate(coll, 8, 2, F.7);;
gap> SetConjugate(coll, 8, 3, F.8*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);