Finite Group DatabasebetaFinite Group Databasebeta| order | 1536 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle | C22 | C22×#1020 |
| commutator, cosocle, hypocenter | #543⋊C3 | C23 |
| Fitting | #62844 | D3 |
| prime radical | #62844⋊C3 | C2 |
| hypercenter | C22⋊C4 | #1020 |
| LCS | G ▷ #543⋊C3 | |
| UCS | C1 ◁ C22 ◁ C22⋊C4 | |
1965 conjugacy classes of subgroups; only the 50 normal subgroups are shown.
This group has 1965 conjugacy classes of subgroups — too many to draw. Only its 50 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) = 64 |
A minimal faithful permutation representation — 3 generators on 48 points:
(1,2)(3,13,16,7,11,14)(4,15,10,8,9,12)(5,6)(17,28,34,19,33,45,23,31,35,20,21,46)(18,42,41,30,40,48,27,37,43,24,25,47)(22,32,36,29)(26,38,44,39);(1,9,4,3)(2,14,8,12)(5,16,11,10)(6,15,13,7)(17,19,23,20)(18,24,27,30)(21,29)(22,31)(25,38)(26,42)(28,36)(32,33)(34,45,35,46)(37,44)(39,40)(41,47,43,48);(2,6)(7,12)(8,13)(14,15)(17,18)(19,24)(20,30)(21,25)(22,26)(23,27)(28,37)(29,38)(31,42)(32,39)(33,40)(34,41)(35,43)(36,44)(45,47)(46,48)Polycyclic presentation:
2^3,3,2^6|b^2=ij,c^2=ij,g^2=ij,h^2=ij,c^a=ci,c^b=cj,d^b=d^2,e^b=ef,e^d=f,f^d=ef,g^b=hij,g^d=h,g^e=gi,g^f=gi,h^b=gij,h^d=ghij,h^f=hi,h^g=hij54 conjugacy classes — view the full character table.
| maximal quotient of | #72268158 |
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, 3, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 2, F.9*F.10);;
gap> SetPower(coll, 3, F.9*F.10);;
gap> SetPower(coll, 7, F.9*F.10);;
gap> SetPower(coll, 8, F.9*F.10);;
gap> SetConjugate(coll, 3, 1, F.3*F.9);;
gap> SetConjugate(coll, 3, 2, F.3*F.10);;
gap> SetConjugate(coll, 4, 2, F.4^2);;
gap> SetConjugate(coll, 5, 2, F.5*F.6);;
gap> SetConjugate(coll, 5, 4, F.6);;
gap> SetConjugate(coll, 6, 4, F.5*F.6);;
gap> SetConjugate(coll, 7, 2, F.8*F.9*F.10);;
gap> SetConjugate(coll, 7, 4, F.8);;
gap> SetConjugate(coll, 7, 5, F.7*F.9);;
gap> SetConjugate(coll, 7, 6, F.7*F.9);;
gap> SetConjugate(coll, 8, 2, F.7*F.9*F.10);;
gap> SetConjugate(coll, 8, 4, F.7*F.8*F.9*F.10);;
gap> SetConjugate(coll, 8, 6, F.8*F.9);;
gap> SetConjugate(coll, 8, 7, F.8*F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);