Finite Group DatabasebetaFinite Group Databasebeta| order | 6144 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C23×C42 |
No aliases for this group.
| Fitting, Sylow | #72226613 ⋊ C3 |
| derived | #72219811 ⋊ C2×C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22 | #72209713 |
| commutator, cosocle, hypocenter | #72219811 | C2×C3 |
| Frattini | C26 | C2×C22⋊A4 |
| Fitting | #72226613 | C3 |
| socle | C24 | #85645 |
| LCS | G ▷ #72219811 | |
| UCS | C1 ◁ C22 | |
At least 10039 conjugacy classes of subgroups; only the 54 normal subgroups are shown.
This group has at least 10039 conjugacy classes of subgroups — too many to draw. Only its 54 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 3 generators on 32 points:
(2,7,6,5,3,8)(11,14,15)(12,16,13)(18,19,23)(20,22,21)(24,29,31)(25,27,32)(26,28,30);(1,6,4,8)(2,7,5,3)(9,16,10,15)(11,12,13,14)(18,25)(19,28)(23,31)(24,26)(27,29)(30,32);(3,7)(6,8)(17,23)(18,19)(20,30)(21,32)(22,31)(24,28)(25,29)(26,27)Polycyclic presentation:
2,3,2^10|e^2=ik,f^2=ik,c^b=cd,d^b=c,e^a=eh,e^b=ef,e^c=ei,e^d=el,f^a=fg,f^b=e,f^c=fij,f^d=fil,f^e=fik,g^b=h,g^e=gijl,h^b=ghijl,h^f=hijl,i^b=ij,j^b=jl,k^b=jk,l^b=j88 conjugacy classes — view the full character table.
| maximal subgroup of | #72226616 |
| maximal quotient of | #72267589, #72267629 |
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", "k", "l");;
gap> coll := SingleCollector(F, [2, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 5, F.9*F.11);;
gap> SetPower(coll, 6, F.9*F.11);;
gap> SetConjugate(coll, 3, 2, F.3*F.4);;
gap> SetConjugate(coll, 4, 2, F.3);;
gap> SetConjugate(coll, 5, 1, F.5*F.8);;
gap> SetConjugate(coll, 5, 2, F.5*F.6);;
gap> SetConjugate(coll, 5, 3, F.5*F.9);;
gap> SetConjugate(coll, 5, 4, F.5*F.12);;
gap> SetConjugate(coll, 6, 1, F.6*F.7);;
gap> SetConjugate(coll, 6, 2, F.5);;
gap> SetConjugate(coll, 6, 3, F.6*F.9*F.10);;
gap> SetConjugate(coll, 6, 4, F.6*F.9*F.12);;
gap> SetConjugate(coll, 6, 5, F.6*F.9*F.11);;
gap> SetConjugate(coll, 7, 2, F.8);;
gap> SetConjugate(coll, 7, 5, F.7*F.9*F.10*F.12);;
gap> SetConjugate(coll, 8, 2, F.7*F.8*F.9*F.10*F.12);;
gap> SetConjugate(coll, 8, 6, F.8*F.9*F.10*F.12);;
gap> SetConjugate(coll, 9, 2, F.9*F.10);;
gap> SetConjugate(coll, 10, 2, F.10*F.12);;
gap> SetConjugate(coll, 11, 2, F.10*F.11);;
gap> SetConjugate(coll, 12, 2, F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);