Finite Group DatabasebetaFinite Group Databasebeta| order | 3072 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Fitting, Sylow | #72227402 ⋊ C3 |
| derived | #7682542 ⋊ C2×C3 |
| subgroup | G/H | |
|---|---|---|
| center, socle, hypercenter | C23 | #85645 |
| commutator, cosocle, hypocenter | #7682542 | C2×C3 |
| Frattini | C25 | C2×C22⋊A4 |
| Fitting | #72227402 | C3 |
| LCS | G ▷ #7682542 | |
| UCS | C1 ◁ C23 | |
5037 conjugacy classes of subgroups; only the 58 normal subgroups are shown.
This group has 5037 conjugacy classes of subgroups — too many to draw. Only its 58 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 24 points:
(2,5,3)(6,8,7)(10,16,11,13,14,15)(18,21,19)(22,23,24);(3,7)(5,8)(17,23,20,21)(18,19,22,24);(1,5)(2,3)(4,8)(6,7)(9,14)(10,11)(12,16)(13,15)(17,23,20,21)(18,19,22,24)Polycyclic presentation:
2,3,2^9|c^2=k,d^2=k,e^2=k,f^2=k,c^a=cgh,c^b=d,d^a=dg,d^b=cdk,d^c=dk,e^b=f,e^c=ei,e^d=eik,f^b=efk,f^c=fk,f^d=fi,f^e=fk,g^b=hj,g^c=gj,h^b=ghj,h^c=hj,h^d=hj80 conjugacy classes — view the full character table.
| maximal subgroup of | #72227547 |
| maximal normal subgroup of | #72227547 |
| maximal quotient of | #72267915, #72268096, #72268124, … |
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");;
gap> coll := SingleCollector(F, [2, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 3, F.11);;
gap> SetPower(coll, 4, F.11);;
gap> SetPower(coll, 5, F.11);;
gap> SetPower(coll, 6, F.11);;
gap> SetConjugate(coll, 3, 1, F.3*F.7*F.8);;
gap> SetConjugate(coll, 3, 2, F.4);;
gap> SetConjugate(coll, 4, 1, F.4*F.7);;
gap> SetConjugate(coll, 4, 2, F.3*F.4*F.11);;
gap> SetConjugate(coll, 4, 3, F.4*F.11);;
gap> SetConjugate(coll, 5, 2, F.6);;
gap> SetConjugate(coll, 5, 3, F.5*F.9);;
gap> SetConjugate(coll, 5, 4, F.5*F.9*F.11);;
gap> SetConjugate(coll, 6, 2, F.5*F.6*F.11);;
gap> SetConjugate(coll, 6, 3, F.6*F.11);;
gap> SetConjugate(coll, 6, 4, F.6*F.9);;
gap> SetConjugate(coll, 6, 5, F.6*F.11);;
gap> SetConjugate(coll, 7, 2, F.8*F.10);;
gap> SetConjugate(coll, 7, 3, F.7*F.10);;
gap> SetConjugate(coll, 8, 2, F.7*F.8*F.10);;
gap> SetConjugate(coll, 8, 3, F.8*F.10);;
gap> SetConjugate(coll, 8, 4, F.8*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);