Finite Group DatabasebetaFinite Group Databasebeta| order | 1536 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C26 |
No aliases for this group.
| Fitting | #62832 ⋊ D3 |
| derived | C22⋊(C42⋊C3) ⋊ C23 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | C22⋊(C42⋊C3) | C23 |
| Frattini, socle | C22 | C22×#1020 |
| Fitting | #62832 | D3 |
| LCS | G ▷ C22⋊(C42⋊C3) | |
| UCS | C1 | |
2887 conjugacy classes of subgroups; only the 42 normal subgroups are shown.
This group has 2887 conjugacy classes of subgroups — too many to draw. Only its 42 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) = 24 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,4,6,5)(2,9,10,8)(3,21,14,20,12,19,13,11)(7,24,17,15,16,22,18,23);(1,3,11)(2,7,15)(4,14,19,5,13,21)(6,12,20)(8,18,22,9,17,24)(10,16,23);(1,2)(3,7)(4,8)(5,9)(6,10)(11,15)(12,16)(13,17)(14,18)(19,22)(20,23)(21,24)Polycyclic presentation:
2^3,3,2^6|e^2=i,f^2=j,d^c=d^2,e^b=ei,e^c=fij,e^d=fi,f^b=fj,f^c=eij,f^d=ef,g^a=gj,g^c=h,g^d=gh,h^a=hi,h^c=g,h^d=g,i^c=j,i^d=j,j^c=i,j^d=ij48 conjugacy classes — view the full character table.
| maximal subgroup of | #72258063, #72261304, #72262239 |
| maximal normal subgroup of | #72262239 |
| maximal quotient of | #72237390, #72262234, #72340823, … |
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, 5, F.9);;
gap> SetPower(coll, 6, F.10);;
gap> SetConjugate(coll, 4, 3, F.4^2);;
gap> SetConjugate(coll, 5, 2, F.5*F.9);;
gap> SetConjugate(coll, 5, 3, F.6*F.9*F.10);;
gap> SetConjugate(coll, 5, 4, F.6*F.9);;
gap> SetConjugate(coll, 6, 2, F.6*F.10);;
gap> SetConjugate(coll, 6, 3, F.5*F.9*F.10);;
gap> SetConjugate(coll, 6, 4, F.5*F.6);;
gap> SetConjugate(coll, 7, 1, F.7*F.10);;
gap> SetConjugate(coll, 7, 3, F.8);;
gap> SetConjugate(coll, 7, 4, F.7*F.8);;
gap> SetConjugate(coll, 8, 1, F.8*F.9);;
gap> SetConjugate(coll, 8, 3, F.7);;
gap> SetConjugate(coll, 8, 4, F.7);;
gap> SetConjugate(coll, 9, 3, F.10);;
gap> SetConjugate(coll, 9, 4, F.10);;
gap> SetConjugate(coll, 10, 3, F.9);;
gap> SetConjugate(coll, 10, 4, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);