Finite Group DatabasebetaFinite Group Databasebeta| order | 1536 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Fitting | #63011 ⋊ D3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, cosocle | #3549⋊C3 | C22 |
| Frattini | C23 | C2×#1020 |
| Fitting | #63011 | D3 |
| socle | C22 | #87524 |
| hypocenter | #5464 | D4 |
| LCS | G ▷ #3549⋊C3 ▷ #5464 | |
| UCS | C1 | |
1365 conjugacy classes of subgroups; only the 22 normal subgroups are shown.
This group has 1365 conjugacy classes of subgroups — too many to draw. Only its 22 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 — 2 generators on 24 points:
(1,2,3,9)(4,23,17,19)(5,11,7,6)(8,24,15,12)(10,14)(13,21)(16,22)(18,20);(1,15,21,5,4,12)(2,17,24,6,8,14)(3,16,23,9,10,20)(7,13,22,11,18,19)Polycyclic presentation:
2^3,3,2^6|b^2=c,e^2=i,f^2=ij,g^2=i,h^2=ij,b^a=bc,d^a=di,d^b=d^2ij,d^c=dij,e^a=ei,e^b=fi,e^d=fi,f^a=fij,f^b=eij,f^d=ef,g^a=gj,g^b=hij,g^c=gi,g^d=hi,g^e=gi,g^f=gj,h^a=hi,h^b=g,h^c=hij,h^d=gh,h^e=hj,h^f=hij,i^b=ij,i^d=ij,j^d=i33 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#72224760, #72261629, #72262232, … |
| maximal normal subgroup of | C2×#72224760, #72340864 |
| maximal quotient of | #72231937, #72232086, C2×#72224760, … |
| automorphism group of | #338 |
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.3);;
gap> SetPower(coll, 5, F.9);;
gap> SetPower(coll, 6, F.9*F.10);;
gap> SetPower(coll, 7, F.9);;
gap> SetPower(coll, 8, F.9*F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 4, 1, F.4*F.9);;
gap> SetConjugate(coll, 4, 2, F.4^2*F.9*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.9*F.10);;
gap> SetConjugate(coll, 5, 1, F.5*F.9);;
gap> SetConjugate(coll, 5, 2, F.6*F.9);;
gap> SetConjugate(coll, 5, 4, F.6*F.9);;
gap> SetConjugate(coll, 6, 1, F.6*F.9*F.10);;
gap> SetConjugate(coll, 6, 2, 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, 2, F.8*F.9*F.10);;
gap> SetConjugate(coll, 7, 3, F.7*F.9);;
gap> SetConjugate(coll, 7, 4, F.8*F.9);;
gap> SetConjugate(coll, 7, 5, F.7*F.9);;
gap> SetConjugate(coll, 7, 6, F.7*F.10);;
gap> SetConjugate(coll, 8, 1, F.8*F.9);;
gap> SetConjugate(coll, 8, 2, F.7);;
gap> SetConjugate(coll, 8, 3, F.8*F.9*F.10);;
gap> SetConjugate(coll, 8, 4, F.7*F.8);;
gap> SetConjugate(coll, 8, 5, F.8*F.10);;
gap> SetConjugate(coll, 8, 6, F.8*F.9*F.10);;
gap> SetConjugate(coll, 9, 2, F.9*F.10);;
gap> SetConjugate(coll, 9, 4, F.9*F.10);;
gap> SetConjugate(coll, 10, 4, F.9);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);