Finite Group DatabasebetaFinite Group Databasebeta| order | 1536 |
| automorphism | ? |
| homology | Abelianization (H1) = C3 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting, derived, Sylow | #346276 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, Fitting, cosocle, hypocenter | #346276 | C3 |
| Frattini | C25 | C22⋊A4 |
| socle | C22 | #73293 |
| LCS | G ▷ #346276 | |
| UCS | C1 | |
455 conjugacy classes of subgroups; only the 25 normal subgroups are shown.
This group has 455 conjugacy classes of subgroups — too many to draw. Only its 25 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) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(2,9,22)(3,16,15)(4,21,32)(5,12,6)(7,29,8)(10,28,27)(11,26,19)(13,18,31)(14,20,17)(24,25,30);(1,10,5,21)(2,27,26,9)(3,14,19,13)(4,15,18,28)(6,16,8,30)(7,31,12,22)(11,23,17,29)(20,32,25,24);(1,30,12,16)(2,25,28,9)(3,14,18,24)(4,15,19,26)(5,17,7,11)(6,22,29,31)(8,10,23,21)(13,20,32,27)Polycyclic presentation:
3,2^9|b^2=g,c^2=gi,d^2=fg,e^2=fgi,b^a=bc,c^a=bij,c^b=cgj,d^a=eij,d^b=df,d^c=dfgh,e^a=defgij,e^b=efghij,e^c=egh,e^d=efgij,f^a=fj,f^b=fi,f^c=fj,f^d=fi,f^e=fij,g^a=gij,g^d=gi,g^e=gij,h^a=hj,h^b=hj,h^c=hij,h^d=hj,h^e=hi,i^a=ij,j^a=i32 conjugacy classes — view the full character table.
| maximal subgroup of | #72340916, #72340953, C2×#72340913, … |
| maximal normal subgroup of | #72340953, C2×#72340913 |
| maximal quotient of | C2×#72340913 |
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, [3, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 2, F.7);;
gap> SetPower(coll, 3, F.7*F.9);;
gap> SetPower(coll, 4, F.6*F.7);;
gap> SetPower(coll, 5, F.6*F.7*F.9);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 3, 1, F.2*F.9*F.10);;
gap> SetConjugate(coll, 3, 2, F.3*F.7*F.10);;
gap> SetConjugate(coll, 4, 1, F.5*F.9*F.10);;
gap> SetConjugate(coll, 4, 2, F.4*F.6);;
gap> SetConjugate(coll, 4, 3, F.4*F.6*F.7*F.8);;
gap> SetConjugate(coll, 5, 1, F.4*F.5*F.6*F.7*F.9*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.6*F.7*F.8*F.9*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.7*F.8);;
gap> SetConjugate(coll, 5, 4, F.5*F.6*F.7*F.9*F.10);;
gap> SetConjugate(coll, 6, 1, F.6*F.10);;
gap> SetConjugate(coll, 6, 2, F.6*F.9);;
gap> SetConjugate(coll, 6, 3, F.6*F.10);;
gap> SetConjugate(coll, 6, 4, F.6*F.9);;
gap> SetConjugate(coll, 6, 5, F.6*F.9*F.10);;
gap> SetConjugate(coll, 7, 1, F.7*F.9*F.10);;
gap> SetConjugate(coll, 7, 4, F.7*F.9);;
gap> SetConjugate(coll, 7, 5, F.7*F.9*F.10);;
gap> SetConjugate(coll, 8, 1, F.8*F.10);;
gap> SetConjugate(coll, 8, 2, F.8*F.10);;
gap> SetConjugate(coll, 8, 3, F.8*F.9*F.10);;
gap> SetConjugate(coll, 8, 4, F.8*F.10);;
gap> SetConjugate(coll, 8, 5, F.8*F.9);;
gap> SetConjugate(coll, 9, 1, F.9*F.10);;
gap> SetConjugate(coll, 10, 1, F.9);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);