Finite Group DatabasebetaFinite Group Databasebeta| order | 1536 |
| automorphism | ? |
| homology | Abelianization (H1) = C3 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting, derived, Sylow | #346283 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, Fitting, cosocle, hypocenter | #346283 | C3 |
| Frattini | C25 | C22⋊A4 |
| socle | C22 | #73293 |
| LCS | G ▷ #346283 | |
| UCS | C1 | |
371 conjugacy classes of subgroups; only the 25 normal subgroups are shown.
This group has 371 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) = 48 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 3 generators on 48 points:
(1,2,7)(3,9,30)(4,16,37)(5,10,25)(6,12,24)(8,26,36)(11,33,21)(13,29,23)(14,34,48)(15,32,45)(17,38,44)(18,40,22)(19,31,47)(20,39,42)(27,46,41)(28,43,35);(1,21,6,5)(3,14,15,19)(4,20,18,17)(7,25,24,33)(9,32)(13,41,36,35)(22,42,37,44)(23,43,26,46)(30,48,45,47)(31,34);(1,13)(2,40,12,16)(3,18)(4,15)(5,41,21,35)(6,36)(7,45,24,30)(8,32,29,9)(10,39)(11,38)(14,17,19,20)(22,23,37,26)(25,47)(27,31)(28,34)(33,48)(42,43)(44,46)Polycyclic presentation:
3,2^9|b^2=f,c^2=fj,d^2=j,e^2=i,b^a=bc,c^a=bij,c^b=cfi,d^a=de,d^b=dfg,d^c=dfgh,e^a=di,e^b=ehij,e^c=efgj,f^a=fij,f^d=fj,f^e=fi,g^b=gj,g^c=gi,g^d=gj,g^e=gi,h^b=hi,h^c=hij,i^a=j,j^a=ij32 conjugacy classes — view the full character table.
| derived subgroup of | #72227560, #72227562, #72340922 |
| maximal subgroup of | #72227560, #72227561, #72340896, … |
| maximal normal subgroup of | #72227560, #72227561, #72340896, … |
| maximal quotient of | C2×#72227559 |
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.6);;
gap> SetPower(coll, 3, F.6*F.10);;
gap> SetPower(coll, 4, F.10);;
gap> SetPower(coll, 5, 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.6*F.9);;
gap> SetConjugate(coll, 4, 1, F.4*F.5);;
gap> SetConjugate(coll, 4, 2, F.4*F.6*F.7);;
gap> SetConjugate(coll, 4, 3, F.4*F.6*F.7*F.8);;
gap> SetConjugate(coll, 5, 1, F.4*F.9);;
gap> SetConjugate(coll, 5, 2, F.5*F.8*F.9*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.6*F.7*F.10);;
gap> SetConjugate(coll, 6, 1, F.6*F.9*F.10);;
gap> SetConjugate(coll, 6, 4, F.6*F.10);;
gap> SetConjugate(coll, 6, 5, F.6*F.9);;
gap> SetConjugate(coll, 7, 2, F.7*F.10);;
gap> SetConjugate(coll, 7, 3, F.7*F.9);;
gap> SetConjugate(coll, 7, 4, F.7*F.10);;
gap> SetConjugate(coll, 7, 5, F.7*F.9);;
gap> SetConjugate(coll, 8, 2, F.8*F.9);;
gap> SetConjugate(coll, 8, 3, F.8*F.9*F.10);;
gap> SetConjugate(coll, 9, 1, F.10);;
gap> SetConjugate(coll, 10, 1, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);