Finite Group DatabasebetaFinite Group Databasebeta| order | 2048 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #97512 |
| commutator | C4×#95 | C22×C4 |
| Frattini, cosocle | C2×C4×#95 | C23 |
| prime radical | C25 | B(4,𝔽2) |
| LCS | G ▷ C4×#95 ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C2×C4×#95 ◁ G | |
| LECS | G ▷ C2×C4×#95 ▷ C23×C4 ▷ C22 ▷ C1 | |
2672 conjugacy classes of subgroups; only the 116 normal subgroups are shown.
This group has 2672 conjugacy classes of subgroups — too many to draw. Only its 116 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) = 64 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2)(3,11)(4,14)(5,9)(6,16)(7,12)(8,13)(10,15)(22,30,31,32);(1,3,7,6)(2,8,12,14)(4,15,13,5)(9,11,10,16)(17,18,21,24)(19,22,26,31)(20,28,29,23)(25,32,27,30);(1,4,7,13)(2,9,12,10)(3,5,6,15)(8,16,14,11)(17,19,21,26)(18,22,24,31)(20,27,29,25)(23,32,28,30)Polycyclic presentation:
2^11|a^2=d,b^2=jk,c^2=jk,e^2=gj,f^2=hj,i^2=k,b^a=be,c^a=cf,c^b=cj,d^b=dg,d^c=dh,e^a=ej,e^b=egj,e^c=egik,f^a=fj,f^b=fgijk,f^c=fhj,f^e=fj,g^c=gk,h^b=hk,i^a=ij,i^b=ik122 conjugacy classes — view the full character table.
| Fitting subgroup of | #72227442 |
| maximal subgroup of | #72227442 |
| maximal normal subgroup of | #72227442 |
| maximal quotient of | #72268640 |
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, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 1, F.4);;
gap> SetPower(coll, 2, F.10*F.11);;
gap> SetPower(coll, 3, F.10*F.11);;
gap> SetPower(coll, 5, F.7*F.10);;
gap> SetPower(coll, 6, F.8*F.10);;
gap> SetPower(coll, 9, F.11);;
gap> SetConjugate(coll, 2, 1, F.2*F.5);;
gap> SetConjugate(coll, 3, 1, F.3*F.6);;
gap> SetConjugate(coll, 3, 2, F.3*F.10);;
gap> SetConjugate(coll, 4, 2, F.4*F.7);;
gap> SetConjugate(coll, 4, 3, F.4*F.8);;
gap> SetConjugate(coll, 5, 1, F.5*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.7*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.7*F.9*F.11);;
gap> SetConjugate(coll, 6, 1, F.6*F.10);;
gap> SetConjugate(coll, 6, 2, F.6*F.7*F.9*F.10*F.11);;
gap> SetConjugate(coll, 6, 3, F.6*F.8*F.10);;
gap> SetConjugate(coll, 6, 5, F.6*F.10);;
gap> SetConjugate(coll, 7, 3, F.7*F.11);;
gap> SetConjugate(coll, 8, 2, F.8*F.11);;
gap> SetConjugate(coll, 9, 1, F.9*F.10);;
gap> SetConjugate(coll, 9, 2, F.9*F.11);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);