Finite Group DatabasebetaFinite Group Databasebeta| order | 2048 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C42 |
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 | #72227424 | C2 |
| 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 | |
3503 conjugacy classes of subgroups; only the 116 normal subgroups are shown.
This group has 3503 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)(3,10,11,7)(4,6)(5,9)(8,13)(12,16,14,15)(22,31,26,32);(2,6)(4,13)(7,15)(8,9)(10,16)(12,14)(17,18)(19,26)(20,23)(21,24)(22,28)(25,30)(27,31)(29,32);(1,3)(2,7)(4,14)(5,11)(6,15)(8,16)(9,10)(12,13)(17,19,21,28)(18,22,24,26)(20,27,30,29)(23,32,25,31)Polycyclic presentation:
2^11|a^2=d,c^2=k,e^2=hj,f^2=gj,i^2=k,b^a=be,c^a=cf,c^b=ck,d^b=dh,d^c=dg,e^a=ej,e^b=ehj,e^c=ei,f^a=fj,f^b=fij,f^c=fgj,f^e=fj,g^b=gk,h^c=hk,i^a=ij,i^b=ik,i^c=ik116 conjugacy classes — view the full character table.
| Fitting subgroup of | #72227443 |
| maximal subgroup of | #72227443 |
| maximal normal subgroup of | #72227443 |
| 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, 3, F.11);;
gap> SetPower(coll, 5, F.8*F.10);;
gap> SetPower(coll, 6, F.7*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.11);;
gap> SetConjugate(coll, 4, 2, F.4*F.8);;
gap> SetConjugate(coll, 4, 3, F.4*F.7);;
gap> SetConjugate(coll, 5, 1, F.5*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.8*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.9);;
gap> SetConjugate(coll, 6, 1, F.6*F.10);;
gap> SetConjugate(coll, 6, 2, F.6*F.9*F.10);;
gap> SetConjugate(coll, 6, 3, F.6*F.7*F.10);;
gap> SetConjugate(coll, 6, 5, F.6*F.10);;
gap> SetConjugate(coll, 7, 2, F.7*F.11);;
gap> SetConjugate(coll, 8, 3, F.8*F.11);;
gap> SetConjugate(coll, 9, 1, F.9*F.10);;
gap> SetConjugate(coll, 9, 2, F.9*F.11);;
gap> SetConjugate(coll, 9, 3, F.9*F.11);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);