Finite Group DatabasebetaFinite Group Databasebeta| order | 8096 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C11 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | #41×C23 ⋊ C2×C11 |
| Hall | #10625120 ⋊ C11 |
| Hall | C23⋊C11 ⋊ AGL(1,ℤ/8) |
| Sylow | C23 ⋊ C11×AGL(1,ℤ/8) |
136 conjugacy classes of subgroups; only the 49 normal subgroups are shown.
This group has 136 conjugacy classes of subgroups — too many to draw. Only its 49 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 184 |
Polycyclic presentation:
2^3,11,2^2,23|a^2=e,e^2=f,b^a=bef,c^a=cf,e^b=ef,g^a=g^22,g^b=g^22,g^d=g^18131 conjugacy classes — view the full character table.
| maximal subgroup of | AGL(1,ℤ/8)×AGL(1,𝔽23), #72247930, #72247944 |
| maximal normal subgroup of | AGL(1,ℤ/8)×AGL(1,𝔽23), #72247930, #72247944 |
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");;
gap> coll := SingleCollector(F, [2, 2, 2, 11, 2, 2, 23]);;
gap> SetPower(coll, 1, F.5);;
gap> SetPower(coll, 5, F.6);;
gap> SetConjugate(coll, 2, 1, F.2*F.5*F.6);;
gap> SetConjugate(coll, 3, 1, F.3*F.6);;
gap> SetConjugate(coll, 5, 2, F.5*F.6);;
gap> SetConjugate(coll, 7, 1, F.7^22);;
gap> SetConjugate(coll, 7, 2, F.7^22);;
gap> SetConjugate(coll, 7, 4, F.7^18);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);