Finite Group DatabasebetaFinite Group Databasebeta| order | 2160 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Hall | C5×3−1+2 ⋊ C4⋊C4 |
| Sylow | 3−1+2 ⋊ C4⋊AGL(1,𝔽5) |
| Sylow | C5 ⋊ (3−1+2⋊C4)⋊C4 |
172 conjugacy classes of subgroups; only the 52 normal subgroups are shown.
This group has 172 conjugacy classes of subgroups — too many to draw. Only its 52 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 180 |
Polycyclic presentation:
2^2,3,2^2,3,5,3|a^2=e,b^2=d,f^3=h,b^a=be,f^a=f^2h^2,f^c=fh,g^b=g^3,g^d=g^4,h^a=h^282 conjugacy classes — view the full character table.
| maximal subgroup of | AGL(1,𝔽5)×#611⋊C3, AGL(1,ℤ/9)×C4⋊AGL(1,𝔽5), #72247459, … |
| maximal normal subgroup of | AGL(1,𝔽5)×#611⋊C3, AGL(1,ℤ/9)×C4⋊AGL(1,𝔽5), #72247459 |
| maximal quotient of | #72254710, #72254727 |
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");;
gap> coll := SingleCollector(F, [2, 2, 3, 2, 2, 3, 5, 3]);;
gap> SetPower(coll, 1, F.5);;
gap> SetPower(coll, 2, F.4);;
gap> SetPower(coll, 6, F.8);;
gap> SetConjugate(coll, 2, 1, F.2*F.5);;
gap> SetConjugate(coll, 6, 1, F.6^2*F.8^2);;
gap> SetConjugate(coll, 6, 3, F.6*F.8);;
gap> SetConjugate(coll, 7, 2, F.7^3);;
gap> SetConjugate(coll, 7, 4, F.7^4);;
gap> SetConjugate(coll, 8, 1, F.8^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);