Finite Group DatabasebetaFinite Group Databasebeta| order | 12168 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Fitting, socle, derived | C3×C132 ⋊ C2×C3×C4 |
| Hall | C3×#92788 ⋊ C2×C4 |
| Sylow | C132 ⋊ C2×C3×Dic3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, hypocenter | C3×C132 | C2×C3×C4 |
| prime radical | C3×#11759815⋊C2 | C2 |
| cosocle | C3×C13×D13 | C22×C3 |
| LCS | G ▷ C3×C132 | |
| UCS | C1 | |
184 conjugacy classes of subgroups; only the 37 normal subgroups are shown.
This group has 184 conjugacy classes of subgroups — too many to draw. Only its 37 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 156 |
Polycyclic presentation:
2^2,3,2,3,13^2|a^2=d,e^a=e^2,f^a=f^8,f^b=f^12,f^c=f^3,f^d=f^12,g^a=f^8g,g^b=g^12,g^c=g^3,g^d=f^7g72 conjugacy classes — view the full character table.
| maximal subgroup of | #72242001, D3×#72214102, #72242014 |
| maximal normal subgroup of | #72242001, D3×#72214102, #72242014 |
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, 3, 2, 3, 13, 13]);;
gap> SetPower(coll, 1, F.4);;
gap> SetConjugate(coll, 5, 1, F.5^2);;
gap> SetConjugate(coll, 6, 1, F.6^8);;
gap> SetConjugate(coll, 6, 2, F.6^12);;
gap> SetConjugate(coll, 6, 3, F.6^3);;
gap> SetConjugate(coll, 6, 4, F.6^12);;
gap> SetConjugate(coll, 7, 1, F.6^8*F.7);;
gap> SetConjugate(coll, 7, 2, F.7^12);;
gap> SetConjugate(coll, 7, 3, F.7^3);;
gap> SetConjugate(coll, 7, 4, F.6^7*F.7);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);