Finite Group DatabasebetaFinite Group Databasebeta| order | 13182 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C13 · Schur multiplier (H2) = C1 |
No aliases for this group.
| Fitting, Sylow | 13−1+2 ⋊ C2×C3 |
| derived | C169 ⋊ C2×C3×C13 |
| Hall | #72244717 ⋊ C2 |
| Hall | #72244716 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | C169 | C2×C3×C13 |
| Frattini, socle | C13 | C13×D13⋊C3 |
| Fitting | 13−1+2 | C2×C3 |
| LCS | G ▷ C169 | |
| UCS | C1 | |
26 conjugacy classes of subgroups.
| transitive degree | μt(G) = 169 |
Polycyclic presentation:
2,3,13^3|d^13=e,d^a=d^12e^12,d^b=d^3e^11,d^c=de^12,e^a=e^12,e^b=e^3106 conjugacy classes — view the full character table.
| maximal subgroup of | #72244721 |
| maximal normal subgroup of | #72244721 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:
gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e");;
gap> coll := SingleCollector(F, [2, 3, 13, 13, 13]);;
gap> SetPower(coll, 4, F.5);;
gap> SetConjugate(coll, 4, 1, F.4^12*F.5^12);;
gap> SetConjugate(coll, 4, 2, F.4^3*F.5^11);;
gap> SetConjugate(coll, 4, 3, F.4*F.5^12);;
gap> SetConjugate(coll, 5, 1, F.5^12);;
gap> SetConjugate(coll, 5, 2, F.5^3);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);