Finite Group DatabasebetaFinite Group Databasebeta| order | 2704 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C132 ⋊ #41 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C2×D132 |
| commutator, socle, cosocle | C2×C132 | C23 |
| Fitting | C22×C132 | C22 |
| hypercenter | C22 | D132 |
| hypocenter | C132 | #41 |
| LCS | G ▷ C2×C132 ▷ C132 | |
| UCS | C1 ◁ C2 ◁ C22 | |
128 conjugacy classes of subgroups.
| transitive degree | μt(G) = 104 |
Polycyclic presentation:
2^4,13^2|a^2=d,b^a=bd,c^b=cd,e^a=e^12,e^c=e^12,f^a=e^2f,f^c=f^12232 conjugacy classes — view the full character table.
| maximal subgroup of | #72259252, #72259284, #72259291, … |
| maximal normal subgroup of | #72259252, #72259284, #72259291, … |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:
gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 13, 13]);;
gap> SetPower(coll, 1, F.4);;
gap> SetConjugate(coll, 2, 1, F.2*F.4);;
gap> SetConjugate(coll, 3, 2, F.3*F.4);;
gap> SetConjugate(coll, 5, 1, F.5^12);;
gap> SetConjugate(coll, 5, 3, F.5^12);;
gap> SetConjugate(coll, 6, 1, F.5^2*F.6);;
gap> SetConjugate(coll, 6, 3, F.6^12);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);