Finite Group DatabasebetaFinite Group Databasebeta| order | 4056 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Fitting, socle, derived, Sylow | C132 ⋊ C2×C3×C4 |
| Hall | D13⋊#261 ⋊ C3 |
| Hall | #92788 ⋊ C2×C4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, hypocenter | C132 | C2×C3×C4 |
| prime radical | #11759815⋊C2 | C2 |
| cosocle | C13⋊D13 | C22×C3 |
| LCS | G ▷ C132 | |
| UCS | C1 | |
100 conjugacy classes of subgroups.
Polycyclic presentation:
2^2,3,2,13^2|b^2=d,e^a=e^12,e^b=e^8,e^c=e^3,e^d=e^12,f^a=e^5f,f^b=f^8,f^c=f^3,f^d=f^1234 conjugacy classes — view the full character table.
| maximal subgroup of | #72214122, C2×#72210302, #72220623, … |
| maximal normal subgroup of | #72214122, C2×#72210302, #72220623, … |
| maximal quotient of | #72214067, #72214141, C2×#72210302, … |
| holomorph of | D13 |
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, 3, 2, 13, 13]);;
gap> SetPower(coll, 2, F.4);;
gap> SetConjugate(coll, 5, 1, F.5^12);;
gap> SetConjugate(coll, 5, 2, F.5^8);;
gap> SetConjugate(coll, 5, 3, F.5^3);;
gap> SetConjugate(coll, 5, 4, F.5^12);;
gap> SetConjugate(coll, 6, 1, F.5^5*F.6);;
gap> SetConjugate(coll, 6, 2, F.6^8);;
gap> SetConjugate(coll, 6, 3, F.6^3);;
gap> SetConjugate(coll, 6, 4, F.6^12);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);