Finite Group DatabasebetaFinite Group Databasebeta| order | 3888 |
| automorphism | ? |
| homology | Abelianization (H1) = C2 · Schur multiplier (H2) = C1 |
No aliases for this group.
| Fitting, socle | C34 ⋊ GL(2,𝔽3) |
| derived | #14519481 ⋊ C2 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | #14519481 | C2 |
| Fitting, socle | C34 | GL(2,𝔽3) |
| LCS | G ▷ #14519481 | |
| UCS | C1 | |
316 conjugacy classes of subgroups; only the 10 normal subgroups are shown.
This group has 316 conjugacy classes of subgroups — too many to draw. Only its 10 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
Polycyclic presentation:
2,3,2^3,3^4|c^2=e,d^2=e,b^a=b^2,c^a=de,c^b=d,d^a=ce,d^b=cde,d^c=de,f^a=g,f^b=g^2,f^c=fg,f^d=fg^2,f^e=f^2,g^a=f,g^b=fg^2,g^c=fg^2,g^d=f^2g^2,g^e=g^2,h^a=hi,h^b=i,h^c=i,h^d=hi^2,h^e=h^2,i^a=i^2,i^b=h^2i^2,i^c=h^2,i^d=h^2i^2,i^e=i^221 conjugacy classes — view the full character table.
| maximal subgroup of | #72210382, C2×#72210292, C3×#72210292, … |
| maximal normal subgroup of | #72210382, C2×#72210292, C3×#72210292, … |
| maximal quotient of | #72210405, C2×#72210292, C3×#72210292, … |
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", "i");;
gap> coll := SingleCollector(F, [2, 3, 2, 2, 2, 3, 3, 3, 3]);;
gap> SetPower(coll, 3, F.5);;
gap> SetPower(coll, 4, F.5);;
gap> SetConjugate(coll, 2, 1, F.2^2);;
gap> SetConjugate(coll, 3, 1, F.4*F.5);;
gap> SetConjugate(coll, 3, 2, F.4);;
gap> SetConjugate(coll, 4, 1, F.3*F.5);;
gap> SetConjugate(coll, 4, 2, F.3*F.4*F.5);;
gap> SetConjugate(coll, 4, 3, F.4*F.5);;
gap> SetConjugate(coll, 6, 1, F.7);;
gap> SetConjugate(coll, 6, 2, F.7^2);;
gap> SetConjugate(coll, 6, 3, F.6*F.7);;
gap> SetConjugate(coll, 6, 4, F.6*F.7^2);;
gap> SetConjugate(coll, 6, 5, F.6^2);;
gap> SetConjugate(coll, 7, 1, F.6);;
gap> SetConjugate(coll, 7, 2, F.6*F.7^2);;
gap> SetConjugate(coll, 7, 3, F.6*F.7^2);;
gap> SetConjugate(coll, 7, 4, F.6^2*F.7^2);;
gap> SetConjugate(coll, 7, 5, F.7^2);;
gap> SetConjugate(coll, 8, 1, F.8*F.9);;
gap> SetConjugate(coll, 8, 2, F.9);;
gap> SetConjugate(coll, 8, 3, F.9);;
gap> SetConjugate(coll, 8, 4, F.8*F.9^2);;
gap> SetConjugate(coll, 8, 5, F.8^2);;
gap> SetConjugate(coll, 9, 1, F.9^2);;
gap> SetConjugate(coll, 9, 2, F.8^2*F.9^2);;
gap> SetConjugate(coll, 9, 3, F.8^2);;
gap> SetConjugate(coll, 9, 4, F.8^2*F.9^2);;
gap> SetConjugate(coll, 9, 5, F.9^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);