Finite Group DatabasebetaFinite Group Databasebeta| order | 7776 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Fitting, socle | C34 ⋊ C2×GL(2,𝔽3) |
| derived | #14519481 ⋊ C22 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | #14519481 | C22 |
| Fitting, socle | C34 | C2×GL(2,𝔽3) |
| LCS | G ▷ #14519481 | |
| UCS | C1 | |
567 conjugacy classes of subgroups; only the 17 normal subgroups are shown.
This group has 567 conjugacy classes of subgroups — too many to draw. Only its 17 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
Polycyclic presentation:
2^2,3,2^3,3^4|d^2=f,e^2=f,c^b=c^2,d^b=ef,d^c=e,e^b=df,e^c=def,e^d=ef,g^a=g^2,g^b=h,g^c=h^2,g^d=gh,g^e=gh^2,g^f=g^2,h^a=h^2,h^b=g,h^c=gh^2,h^d=gh^2,h^e=g^2h^2,h^f=h^2,i^a=h^2i,i^b=ij,i^c=j,i^d=j,i^e=ij^2,i^f=i^2,j^a=g^2hj,j^b=j^2,j^c=i^2j^2,j^d=i^2,j^e=i^2j^2,j^f=j^233 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#72210382, C3×#72210382, #72220496, … |
| maximal normal subgroup of | C2×#72210382, C3×#72210382 |
| maximal quotient of | #72213815, C2×#72210382, C3×#72210382, … |
| holomorph of | C3⋊D3 |
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", "j");;
gap> coll := SingleCollector(F, [2, 2, 3, 2, 2, 2, 3, 3, 3, 3]);;
gap> SetPower(coll, 4, F.6);;
gap> SetPower(coll, 5, F.6);;
gap> SetConjugate(coll, 3, 2, F.3^2);;
gap> SetConjugate(coll, 4, 2, F.5*F.6);;
gap> SetConjugate(coll, 4, 3, F.5);;
gap> SetConjugate(coll, 5, 2, F.4*F.6);;
gap> SetConjugate(coll, 5, 3, F.4*F.5*F.6);;
gap> SetConjugate(coll, 5, 4, F.5*F.6);;
gap> SetConjugate(coll, 7, 1, F.7^2);;
gap> SetConjugate(coll, 7, 2, F.8);;
gap> SetConjugate(coll, 7, 3, F.8^2);;
gap> SetConjugate(coll, 7, 4, F.7*F.8);;
gap> SetConjugate(coll, 7, 5, F.7*F.8^2);;
gap> SetConjugate(coll, 7, 6, F.7^2);;
gap> SetConjugate(coll, 8, 1, F.8^2);;
gap> SetConjugate(coll, 8, 2, F.7);;
gap> SetConjugate(coll, 8, 3, F.7*F.8^2);;
gap> SetConjugate(coll, 8, 4, F.7*F.8^2);;
gap> SetConjugate(coll, 8, 5, F.7^2*F.8^2);;
gap> SetConjugate(coll, 8, 6, F.8^2);;
gap> SetConjugate(coll, 9, 1, F.8^2*F.9);;
gap> SetConjugate(coll, 9, 2, F.9*F.10);;
gap> SetConjugate(coll, 9, 3, F.10);;
gap> SetConjugate(coll, 9, 4, F.10);;
gap> SetConjugate(coll, 9, 5, F.9*F.10^2);;
gap> SetConjugate(coll, 9, 6, F.9^2);;
gap> SetConjugate(coll, 10, 1, F.7^2*F.8*F.10);;
gap> SetConjugate(coll, 10, 2, F.10^2);;
gap> SetConjugate(coll, 10, 3, F.9^2*F.10^2);;
gap> SetConjugate(coll, 10, 4, F.9^2);;
gap> SetConjugate(coll, 10, 5, F.9^2*F.10^2);;
gap> SetConjugate(coll, 10, 6, F.10^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);