Finite Group DatabasebetaFinite Group Databasebeta| order | 3528 |
| automorphism | ? |
| homology | Abelianization (H1) = C32×C8 · Schur multiplier (H2) = C3 |
No aliases for this group.
| derived, Sylow | C72 ⋊ C32×C8 |
| Hall | ΓL(1,𝔽8)2 ⋊ C8 |
| Hall | C7⋊(C7⋊C8) ⋊ C32 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C4 | ΓL(1,𝔽8)⋊AGL(1,𝔽7) |
| commutator, hypocenter | C72 | C32×C8 |
| Fitting, cosocle | C4×C72 | C2×C32 |
| socle | C2×C72 | C32×C4 |
| prime radical | C2×ΓL(1,𝔽8)2 | C4 |
| LCS | G ▷ C72 | |
| UCS | C1 ◁ C4 | |
112 conjugacy classes of subgroups.
| transitive degree | μt(G) = 168 |
Polycyclic presentation:
2,3^2,2^2,7^2|a^2=d,d^2=e,f^a=f^6,f^b=g,f^c=f^2,g^a=g^6,g^b=f^5g^3,g^c=g^2104 conjugacy classes — view the full character table.
| maximal subgroup of | C8×ΓL(1,𝔽8)⋊AGL(1,𝔽7), #72245290, #72245292, … |
| maximal normal subgroup of | C8×ΓL(1,𝔽8)⋊AGL(1,𝔽7), #72245290, #72245292, … |
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");;
gap> coll := SingleCollector(F, [2, 3, 3, 2, 2, 7, 7]);;
gap> SetPower(coll, 1, F.4);;
gap> SetPower(coll, 4, F.5);;
gap> SetConjugate(coll, 6, 1, F.6^6);;
gap> SetConjugate(coll, 6, 2, F.7);;
gap> SetConjugate(coll, 6, 3, F.6^2);;
gap> SetConjugate(coll, 7, 1, F.7^6);;
gap> SetConjugate(coll, 7, 2, F.6^5*F.7^3);;
gap> SetConjugate(coll, 7, 3, F.7^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);