Finite Group DatabasebetaFinite Group Databasebeta| order | 4200 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C2 |
No aliases for this group.
| derived, Hall | C52×C7 ⋊ C2×C3×C4 |
| Hall | #13063793 ⋊ C3 |
| Hall | C52×ΓL(1,𝔽8) ⋊ C2×C4 |
| Sylow | C52 ⋊ C2×ΓL(1,𝔽8)⋊C4 |
| Hall | ΓL(1,𝔽8) ⋊ #6036 |
| Sylow | C7 ⋊ C3×#6036 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | (C5⋊AGL(1,𝔽7))⋊D5 |
| commutator, hypocenter | C52×C7 | C2×C3×C4 |
| Fitting, socle, cosocle | C2×C52×C7 | C22×C3 |
| prime radical | C2×ΓL(1,𝔽8)×C5⋊D5 | C2 |
| LCS | G ▷ C52×C7 | |
| UCS | C1 ◁ C2 | |
168 conjugacy classes of subgroups.
| transitive degree | μt(G) = 140 |
Polycyclic presentation:
2^2,3,2,5^2,7|a^2=d,e^a=e^4,e^b=e^4,f^a=e^3f,f^b=f^4,g^a=g^6,g^c=g^4124 conjugacy classes — view the full character table.
| maximal subgroup of | #72243125, AGL(1,𝔽7)×#6036, #72243136, … |
| maximal normal subgroup of | #72243125, AGL(1,𝔽7)×#6036, #72243136, … |
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, 2, 3, 2, 5, 5, 7]);;
gap> SetPower(coll, 1, F.4);;
gap> SetConjugate(coll, 5, 1, F.5^4);;
gap> SetConjugate(coll, 5, 2, F.5^4);;
gap> SetConjugate(coll, 6, 1, F.5^3*F.6);;
gap> SetConjugate(coll, 6, 2, F.6^4);;
gap> SetConjugate(coll, 7, 1, F.7^6);;
gap> SetConjugate(coll, 7, 3, F.7^4);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);