Finite Group DatabasebetaFinite Group Databasebeta| order | 8428 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | (C7⋊D7)⋊D43 |
| unambiguous semidirect product | D301⋊D7 |
| Fitting, socle, derived, Hall | C72×C43 ⋊ C22 |
| Sylow | C72 ⋊ C2×D43 |
| Sylow | C43 ⋊ D72 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, hypocenter | C72×C43 | C22 |
| LCS | G ▷ C72×C43 | |
| UCS | C1 | |
52 conjugacy classes of subgroups.
Polycyclic presentation:
2^2,7^2,43|c^a=c^6,e^a=e^42,c^b=c^6,d^b=d^6571 conjugacy classes — view the full character table.
| maximal subgroup of | #38760426, #14268594⋊D7, #14268593⋊D7, … |
| maximal normal subgroup of | #14268594⋊D7, #14268593⋊D7, (ΓL(1,𝔽8)⋊D7)⋊D43, … |
| maximal quotient of | C3×(C7⋊D7)⋊D43, D301⋊D21, #38760556 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(8428, 34);