Finite Group DatabasebetaFinite Group Databasebeta| order | 1428 |
| isoclinism | isoclinic class of C22⋊ΓL(1,𝔽8) |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C17 · Schur multiplier (H2) = C2 |
| Fitting, socle, Hall | C22×C7×C17 ⋊ C3 |
| derived, Hall | C22×C7 ⋊ C3×C17 |
| Hall | C7×C17 ⋊ A4 |
| Hall | C22×C17 ⋊ ΓL(1,𝔽8) |
| Sylow | C7 ⋊ A4×C17 |
| Sylow | C22 ⋊ C17×ΓL(1,𝔽8) |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C17 | C22⋊ΓL(1,𝔽8) |
| commutator, cosocle, hypocenter | C22×C7 | C3×C17 |
| Frattini | C1 | G |
| Fitting, socle | C22×C7×C17 | C3 |
| LCS | G ▷ C22×C7 | |
| UCS | C1 ◁ C17 | |
20 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 476 |
A minimal faithful permutation representation — 2 generators on 28 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17)(19,20,21)(23,24,25)(26,28,27);(18,21)(19,20)(22,23,24,26,25,27,28)204 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1428, 32);