Finite Group DatabasebetaFinite Group Databasebeta| order | 2100 |
| isoclinism | isoclinic class of C22⋊ΓL(1,𝔽8) |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C52 · Schur multiplier (H2) = C2×C5 |
| Fitting, socle, Hall | C22×C52×C7 ⋊ C3 |
| derived, Hall | C22×C7 ⋊ C3×C52 |
| Hall | C52×C7 ⋊ A4 |
| Hall | C22×C52 ⋊ ΓL(1,𝔽8) |
| Sylow | C7 ⋊ C52×A4 |
| Sylow | C22 ⋊ C52×ΓL(1,𝔽8) |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C52 | C22⋊ΓL(1,𝔽8) |
| commutator, cosocle, hypocenter | C22×C7 | C3×C52 |
| Frattini | C1 | G |
| Fitting, socle | C22×C52×C7 | C3 |
| LCS | G ▷ C22×C7 | |
| UCS | C1 ◁ C52 | |
80 conjugacy classes of subgroups.
| minimal degree | μ(G) = 21 |
| transitive degree | μt(G) = 700 |
A minimal faithful permutation representation — 2 generators on 21 points:
(1,2,3,4,5)(12,13,14)(16,17,18)(19,21,20);(6,7,8,9,10)(11,14)(12,13)(15,16,17,19,18,20,21)300 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(2100, 151);