Finite Group DatabasebetaFinite Group Databasebeta| order | 2100 |
| isoclinism | isoclinic class of #1043⋊C3 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C4×C7 · Schur multiplier (H2) = C1 |
| direct product | C7×#1043⋊C3 |
| unambiguous semidirect product | #1043⋊(C3×C7) |
| Fitting, socle, Hall | C52×C7 ⋊ C3×C4 |
| derived, Sylow | C52 ⋊ C3×C4×C7 |
| Hall | C7×#1043 ⋊ C3 |
| Hall | C7×C52⋊C3 ⋊ C4 |
| Hall | #1043 ⋊ C3×C7 |
| Hall | C52⋊C3 ⋊ C4×C7 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C7 | #1043⋊C3 |
| commutator, hypocenter | C52 | C3×C4×C7 |
| Frattini | C1 | G |
| Fitting, socle | C52×C7 | C3×C4 |
| prime radical | C7×(C5⋊D5)⋊C3 | C2 |
| cosocle | C5⋊D5 | C2×C3×C7 |
| LCS | G ▷ C52 | |
| UCS | C1 ◁ C7 | |
36 conjugacy classes of subgroups.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 105 |
A minimal faithful permutation representation — 2 generators on 22 points:
(1,2,3,4,5,6,7)(8,9,11)(10,19,17,21,13,22,15,12,16,14,20,18);(8,10,14,21,15)(9,12,13,20,19)(11,16,22,18,17)98 conjugacy classes — view the full character table.
| maximal subgroup of | C49×#1043⋊C3, C72×#1043⋊C3, C7×#14523608, … |
| maximal normal subgroup of | C49×#1043⋊C3, C72×#1043⋊C3, C7×C11×#1043⋊C3, … |
| maximal quotient of | C49×#1043⋊C3, C72×#1043⋊C3, C7×#14523608, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(2100, 87);