Finite Group DatabasebetaFinite Group Databasebeta| order | 294 |
| isoclinism | isoclinic class of D21 |
| automorphism | Aut(G) = C2×C3×D3×AGL(1,𝔽7) (order 1512) · Inn(G) = D21 · Out(G) = C22×C32 |
| homology | Abelianization (H1) = C2×C7 · Schur multiplier (H2) = C1 |
| direct product | C7×D21 |
| unambiguous semidirect product | C3⋊(C7×D7) |
| unambiguous semidirect product | C7⋊(D3×C7) |
| Fitting, socle, Hall | C3×C72 ⋊ C2 |
| derived | C3×C7 ⋊ C2×C7 |
| Sylow | C72 ⋊ D3 |
| Sylow | C3 ⋊ C7×D7 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C7 | D21 |
| commutator, cosocle, hypocenter | C3×C7 | C2×C7 |
| Frattini | C1 | G |
| Fitting, socle | C3×C72 | C2 |
| LCS | G ▷ C3×C7 | |
| UCS | C1 ◁ C7 | |
22 conjugacy classes of subgroups.
| minimal degree | μ(G) = 17 |
| transitive degree | μt(G) = 42 |
A minimal faithful permutation representation — 2 generators on 17 points:
(2,3)(4,5,6,7,8,9,10)(12,13)(14,15)(16,17);(1,2,3)(11,12,14,16,17,15,13)84 conjugacy classes — view the full character table.
| maximal subgroup of | D3×C7×D7, D7×D21, D21⋊D7, … |
| maximal normal subgroup of | D3×C7×D7, D7×D21, D21⋊D7, … |
| maximal quotient of | C7×Dic21, C2×C7×D21, C7×D63, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(294, 21);