Finite Group DatabasebetaFinite Group Databasebeta| order | 204 |
| isoclinism | isoclinic class of D51 |
| automorphism | Aut(G) = C2×D3×AGL(1,𝔽17) (order 3264) · Inn(G) = D51 · Out(G) = C2×C16 |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2 |
| dihedral | D102 |
| direct product | C2×D51 |
| unambiguous semidirect product | (C2×C3)⋊D17 |
| unambiguous semidirect product | (C2×C17)⋊D3 |
| Fitting, socle | C2×C3×C17 ⋊ C2 |
| derived, Hall | C3×C17 ⋊ C22 |
| Sylow | C17 ⋊ C2×D3 |
| Sylow | C3 ⋊ C2×D17 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | D51 |
| commutator, cosocle, hypocenter | C3×C17 | C22 |
| Frattini | C1 | G |
| Fitting, socle | C2×C3×C17 | C2 |
| LCS | G ▷ C3×C17 | |
| UCS | C1 ◁ C2 | |
20 conjugacy classes of subgroups.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 102 |
A minimal faithful permutation representation — 2 generators on 22 points:
(1,2)(3,4,5)(6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22);(4,5)(7,22)(8,21)(9,20)(10,19)(11,18)(12,17)(13,16)(14,15)54 conjugacy classes — view the full character table.
| maximal subgroup of | #87972, #87974, #87975, … |
| maximal normal subgroup of | #87972, #87974, #87975, … |
| maximal quotient of | Dic102, C4×D51, D204, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(204, 11);