Finite Group DatabasebetaFinite Group Databasebeta| order | 468 |
| isoclinism | isoclinic class of D13⋊C3 |
| automorphism | Aut(G) = C2×D3×AGL(1,𝔽13) (order 1872) · Inn(G) = D13⋊C3 · Out(G) = C22×D3 |
| homology | Abelianization (H1) = C32×C4 · Schur multiplier (H2) = C3 |
| direct product | C3×Dic13⋊C3 |
| unambiguous semidirect product | Dic13⋊C32 |
| unambiguous semidirect product | (C3×Dic13)⋊C3 |
| derived, Sylow | C13 ⋊ C32×C4 |
| Hall | C3×C13⋊C3 ⋊ C4 |
| Hall | Dic13 ⋊ C32 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C3 | D13⋊C3 |
| commutator, hypocenter | C13 | C32×C4 |
| Frattini | C2 | C3×D13⋊C3 |
| Fitting, socle | C2×C3×C13 | C2×C3 |
| prime radical | C2×C3×C13⋊C3 | C2 |
| cosocle | C2×C13 | C2×C32 |
| LCS | G ▷ C13 | |
| UCS | C1 ◁ C2×C3 | |
36 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 156 |
A minimal faithful permutation representation — 2 generators on 20 points:
(4,7,6,5)(9,13,14,10,11,15)(12,19,18,16,17,20);(1,2,3)(8,9,12,11,15,18,17,19,20,14,13,16,10)48 conjugacy classes — view the full character table.
| maximal subgroup of | C3×#1057⋊C3, C3⋊(#1057⋊C3), D3×Dic13⋊C3, … |
| maximal normal subgroup of | C3×#1057⋊C3, C3⋊(#1057⋊C3), D3×Dic13⋊C3, … |
| maximal quotient of | C3×#1055⋊C3, C2×C3×Dic13⋊C3, C9×Dic13⋊C3, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(468, 19);