Finite Group DatabasebetaFinite Group Databasebeta| order | 468 |
| isoclinism | isoclinic class of C3⋊(D13⋊C3) |
| automorphism | Aut(G) = C2×D3×AGL(1,𝔽13) (order 1872) · Inn(G) = C3⋊(D13⋊C3) · Out(G) = C2×C4 |
| homology | Abelianization (H1) = C3×C4 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | C3⋊(Dic13⋊C3) |
| unambiguous semidirect product | Dic39⋊C3 |
| derived | C3×C13 ⋊ C3×C4 |
| Hall | C3×C13⋊C3 ⋊ C4 |
| Sylow | C13 ⋊ C3×Dic3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | C3⋊(D13⋊C3) |
| commutator, hypocenter | C3×C13 | C3×C4 |
| Fitting, socle, cosocle | C2×C3×C13 | C2×C3 |
| prime radical | C2×C3×C13⋊C3 | C2 |
| LCS | G ▷ C3×C13 | |
| UCS | C1 ◁ C2 | |
28 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 156 |
A minimal faithful permutation representation — 2 generators on 20 points:
(2,3)(4,5,6,7)(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)Polycyclic presentation:
2,3,2,3,13|a^2=c,d^a=d^2,e^a=e^12,e^b=e^330 conjugacy classes — view the full character table.
| maximal subgroup of | Dic3×D13⋊C3, D3×Dic13⋊C3, #64602⋊C3, … |
| maximal normal subgroup of | Dic3×D13⋊C3, D3×Dic13⋊C3, #64602⋊C3, … |
| maximal quotient of | C3⋊(#1055⋊C3), C2×C3⋊(Dic13⋊C3), #13063898, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(468, 21);