direct product, metacyclic, supersoluble, monomial, A-group
Aliases: S3×C19⋊C3, C57⋊3C6, (S3×C19)⋊C3, C19⋊2(C3×S3), C3⋊(C2×C19⋊C3), (C3×C19⋊C3)⋊3C2, SmallGroup(342,10)
Series: Derived ►Chief ►Lower central ►Upper central
| C57 — S3×C19⋊C3 |
Generators and relations for S3×C19⋊C3
G = < a,b,c,d | a3=b2=c19=d3=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c11 >
Character table of S3×C19⋊C3
| class | 1 | 2 | 3A | 3B | 3C | 3D | 3E | 6A | 6B | 19A | 19B | 19C | 19D | 19E | 19F | 38A | 38B | 38C | 38D | 38E | 38F | 57A | 57B | 57C | 57D | 57E | 57F | |
| size | 1 | 3 | 2 | 19 | 19 | 38 | 38 | 57 | 57 | 3 | 3 | 3 | 3 | 3 | 3 | 9 | 9 | 9 | 9 | 9 | 9 | 6 | 6 | 6 | 6 | 6 | 6 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial |
| ρ2 | 1 | -1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 2 |
| ρ3 | 1 | 1 | 1 | ζ3 | ζ32 | ζ3 | ζ32 | ζ3 | ζ32 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 3 |
| ρ4 | 1 | 1 | 1 | ζ32 | ζ3 | ζ32 | ζ3 | ζ32 | ζ3 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 3 |
| ρ5 | 1 | -1 | 1 | ζ3 | ζ32 | ζ3 | ζ32 | ζ65 | ζ6 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 6 |
| ρ6 | 1 | -1 | 1 | ζ32 | ζ3 | ζ32 | ζ3 | ζ6 | ζ65 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 6 |
| ρ7 | 2 | 0 | -1 | 2 | 2 | -1 | -1 | 0 | 0 | 2 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ8 | 2 | 0 | -1 | -1+√-3 | -1-√-3 | ζ65 | ζ6 | 0 | 0 | 2 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | complex lifted from C3×S3 |
| ρ9 | 2 | 0 | -1 | -1-√-3 | -1+√-3 | ζ6 | ζ65 | 0 | 0 | 2 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | complex lifted from C3×S3 |
| ρ10 | 3 | 3 | 3 | 0 | 0 | 0 | 0 | 0 | 0 | ζ1914+ζ193+ζ192 | ζ199+ζ196+ζ194 | ζ1918+ζ1912+ζ198 | ζ1915+ζ1913+ζ1910 | ζ1917+ζ1916+ζ195 | ζ1911+ζ197+ζ19 | ζ1911+ζ197+ζ19 | ζ199+ζ196+ζ194 | ζ1918+ζ1912+ζ198 | ζ1915+ζ1913+ζ1910 | ζ1917+ζ1916+ζ195 | ζ1914+ζ193+ζ192 | ζ199+ζ196+ζ194 | ζ1918+ζ1912+ζ198 | ζ1915+ζ1913+ζ1910 | ζ1917+ζ1916+ζ195 | ζ1911+ζ197+ζ19 | ζ1914+ζ193+ζ192 | complex lifted from C19⋊C3 |
| ρ11 | 3 | 3 | 3 | 0 | 0 | 0 | 0 | 0 | 0 | ζ1917+ζ1916+ζ195 | ζ1915+ζ1913+ζ1910 | ζ1911+ζ197+ζ19 | ζ199+ζ196+ζ194 | ζ1914+ζ193+ζ192 | ζ1918+ζ1912+ζ198 | ζ1918+ζ1912+ζ198 | ζ1915+ζ1913+ζ1910 | ζ1911+ζ197+ζ19 | ζ199+ζ196+ζ194 | ζ1914+ζ193+ζ192 | ζ1917+ζ1916+ζ195 | ζ1915+ζ1913+ζ1910 | ζ1911+ζ197+ζ19 | ζ199+ζ196+ζ194 | ζ1914+ζ193+ζ192 | ζ1918+ζ1912+ζ198 | ζ1917+ζ1916+ζ195 | complex lifted from C19⋊C3 |
| ρ12 | 3 | -3 | 3 | 0 | 0 | 0 | 0 | 0 | 0 | ζ199+ζ196+ζ194 | ζ1918+ζ1912+ζ198 | ζ1917+ζ1916+ζ195 | ζ1911+ζ197+ζ19 | ζ1915+ζ1913+ζ1910 | ζ1914+ζ193+ζ192 | -ζ1914-ζ193-ζ192 | -ζ1918-ζ1912-ζ198 | -ζ1917-ζ1916-ζ195 | -ζ1911-ζ197-ζ19 | -ζ1915-ζ1913-ζ1910 | -ζ199-ζ196-ζ194 | ζ1918+ζ1912+ζ198 | ζ1917+ζ1916+ζ195 | ζ1911+ζ197+ζ19 | ζ1915+ζ1913+ζ1910 | ζ1914+ζ193+ζ192 | ζ199+ζ196+ζ194 | complex lifted from C2×C19⋊C3 |
| ρ13 | 3 | 3 | 3 | 0 | 0 | 0 | 0 | 0 | 0 | ζ1918+ζ1912+ζ198 | ζ1917+ζ1916+ζ195 | ζ1915+ζ1913+ζ1910 | ζ1914+ζ193+ζ192 | ζ1911+ζ197+ζ19 | ζ199+ζ196+ζ194 | ζ199+ζ196+ζ194 | ζ1917+ζ1916+ζ195 | ζ1915+ζ1913+ζ1910 | ζ1914+ζ193+ζ192 | ζ1911+ζ197+ζ19 | ζ1918+ζ1912+ζ198 | ζ1917+ζ1916+ζ195 | ζ1915+ζ1913+ζ1910 | ζ1914+ζ193+ζ192 | ζ1911+ζ197+ζ19 | ζ199+ζ196+ζ194 | ζ1918+ζ1912+ζ198 | complex lifted from C19⋊C3 |
| ρ14 | 3 | -3 | 3 | 0 | 0 | 0 | 0 | 0 | 0 | ζ1915+ζ1913+ζ1910 | ζ1911+ζ197+ζ19 | ζ1914+ζ193+ζ192 | ζ1918+ζ1912+ζ198 | ζ199+ζ196+ζ194 | ζ1917+ζ1916+ζ195 | -ζ1917-ζ1916-ζ195 | -ζ1911-ζ197-ζ19 | -ζ1914-ζ193-ζ192 | -ζ1918-ζ1912-ζ198 | -ζ199-ζ196-ζ194 | -ζ1915-ζ1913-ζ1910 | ζ1911+ζ197+ζ19 | ζ1914+ζ193+ζ192 | ζ1918+ζ1912+ζ198 | ζ199+ζ196+ζ194 | ζ1917+ζ1916+ζ195 | ζ1915+ζ1913+ζ1910 | complex lifted from C2×C19⋊C3 |
| ρ15 | 3 | -3 | 3 | 0 | 0 | 0 | 0 | 0 | 0 | ζ1914+ζ193+ζ192 | ζ199+ζ196+ζ194 | ζ1918+ζ1912+ζ198 | ζ1915+ζ1913+ζ1910 | ζ1917+ζ1916+ζ195 | ζ1911+ζ197+ζ19 | -ζ1911-ζ197-ζ19 | -ζ199-ζ196-ζ194 | -ζ1918-ζ1912-ζ198 | -ζ1915-ζ1913-ζ1910 | -ζ1917-ζ1916-ζ195 | -ζ1914-ζ193-ζ192 | ζ199+ζ196+ζ194 | ζ1918+ζ1912+ζ198 | ζ1915+ζ1913+ζ1910 | ζ1917+ζ1916+ζ195 | ζ1911+ζ197+ζ19 | ζ1914+ζ193+ζ192 | complex lifted from C2×C19⋊C3 |
| ρ16 | 3 | -3 | 3 | 0 | 0 | 0 | 0 | 0 | 0 | ζ1918+ζ1912+ζ198 | ζ1917+ζ1916+ζ195 | ζ1915+ζ1913+ζ1910 | ζ1914+ζ193+ζ192 | ζ1911+ζ197+ζ19 | ζ199+ζ196+ζ194 | -ζ199-ζ196-ζ194 | -ζ1917-ζ1916-ζ195 | -ζ1915-ζ1913-ζ1910 | -ζ1914-ζ193-ζ192 | -ζ1911-ζ197-ζ19 | -ζ1918-ζ1912-ζ198 | ζ1917+ζ1916+ζ195 | ζ1915+ζ1913+ζ1910 | ζ1914+ζ193+ζ192 | ζ1911+ζ197+ζ19 | ζ199+ζ196+ζ194 | ζ1918+ζ1912+ζ198 | complex lifted from C2×C19⋊C3 |
| ρ17 | 3 | 3 | 3 | 0 | 0 | 0 | 0 | 0 | 0 | ζ199+ζ196+ζ194 | ζ1918+ζ1912+ζ198 | ζ1917+ζ1916+ζ195 | ζ1911+ζ197+ζ19 | ζ1915+ζ1913+ζ1910 | ζ1914+ζ193+ζ192 | ζ1914+ζ193+ζ192 | ζ1918+ζ1912+ζ198 | ζ1917+ζ1916+ζ195 | ζ1911+ζ197+ζ19 | ζ1915+ζ1913+ζ1910 | ζ199+ζ196+ζ194 | ζ1918+ζ1912+ζ198 | ζ1917+ζ1916+ζ195 | ζ1911+ζ197+ζ19 | ζ1915+ζ1913+ζ1910 | ζ1914+ζ193+ζ192 | ζ199+ζ196+ζ194 | complex lifted from C19⋊C3 |
| ρ18 | 3 | -3 | 3 | 0 | 0 | 0 | 0 | 0 | 0 | ζ1917+ζ1916+ζ195 | ζ1915+ζ1913+ζ1910 | ζ1911+ζ197+ζ19 | ζ199+ζ196+ζ194 | ζ1914+ζ193+ζ192 | ζ1918+ζ1912+ζ198 | -ζ1918-ζ1912-ζ198 | -ζ1915-ζ1913-ζ1910 | -ζ1911-ζ197-ζ19 | -ζ199-ζ196-ζ194 | -ζ1914-ζ193-ζ192 | -ζ1917-ζ1916-ζ195 | ζ1915+ζ1913+ζ1910 | ζ1911+ζ197+ζ19 | ζ199+ζ196+ζ194 | ζ1914+ζ193+ζ192 | ζ1918+ζ1912+ζ198 | ζ1917+ζ1916+ζ195 | complex lifted from C2×C19⋊C3 |
| ρ19 | 3 | -3 | 3 | 0 | 0 | 0 | 0 | 0 | 0 | ζ1911+ζ197+ζ19 | ζ1914+ζ193+ζ192 | ζ199+ζ196+ζ194 | ζ1917+ζ1916+ζ195 | ζ1918+ζ1912+ζ198 | ζ1915+ζ1913+ζ1910 | -ζ1915-ζ1913-ζ1910 | -ζ1914-ζ193-ζ192 | -ζ199-ζ196-ζ194 | -ζ1917-ζ1916-ζ195 | -ζ1918-ζ1912-ζ198 | -ζ1911-ζ197-ζ19 | ζ1914+ζ193+ζ192 | ζ199+ζ196+ζ194 | ζ1917+ζ1916+ζ195 | ζ1918+ζ1912+ζ198 | ζ1915+ζ1913+ζ1910 | ζ1911+ζ197+ζ19 | complex lifted from C2×C19⋊C3 |
| ρ20 | 3 | 3 | 3 | 0 | 0 | 0 | 0 | 0 | 0 | ζ1911+ζ197+ζ19 | ζ1914+ζ193+ζ192 | ζ199+ζ196+ζ194 | ζ1917+ζ1916+ζ195 | ζ1918+ζ1912+ζ198 | ζ1915+ζ1913+ζ1910 | ζ1915+ζ1913+ζ1910 | ζ1914+ζ193+ζ192 | ζ199+ζ196+ζ194 | ζ1917+ζ1916+ζ195 | ζ1918+ζ1912+ζ198 | ζ1911+ζ197+ζ19 | ζ1914+ζ193+ζ192 | ζ199+ζ196+ζ194 | ζ1917+ζ1916+ζ195 | ζ1918+ζ1912+ζ198 | ζ1915+ζ1913+ζ1910 | ζ1911+ζ197+ζ19 | complex lifted from C19⋊C3 |
| ρ21 | 3 | 3 | 3 | 0 | 0 | 0 | 0 | 0 | 0 | ζ1915+ζ1913+ζ1910 | ζ1911+ζ197+ζ19 | ζ1914+ζ193+ζ192 | ζ1918+ζ1912+ζ198 | ζ199+ζ196+ζ194 | ζ1917+ζ1916+ζ195 | ζ1917+ζ1916+ζ195 | ζ1911+ζ197+ζ19 | ζ1914+ζ193+ζ192 | ζ1918+ζ1912+ζ198 | ζ199+ζ196+ζ194 | ζ1915+ζ1913+ζ1910 | ζ1911+ζ197+ζ19 | ζ1914+ζ193+ζ192 | ζ1918+ζ1912+ζ198 | ζ199+ζ196+ζ194 | ζ1917+ζ1916+ζ195 | ζ1915+ζ1913+ζ1910 | complex lifted from C19⋊C3 |
| ρ22 | 6 | 0 | -3 | 0 | 0 | 0 | 0 | 0 | 0 | 2ζ199+2ζ196+2ζ194 | 2ζ1918+2ζ1912+2ζ198 | 2ζ1917+2ζ1916+2ζ195 | 2ζ1911+2ζ197+2ζ19 | 2ζ1915+2ζ1913+2ζ1910 | 2ζ1914+2ζ193+2ζ192 | 0 | 0 | 0 | 0 | 0 | 0 | -ζ1918-ζ1912-ζ198 | -ζ1917-ζ1916-ζ195 | -ζ1911-ζ197-ζ19 | -ζ1915-ζ1913-ζ1910 | -ζ1914-ζ193-ζ192 | -ζ199-ζ196-ζ194 | complex faithful |
| ρ23 | 6 | 0 | -3 | 0 | 0 | 0 | 0 | 0 | 0 | 2ζ1911+2ζ197+2ζ19 | 2ζ1914+2ζ193+2ζ192 | 2ζ199+2ζ196+2ζ194 | 2ζ1917+2ζ1916+2ζ195 | 2ζ1918+2ζ1912+2ζ198 | 2ζ1915+2ζ1913+2ζ1910 | 0 | 0 | 0 | 0 | 0 | 0 | -ζ1914-ζ193-ζ192 | -ζ199-ζ196-ζ194 | -ζ1917-ζ1916-ζ195 | -ζ1918-ζ1912-ζ198 | -ζ1915-ζ1913-ζ1910 | -ζ1911-ζ197-ζ19 | complex faithful |
| ρ24 | 6 | 0 | -3 | 0 | 0 | 0 | 0 | 0 | 0 | 2ζ1914+2ζ193+2ζ192 | 2ζ199+2ζ196+2ζ194 | 2ζ1918+2ζ1912+2ζ198 | 2ζ1915+2ζ1913+2ζ1910 | 2ζ1917+2ζ1916+2ζ195 | 2ζ1911+2ζ197+2ζ19 | 0 | 0 | 0 | 0 | 0 | 0 | -ζ199-ζ196-ζ194 | -ζ1918-ζ1912-ζ198 | -ζ1915-ζ1913-ζ1910 | -ζ1917-ζ1916-ζ195 | -ζ1911-ζ197-ζ19 | -ζ1914-ζ193-ζ192 | complex faithful |
| ρ25 | 6 | 0 | -3 | 0 | 0 | 0 | 0 | 0 | 0 | 2ζ1915+2ζ1913+2ζ1910 | 2ζ1911+2ζ197+2ζ19 | 2ζ1914+2ζ193+2ζ192 | 2ζ1918+2ζ1912+2ζ198 | 2ζ199+2ζ196+2ζ194 | 2ζ1917+2ζ1916+2ζ195 | 0 | 0 | 0 | 0 | 0 | 0 | -ζ1911-ζ197-ζ19 | -ζ1914-ζ193-ζ192 | -ζ1918-ζ1912-ζ198 | -ζ199-ζ196-ζ194 | -ζ1917-ζ1916-ζ195 | -ζ1915-ζ1913-ζ1910 | complex faithful |
| ρ26 | 6 | 0 | -3 | 0 | 0 | 0 | 0 | 0 | 0 | 2ζ1918+2ζ1912+2ζ198 | 2ζ1917+2ζ1916+2ζ195 | 2ζ1915+2ζ1913+2ζ1910 | 2ζ1914+2ζ193+2ζ192 | 2ζ1911+2ζ197+2ζ19 | 2ζ199+2ζ196+2ζ194 | 0 | 0 | 0 | 0 | 0 | 0 | -ζ1917-ζ1916-ζ195 | -ζ1915-ζ1913-ζ1910 | -ζ1914-ζ193-ζ192 | -ζ1911-ζ197-ζ19 | -ζ199-ζ196-ζ194 | -ζ1918-ζ1912-ζ198 | complex faithful |
| ρ27 | 6 | 0 | -3 | 0 | 0 | 0 | 0 | 0 | 0 | 2ζ1917+2ζ1916+2ζ195 | 2ζ1915+2ζ1913+2ζ1910 | 2ζ1911+2ζ197+2ζ19 | 2ζ199+2ζ196+2ζ194 | 2ζ1914+2ζ193+2ζ192 | 2ζ1918+2ζ1912+2ζ198 | 0 | 0 | 0 | 0 | 0 | 0 | -ζ1915-ζ1913-ζ1910 | -ζ1911-ζ197-ζ19 | -ζ199-ζ196-ζ194 | -ζ1914-ζ193-ζ192 | -ζ1918-ζ1912-ζ198 | -ζ1917-ζ1916-ζ195 | complex faithful |
►On 57 points
Generators in S57
(1 20 39)(2 21 40)(3 22 41)(4 23 42)(5 24 43)(6 25 44)(7 26 45)(8 27 46)(9 28 47)(10 29 48)(11 30 49)(12 31 50)(13 32 51)(14 33 52)(15 34 53)(16 35 54)(17 36 55)(18 37 56)(19 38 57)
(20 39)(21 40)(22 41)(23 42)(24 43)(25 44)(26 45)(27 46)(28 47)(29 48)(30 49)(31 50)(32 51)(33 52)(34 53)(35 54)(36 55)(37 56)(38 57)
(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19)(20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38)(39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57)
(2 8 12)(3 15 4)(5 10 7)(6 17 18)(9 19 13)(11 14 16)(21 27 31)(22 34 23)(24 29 26)(25 36 37)(28 38 32)(30 33 35)(40 46 50)(41 53 42)(43 48 45)(44 55 56)(47 57 51)(49 52 54)
G:=sub<Sym(57)| (1,20,39)(2,21,40)(3,22,41)(4,23,42)(5,24,43)(6,25,44)(7,26,45)(8,27,46)(9,28,47)(10,29,48)(11,30,49)(12,31,50)(13,32,51)(14,33,52)(15,34,53)(16,35,54)(17,36,55)(18,37,56)(19,38,57), (20,39)(21,40)(22,41)(23,42)(24,43)(25,44)(26,45)(27,46)(28,47)(29,48)(30,49)(31,50)(32,51)(33,52)(34,53)(35,54)(36,55)(37,56)(38,57), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19)(20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38)(39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57), (2,8,12)(3,15,4)(5,10,7)(6,17,18)(9,19,13)(11,14,16)(21,27,31)(22,34,23)(24,29,26)(25,36,37)(28,38,32)(30,33,35)(40,46,50)(41,53,42)(43,48,45)(44,55,56)(47,57,51)(49,52,54)>;
G:=Group( (1,20,39)(2,21,40)(3,22,41)(4,23,42)(5,24,43)(6,25,44)(7,26,45)(8,27,46)(9,28,47)(10,29,48)(11,30,49)(12,31,50)(13,32,51)(14,33,52)(15,34,53)(16,35,54)(17,36,55)(18,37,56)(19,38,57), (20,39)(21,40)(22,41)(23,42)(24,43)(25,44)(26,45)(27,46)(28,47)(29,48)(30,49)(31,50)(32,51)(33,52)(34,53)(35,54)(36,55)(37,56)(38,57), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19)(20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38)(39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57), (2,8,12)(3,15,4)(5,10,7)(6,17,18)(9,19,13)(11,14,16)(21,27,31)(22,34,23)(24,29,26)(25,36,37)(28,38,32)(30,33,35)(40,46,50)(41,53,42)(43,48,45)(44,55,56)(47,57,51)(49,52,54) );
G=PermutationGroup([[(1,20,39),(2,21,40),(3,22,41),(4,23,42),(5,24,43),(6,25,44),(7,26,45),(8,27,46),(9,28,47),(10,29,48),(11,30,49),(12,31,50),(13,32,51),(14,33,52),(15,34,53),(16,35,54),(17,36,55),(18,37,56),(19,38,57)], [(20,39),(21,40),(22,41),(23,42),(24,43),(25,44),(26,45),(27,46),(28,47),(29,48),(30,49),(31,50),(32,51),(33,52),(34,53),(35,54),(36,55),(37,56),(38,57)], [(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19),(20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38),(39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57)], [(2,8,12),(3,15,4),(5,10,7),(6,17,18),(9,19,13),(11,14,16),(21,27,31),(22,34,23),(24,29,26),(25,36,37),(28,38,32),(30,33,35),(40,46,50),(41,53,42),(43,48,45),(44,55,56),(47,57,51),(49,52,54)]])
Matrix representation of S3×C19⋊C3 ►in GL5(𝔽229)
| 0 | 228 | 0 | 0 | 0 |
| 1 | 228 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 68 | 204 | 44 |
| 0 | 0 | 1 | 0 | 70 |
| 0 | 0 | 0 | 1 | 207 |
| 134 | 0 | 0 | 0 | 0 |
| 0 | 134 | 0 | 0 | 0 |
| 0 | 0 | 73 | 181 | 42 |
| 0 | 0 | 125 | 48 | 44 |
| 0 | 0 | 166 | 188 | 108 |
G:=sub<GL(5,GF(229))| [0,1,0,0,0,228,228,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[0,1,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,68,1,0,0,0,204,0,1,0,0,44,70,207],[134,0,0,0,0,0,134,0,0,0,0,0,73,125,166,0,0,181,48,188,0,0,42,44,108] >;
S3×C19⋊C3 in GAP, Magma, Sage, TeX
S_3\times C_{19}\rtimes C_3 % in TeX
G:=Group("S3xC19:C3"); // GroupNames label
G:=SmallGroup(342,10);
// by ID
G=gap.SmallGroup(342,10);
# by ID
G:=PCGroup([4,-2,-3,-3,-19,146,1015]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^2=c^19=d^3=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^11>;
// generators/relations
Export
Subgroup lattice of S3×C19⋊C3 in TeX
Character table of S3×C19⋊C3 in TeX