metabelian, soluble, monomial, A-group
Aliases: C19⋊A4, (C2×C38)⋊2C3, C22⋊(C19⋊C3), SmallGroup(228,11)
Series: Derived ►Chief ►Lower central ►Upper central
| C2×C38 — C19⋊A4 |
Generators and relations for C19⋊A4
G = < a,b,c,d | a19=b2=c2=d3=1, ab=ba, ac=ca, dad-1=a11, dbd-1=bc=cb, dcd-1=b >
Character table of C19⋊A4
| class | 1 | 2 | 3A | 3B | 19A | 19B | 19C | 19D | 19E | 19F | 38A | 38B | 38C | 38D | 38E | 38F | 38G | 38H | 38I | 38J | 38K | 38L | 38M | 38N | 38O | 38P | 38Q | 38R | |
| size | 1 | 3 | 76 | 76 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | |
| 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 | 1 | trivial |
| ρ2 | 1 | 1 | ζ3 | ζ32 | 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 3 |
| ρ3 | 1 | 1 | ζ32 | ζ3 | 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 3 |
| ρ4 | 3 | -1 | 0 | 0 | 3 | 3 | 3 | 3 | 3 | 3 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from A4 |
| ρ5 | 3 | -1 | 0 | 0 | ζ199+ζ196+ζ194 | ζ1914+ζ193+ζ192 | ζ1918+ζ1912+ζ198 | ζ1917+ζ1916+ζ195 | ζ1915+ζ1913+ζ1910 | ζ1911+ζ197+ζ19 | -ζ1915+ζ1913-ζ1910 | ζ1911-ζ197-ζ19 | ζ199-ζ196-ζ194 | -ζ1911+ζ197-ζ19 | -ζ1917-ζ1916+ζ195 | -ζ1914+ζ193-ζ192 | -ζ1911-ζ197+ζ19 | ζ1918-ζ1912-ζ198 | -ζ1917+ζ1916-ζ195 | ζ1914-ζ193-ζ192 | -ζ1918+ζ1912-ζ198 | -ζ1915-ζ1913+ζ1910 | -ζ1918-ζ1912+ζ198 | -ζ199+ζ196-ζ194 | -ζ199-ζ196+ζ194 | -ζ1914-ζ193+ζ192 | ζ1917-ζ1916-ζ195 | ζ1915-ζ1913-ζ1910 | complex faithful |
| ρ6 | 3 | -1 | 0 | 0 | ζ1915+ζ1913+ζ1910 | ζ1917+ζ1916+ζ195 | ζ1911+ζ197+ζ19 | ζ1914+ζ193+ζ192 | ζ199+ζ196+ζ194 | ζ1918+ζ1912+ζ198 | -ζ199-ζ196+ζ194 | ζ1918-ζ1912-ζ198 | -ζ1915+ζ1913-ζ1910 | -ζ1918-ζ1912+ζ198 | -ζ1914+ζ193-ζ192 | ζ1917-ζ1916-ζ195 | -ζ1918+ζ1912-ζ198 | -ζ1911+ζ197-ζ19 | -ζ1914-ζ193+ζ192 | -ζ1917+ζ1916-ζ195 | ζ1911-ζ197-ζ19 | -ζ199+ζ196-ζ194 | -ζ1911-ζ197+ζ19 | ζ1915-ζ1913-ζ1910 | -ζ1915-ζ1913+ζ1910 | -ζ1917-ζ1916+ζ195 | ζ1914-ζ193-ζ192 | ζ199-ζ196-ζ194 | complex faithful |
| ρ7 | 3 | -1 | 0 | 0 | ζ1914+ζ193+ζ192 | ζ1911+ζ197+ζ19 | ζ199+ζ196+ζ194 | ζ1918+ζ1912+ζ198 | ζ1917+ζ1916+ζ195 | ζ1915+ζ1913+ζ1910 | -ζ1917+ζ1916-ζ195 | ζ1915-ζ1913-ζ1910 | ζ1914-ζ193-ζ192 | -ζ1915+ζ1913-ζ1910 | -ζ1918+ζ1912-ζ198 | ζ1911-ζ197-ζ19 | -ζ1915-ζ1913+ζ1910 | ζ199-ζ196-ζ194 | -ζ1918-ζ1912+ζ198 | -ζ1911+ζ197-ζ19 | -ζ199+ζ196-ζ194 | -ζ1917-ζ1916+ζ195 | -ζ199-ζ196+ζ194 | -ζ1914+ζ193-ζ192 | -ζ1914-ζ193+ζ192 | -ζ1911-ζ197+ζ19 | ζ1918-ζ1912-ζ198 | ζ1917-ζ1916-ζ195 | complex faithful |
| ρ8 | 3 | -1 | 0 | 0 | ζ1915+ζ1913+ζ1910 | ζ1917+ζ1916+ζ195 | ζ1911+ζ197+ζ19 | ζ1914+ζ193+ζ192 | ζ199+ζ196+ζ194 | ζ1918+ζ1912+ζ198 | ζ199-ζ196-ζ194 | -ζ1918+ζ1912-ζ198 | ζ1915-ζ1913-ζ1910 | ζ1918-ζ1912-ζ198 | -ζ1914-ζ193+ζ192 | -ζ1917-ζ1916+ζ195 | -ζ1918-ζ1912+ζ198 | ζ1911-ζ197-ζ19 | ζ1914-ζ193-ζ192 | ζ1917-ζ1916-ζ195 | -ζ1911-ζ197+ζ19 | -ζ199-ζ196+ζ194 | -ζ1911+ζ197-ζ19 | -ζ1915-ζ1913+ζ1910 | -ζ1915+ζ1913-ζ1910 | -ζ1917+ζ1916-ζ195 | -ζ1914+ζ193-ζ192 | -ζ199+ζ196-ζ194 | complex faithful |
| ρ9 | 3 | 3 | 0 | 0 | ζ199+ζ196+ζ194 | ζ1914+ζ193+ζ192 | ζ1918+ζ1912+ζ198 | ζ1917+ζ1916+ζ195 | ζ1915+ζ1913+ζ1910 | ζ1911+ζ197+ζ19 | ζ1915+ζ1913+ζ1910 | ζ1911+ζ197+ζ19 | ζ199+ζ196+ζ194 | ζ1911+ζ197+ζ19 | ζ1917+ζ1916+ζ195 | ζ1914+ζ193+ζ192 | ζ1911+ζ197+ζ19 | ζ1918+ζ1912+ζ198 | ζ1917+ζ1916+ζ195 | ζ1914+ζ193+ζ192 | ζ1918+ζ1912+ζ198 | ζ1915+ζ1913+ζ1910 | ζ1918+ζ1912+ζ198 | ζ199+ζ196+ζ194 | ζ199+ζ196+ζ194 | ζ1914+ζ193+ζ192 | ζ1917+ζ1916+ζ195 | ζ1915+ζ1913+ζ1910 | complex lifted from C19⋊C3 |
| ρ10 | 3 | -1 | 0 | 0 | ζ1918+ζ1912+ζ198 | ζ199+ζ196+ζ194 | ζ1917+ζ1916+ζ195 | ζ1915+ζ1913+ζ1910 | ζ1911+ζ197+ζ19 | ζ1914+ζ193+ζ192 | -ζ1911+ζ197-ζ19 | -ζ1914+ζ193-ζ192 | ζ1918-ζ1912-ζ198 | ζ1914-ζ193-ζ192 | -ζ1915-ζ1913+ζ1910 | -ζ199+ζ196-ζ194 | -ζ1914-ζ193+ζ192 | ζ1917-ζ1916-ζ195 | -ζ1915+ζ1913-ζ1910 | ζ199-ζ196-ζ194 | -ζ1917-ζ1916+ζ195 | -ζ1911-ζ197+ζ19 | -ζ1917+ζ1916-ζ195 | -ζ1918+ζ1912-ζ198 | -ζ1918-ζ1912+ζ198 | -ζ199-ζ196+ζ194 | ζ1915-ζ1913-ζ1910 | ζ1911-ζ197-ζ19 | complex faithful |
| ρ11 | 3 | -1 | 0 | 0 | ζ1918+ζ1912+ζ198 | ζ199+ζ196+ζ194 | ζ1917+ζ1916+ζ195 | ζ1915+ζ1913+ζ1910 | ζ1911+ζ197+ζ19 | ζ1914+ζ193+ζ192 | -ζ1911-ζ197+ζ19 | ζ1914-ζ193-ζ192 | -ζ1918-ζ1912+ζ198 | -ζ1914-ζ193+ζ192 | ζ1915-ζ1913-ζ1910 | ζ199-ζ196-ζ194 | -ζ1914+ζ193-ζ192 | -ζ1917+ζ1916-ζ195 | -ζ1915-ζ1913+ζ1910 | -ζ199-ζ196+ζ194 | ζ1917-ζ1916-ζ195 | ζ1911-ζ197-ζ19 | -ζ1917-ζ1916+ζ195 | ζ1918-ζ1912-ζ198 | -ζ1918+ζ1912-ζ198 | -ζ199+ζ196-ζ194 | -ζ1915+ζ1913-ζ1910 | -ζ1911+ζ197-ζ19 | complex faithful |
| ρ12 | 3 | 3 | 0 | 0 | ζ1914+ζ193+ζ192 | ζ1911+ζ197+ζ19 | ζ199+ζ196+ζ194 | ζ1918+ζ1912+ζ198 | ζ1917+ζ1916+ζ195 | ζ1915+ζ1913+ζ1910 | ζ1917+ζ1916+ζ195 | ζ1915+ζ1913+ζ1910 | ζ1914+ζ193+ζ192 | ζ1915+ζ1913+ζ1910 | ζ1918+ζ1912+ζ198 | ζ1911+ζ197+ζ19 | ζ1915+ζ1913+ζ1910 | ζ199+ζ196+ζ194 | ζ1918+ζ1912+ζ198 | ζ1911+ζ197+ζ19 | ζ199+ζ196+ζ194 | ζ1917+ζ1916+ζ195 | ζ199+ζ196+ζ194 | ζ1914+ζ193+ζ192 | ζ1914+ζ193+ζ192 | ζ1911+ζ197+ζ19 | ζ1918+ζ1912+ζ198 | ζ1917+ζ1916+ζ195 | complex lifted from C19⋊C3 |
| ρ13 | 3 | -1 | 0 | 0 | ζ1917+ζ1916+ζ195 | ζ1918+ζ1912+ζ198 | ζ1915+ζ1913+ζ1910 | ζ1911+ζ197+ζ19 | ζ1914+ζ193+ζ192 | ζ199+ζ196+ζ194 | -ζ1914+ζ193-ζ192 | -ζ199-ζ196+ζ194 | -ζ1917-ζ1916+ζ195 | -ζ199+ζ196-ζ194 | -ζ1911+ζ197-ζ19 | -ζ1918-ζ1912+ζ198 | ζ199-ζ196-ζ194 | -ζ1915-ζ1913+ζ1910 | ζ1911-ζ197-ζ19 | -ζ1918+ζ1912-ζ198 | -ζ1915+ζ1913-ζ1910 | ζ1914-ζ193-ζ192 | ζ1915-ζ1913-ζ1910 | -ζ1917+ζ1916-ζ195 | ζ1917-ζ1916-ζ195 | ζ1918-ζ1912-ζ198 | -ζ1911-ζ197+ζ19 | -ζ1914-ζ193+ζ192 | complex faithful |
| ρ14 | 3 | -1 | 0 | 0 | ζ1914+ζ193+ζ192 | ζ1911+ζ197+ζ19 | ζ199+ζ196+ζ194 | ζ1918+ζ1912+ζ198 | ζ1917+ζ1916+ζ195 | ζ1915+ζ1913+ζ1910 | ζ1917-ζ1916-ζ195 | -ζ1915-ζ1913+ζ1910 | -ζ1914+ζ193-ζ192 | ζ1915-ζ1913-ζ1910 | -ζ1918-ζ1912+ζ198 | -ζ1911-ζ197+ζ19 | -ζ1915+ζ1913-ζ1910 | -ζ199+ζ196-ζ194 | ζ1918-ζ1912-ζ198 | ζ1911-ζ197-ζ19 | -ζ199-ζ196+ζ194 | -ζ1917+ζ1916-ζ195 | ζ199-ζ196-ζ194 | -ζ1914-ζ193+ζ192 | ζ1914-ζ193-ζ192 | -ζ1911+ζ197-ζ19 | -ζ1918+ζ1912-ζ198 | -ζ1917-ζ1916+ζ195 | complex faithful |
| ρ15 | 3 | 3 | 0 | 0 | ζ1911+ζ197+ζ19 | ζ1915+ζ1913+ζ1910 | ζ1914+ζ193+ζ192 | ζ199+ζ196+ζ194 | ζ1918+ζ1912+ζ198 | ζ1917+ζ1916+ζ195 | ζ1918+ζ1912+ζ198 | ζ1917+ζ1916+ζ195 | ζ1911+ζ197+ζ19 | ζ1917+ζ1916+ζ195 | ζ199+ζ196+ζ194 | ζ1915+ζ1913+ζ1910 | ζ1917+ζ1916+ζ195 | ζ1914+ζ193+ζ192 | ζ199+ζ196+ζ194 | ζ1915+ζ1913+ζ1910 | ζ1914+ζ193+ζ192 | ζ1918+ζ1912+ζ198 | ζ1914+ζ193+ζ192 | ζ1911+ζ197+ζ19 | ζ1911+ζ197+ζ19 | ζ1915+ζ1913+ζ1910 | ζ199+ζ196+ζ194 | ζ1918+ζ1912+ζ198 | complex lifted from C19⋊C3 |
| ρ16 | 3 | -1 | 0 | 0 | ζ199+ζ196+ζ194 | ζ1914+ζ193+ζ192 | ζ1918+ζ1912+ζ198 | ζ1917+ζ1916+ζ195 | ζ1915+ζ1913+ζ1910 | ζ1911+ζ197+ζ19 | ζ1915-ζ1913-ζ1910 | -ζ1911-ζ197+ζ19 | -ζ199+ζ196-ζ194 | ζ1911-ζ197-ζ19 | -ζ1917+ζ1916-ζ195 | -ζ1914-ζ193+ζ192 | -ζ1911+ζ197-ζ19 | -ζ1918+ζ1912-ζ198 | ζ1917-ζ1916-ζ195 | -ζ1914+ζ193-ζ192 | -ζ1918-ζ1912+ζ198 | -ζ1915+ζ1913-ζ1910 | ζ1918-ζ1912-ζ198 | -ζ199-ζ196+ζ194 | ζ199-ζ196-ζ194 | ζ1914-ζ193-ζ192 | -ζ1917-ζ1916+ζ195 | -ζ1915-ζ1913+ζ1910 | complex faithful |
| ρ17 | 3 | 3 | 0 | 0 | ζ1915+ζ1913+ζ1910 | ζ1917+ζ1916+ζ195 | ζ1911+ζ197+ζ19 | ζ1914+ζ193+ζ192 | ζ199+ζ196+ζ194 | ζ1918+ζ1912+ζ198 | ζ199+ζ196+ζ194 | ζ1918+ζ1912+ζ198 | ζ1915+ζ1913+ζ1910 | ζ1918+ζ1912+ζ198 | ζ1914+ζ193+ζ192 | ζ1917+ζ1916+ζ195 | ζ1918+ζ1912+ζ198 | ζ1911+ζ197+ζ19 | ζ1914+ζ193+ζ192 | ζ1917+ζ1916+ζ195 | ζ1911+ζ197+ζ19 | ζ199+ζ196+ζ194 | ζ1911+ζ197+ζ19 | ζ1915+ζ1913+ζ1910 | ζ1915+ζ1913+ζ1910 | ζ1917+ζ1916+ζ195 | ζ1914+ζ193+ζ192 | ζ199+ζ196+ζ194 | complex lifted from C19⋊C3 |
| ρ18 | 3 | -1 | 0 | 0 | ζ1911+ζ197+ζ19 | ζ1915+ζ1913+ζ1910 | ζ1914+ζ193+ζ192 | ζ199+ζ196+ζ194 | ζ1918+ζ1912+ζ198 | ζ1917+ζ1916+ζ195 | -ζ1918-ζ1912+ζ198 | ζ1917-ζ1916-ζ195 | -ζ1911+ζ197-ζ19 | -ζ1917+ζ1916-ζ195 | -ζ199+ζ196-ζ194 | ζ1915-ζ1913-ζ1910 | -ζ1917-ζ1916+ζ195 | ζ1914-ζ193-ζ192 | -ζ199-ζ196+ζ194 | -ζ1915+ζ1913-ζ1910 | -ζ1914+ζ193-ζ192 | -ζ1918+ζ1912-ζ198 | -ζ1914-ζ193+ζ192 | ζ1911-ζ197-ζ19 | -ζ1911-ζ197+ζ19 | -ζ1915-ζ1913+ζ1910 | ζ199-ζ196-ζ194 | ζ1918-ζ1912-ζ198 | complex faithful |
| ρ19 | 3 | -1 | 0 | 0 | ζ1918+ζ1912+ζ198 | ζ199+ζ196+ζ194 | ζ1917+ζ1916+ζ195 | ζ1915+ζ1913+ζ1910 | ζ1911+ζ197+ζ19 | ζ1914+ζ193+ζ192 | ζ1911-ζ197-ζ19 | -ζ1914-ζ193+ζ192 | -ζ1918+ζ1912-ζ198 | -ζ1914+ζ193-ζ192 | -ζ1915+ζ1913-ζ1910 | -ζ199-ζ196+ζ194 | ζ1914-ζ193-ζ192 | -ζ1917-ζ1916+ζ195 | ζ1915-ζ1913-ζ1910 | -ζ199+ζ196-ζ194 | -ζ1917+ζ1916-ζ195 | -ζ1911+ζ197-ζ19 | ζ1917-ζ1916-ζ195 | -ζ1918-ζ1912+ζ198 | ζ1918-ζ1912-ζ198 | ζ199-ζ196-ζ194 | -ζ1915-ζ1913+ζ1910 | -ζ1911-ζ197+ζ19 | complex faithful |
| ρ20 | 3 | 3 | 0 | 0 | ζ1918+ζ1912+ζ198 | ζ199+ζ196+ζ194 | ζ1917+ζ1916+ζ195 | ζ1915+ζ1913+ζ1910 | ζ1911+ζ197+ζ19 | ζ1914+ζ193+ζ192 | ζ1911+ζ197+ζ19 | ζ1914+ζ193+ζ192 | ζ1918+ζ1912+ζ198 | ζ1914+ζ193+ζ192 | ζ1915+ζ1913+ζ1910 | ζ199+ζ196+ζ194 | ζ1914+ζ193+ζ192 | ζ1917+ζ1916+ζ195 | ζ1915+ζ1913+ζ1910 | ζ199+ζ196+ζ194 | ζ1917+ζ1916+ζ195 | ζ1911+ζ197+ζ19 | ζ1917+ζ1916+ζ195 | ζ1918+ζ1912+ζ198 | ζ1918+ζ1912+ζ198 | ζ199+ζ196+ζ194 | ζ1915+ζ1913+ζ1910 | ζ1911+ζ197+ζ19 | complex lifted from C19⋊C3 |
| ρ21 | 3 | -1 | 0 | 0 | ζ1915+ζ1913+ζ1910 | ζ1917+ζ1916+ζ195 | ζ1911+ζ197+ζ19 | ζ1914+ζ193+ζ192 | ζ199+ζ196+ζ194 | ζ1918+ζ1912+ζ198 | -ζ199+ζ196-ζ194 | -ζ1918-ζ1912+ζ198 | -ζ1915-ζ1913+ζ1910 | -ζ1918+ζ1912-ζ198 | ζ1914-ζ193-ζ192 | -ζ1917+ζ1916-ζ195 | ζ1918-ζ1912-ζ198 | -ζ1911-ζ197+ζ19 | -ζ1914+ζ193-ζ192 | -ζ1917-ζ1916+ζ195 | -ζ1911+ζ197-ζ19 | ζ199-ζ196-ζ194 | ζ1911-ζ197-ζ19 | -ζ1915+ζ1913-ζ1910 | ζ1915-ζ1913-ζ1910 | ζ1917-ζ1916-ζ195 | -ζ1914-ζ193+ζ192 | -ζ199-ζ196+ζ194 | complex faithful |
| ρ22 | 3 | -1 | 0 | 0 | ζ1917+ζ1916+ζ195 | ζ1918+ζ1912+ζ198 | ζ1915+ζ1913+ζ1910 | ζ1911+ζ197+ζ19 | ζ1914+ζ193+ζ192 | ζ199+ζ196+ζ194 | ζ1914-ζ193-ζ192 | -ζ199+ζ196-ζ194 | ζ1917-ζ1916-ζ195 | ζ199-ζ196-ζ194 | -ζ1911-ζ197+ζ19 | -ζ1918+ζ1912-ζ198 | -ζ199-ζ196+ζ194 | ζ1915-ζ1913-ζ1910 | -ζ1911+ζ197-ζ19 | ζ1918-ζ1912-ζ198 | -ζ1915-ζ1913+ζ1910 | -ζ1914-ζ193+ζ192 | -ζ1915+ζ1913-ζ1910 | -ζ1917-ζ1916+ζ195 | -ζ1917+ζ1916-ζ195 | -ζ1918-ζ1912+ζ198 | ζ1911-ζ197-ζ19 | -ζ1914+ζ193-ζ192 | complex faithful |
| ρ23 | 3 | -1 | 0 | 0 | ζ1917+ζ1916+ζ195 | ζ1918+ζ1912+ζ198 | ζ1915+ζ1913+ζ1910 | ζ1911+ζ197+ζ19 | ζ1914+ζ193+ζ192 | ζ199+ζ196+ζ194 | -ζ1914-ζ193+ζ192 | ζ199-ζ196-ζ194 | -ζ1917+ζ1916-ζ195 | -ζ199-ζ196+ζ194 | ζ1911-ζ197-ζ19 | ζ1918-ζ1912-ζ198 | -ζ199+ζ196-ζ194 | -ζ1915+ζ1913-ζ1910 | -ζ1911-ζ197+ζ19 | -ζ1918-ζ1912+ζ198 | ζ1915-ζ1913-ζ1910 | -ζ1914+ζ193-ζ192 | -ζ1915-ζ1913+ζ1910 | ζ1917-ζ1916-ζ195 | -ζ1917-ζ1916+ζ195 | -ζ1918+ζ1912-ζ198 | -ζ1911+ζ197-ζ19 | ζ1914-ζ193-ζ192 | complex faithful |
| ρ24 | 3 | -1 | 0 | 0 | ζ199+ζ196+ζ194 | ζ1914+ζ193+ζ192 | ζ1918+ζ1912+ζ198 | ζ1917+ζ1916+ζ195 | ζ1915+ζ1913+ζ1910 | ζ1911+ζ197+ζ19 | -ζ1915-ζ1913+ζ1910 | -ζ1911+ζ197-ζ19 | -ζ199-ζ196+ζ194 | -ζ1911-ζ197+ζ19 | ζ1917-ζ1916-ζ195 | ζ1914-ζ193-ζ192 | ζ1911-ζ197-ζ19 | -ζ1918-ζ1912+ζ198 | -ζ1917-ζ1916+ζ195 | -ζ1914-ζ193+ζ192 | ζ1918-ζ1912-ζ198 | ζ1915-ζ1913-ζ1910 | -ζ1918+ζ1912-ζ198 | ζ199-ζ196-ζ194 | -ζ199+ζ196-ζ194 | -ζ1914+ζ193-ζ192 | -ζ1917+ζ1916-ζ195 | -ζ1915+ζ1913-ζ1910 | complex faithful |
| ρ25 | 3 | -1 | 0 | 0 | ζ1911+ζ197+ζ19 | ζ1915+ζ1913+ζ1910 | ζ1914+ζ193+ζ192 | ζ199+ζ196+ζ194 | ζ1918+ζ1912+ζ198 | ζ1917+ζ1916+ζ195 | ζ1918-ζ1912-ζ198 | -ζ1917-ζ1916+ζ195 | ζ1911-ζ197-ζ19 | ζ1917-ζ1916-ζ195 | -ζ199-ζ196+ζ194 | -ζ1915-ζ1913+ζ1910 | -ζ1917+ζ1916-ζ195 | -ζ1914+ζ193-ζ192 | ζ199-ζ196-ζ194 | ζ1915-ζ1913-ζ1910 | -ζ1914-ζ193+ζ192 | -ζ1918-ζ1912+ζ198 | ζ1914-ζ193-ζ192 | -ζ1911-ζ197+ζ19 | -ζ1911+ζ197-ζ19 | -ζ1915+ζ1913-ζ1910 | -ζ199+ζ196-ζ194 | -ζ1918+ζ1912-ζ198 | complex faithful |
| ρ26 | 3 | -1 | 0 | 0 | ζ1911+ζ197+ζ19 | ζ1915+ζ1913+ζ1910 | ζ1914+ζ193+ζ192 | ζ199+ζ196+ζ194 | ζ1918+ζ1912+ζ198 | ζ1917+ζ1916+ζ195 | -ζ1918+ζ1912-ζ198 | -ζ1917+ζ1916-ζ195 | -ζ1911-ζ197+ζ19 | -ζ1917-ζ1916+ζ195 | ζ199-ζ196-ζ194 | -ζ1915+ζ1913-ζ1910 | ζ1917-ζ1916-ζ195 | -ζ1914-ζ193+ζ192 | -ζ199+ζ196-ζ194 | -ζ1915-ζ1913+ζ1910 | ζ1914-ζ193-ζ192 | ζ1918-ζ1912-ζ198 | -ζ1914+ζ193-ζ192 | -ζ1911+ζ197-ζ19 | ζ1911-ζ197-ζ19 | ζ1915-ζ1913-ζ1910 | -ζ199-ζ196+ζ194 | -ζ1918-ζ1912+ζ198 | complex faithful |
| ρ27 | 3 | 3 | 0 | 0 | ζ1917+ζ1916+ζ195 | ζ1918+ζ1912+ζ198 | ζ1915+ζ1913+ζ1910 | ζ1911+ζ197+ζ19 | ζ1914+ζ193+ζ192 | ζ199+ζ196+ζ194 | ζ1914+ζ193+ζ192 | ζ199+ζ196+ζ194 | ζ1917+ζ1916+ζ195 | ζ199+ζ196+ζ194 | ζ1911+ζ197+ζ19 | ζ1918+ζ1912+ζ198 | ζ199+ζ196+ζ194 | ζ1915+ζ1913+ζ1910 | ζ1911+ζ197+ζ19 | ζ1918+ζ1912+ζ198 | ζ1915+ζ1913+ζ1910 | ζ1914+ζ193+ζ192 | ζ1915+ζ1913+ζ1910 | ζ1917+ζ1916+ζ195 | ζ1917+ζ1916+ζ195 | ζ1918+ζ1912+ζ198 | ζ1911+ζ197+ζ19 | ζ1914+ζ193+ζ192 | complex lifted from C19⋊C3 |
| ρ28 | 3 | -1 | 0 | 0 | ζ1914+ζ193+ζ192 | ζ1911+ζ197+ζ19 | ζ199+ζ196+ζ194 | ζ1918+ζ1912+ζ198 | ζ1917+ζ1916+ζ195 | ζ1915+ζ1913+ζ1910 | -ζ1917-ζ1916+ζ195 | -ζ1915+ζ1913-ζ1910 | -ζ1914-ζ193+ζ192 | -ζ1915-ζ1913+ζ1910 | ζ1918-ζ1912-ζ198 | -ζ1911+ζ197-ζ19 | ζ1915-ζ1913-ζ1910 | -ζ199-ζ196+ζ194 | -ζ1918+ζ1912-ζ198 | -ζ1911-ζ197+ζ19 | ζ199-ζ196-ζ194 | ζ1917-ζ1916-ζ195 | -ζ199+ζ196-ζ194 | ζ1914-ζ193-ζ192 | -ζ1914+ζ193-ζ192 | ζ1911-ζ197-ζ19 | -ζ1918-ζ1912+ζ198 | -ζ1917+ζ1916-ζ195 | complex faithful |
►On 76 points
Generators in S76
(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)(58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76)
(1 41)(2 42)(3 43)(4 44)(5 45)(6 46)(7 47)(8 48)(9 49)(10 50)(11 51)(12 52)(13 53)(14 54)(15 55)(16 56)(17 57)(18 39)(19 40)(20 71)(21 72)(22 73)(23 74)(24 75)(25 76)(26 58)(27 59)(28 60)(29 61)(30 62)(31 63)(32 64)(33 65)(34 66)(35 67)(36 68)(37 69)(38 70)
(1 31)(2 32)(3 33)(4 34)(5 35)(6 36)(7 37)(8 38)(9 20)(10 21)(11 22)(12 23)(13 24)(14 25)(15 26)(16 27)(17 28)(18 29)(19 30)(39 61)(40 62)(41 63)(42 64)(43 65)(44 66)(45 67)(46 68)(47 69)(48 70)(49 71)(50 72)(51 73)(52 74)(53 75)(54 76)(55 58)(56 59)(57 60)
(2 8 12)(3 15 4)(5 10 7)(6 17 18)(9 19 13)(11 14 16)(20 62 53)(21 69 45)(22 76 56)(23 64 48)(24 71 40)(25 59 51)(26 66 43)(27 73 54)(28 61 46)(29 68 57)(30 75 49)(31 63 41)(32 70 52)(33 58 44)(34 65 55)(35 72 47)(36 60 39)(37 67 50)(38 74 42)
G:=sub<Sym(76)| (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)(58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76), (1,41)(2,42)(3,43)(4,44)(5,45)(6,46)(7,47)(8,48)(9,49)(10,50)(11,51)(12,52)(13,53)(14,54)(15,55)(16,56)(17,57)(18,39)(19,40)(20,71)(21,72)(22,73)(23,74)(24,75)(25,76)(26,58)(27,59)(28,60)(29,61)(30,62)(31,63)(32,64)(33,65)(34,66)(35,67)(36,68)(37,69)(38,70), (1,31)(2,32)(3,33)(4,34)(5,35)(6,36)(7,37)(8,38)(9,20)(10,21)(11,22)(12,23)(13,24)(14,25)(15,26)(16,27)(17,28)(18,29)(19,30)(39,61)(40,62)(41,63)(42,64)(43,65)(44,66)(45,67)(46,68)(47,69)(48,70)(49,71)(50,72)(51,73)(52,74)(53,75)(54,76)(55,58)(56,59)(57,60), (2,8,12)(3,15,4)(5,10,7)(6,17,18)(9,19,13)(11,14,16)(20,62,53)(21,69,45)(22,76,56)(23,64,48)(24,71,40)(25,59,51)(26,66,43)(27,73,54)(28,61,46)(29,68,57)(30,75,49)(31,63,41)(32,70,52)(33,58,44)(34,65,55)(35,72,47)(36,60,39)(37,67,50)(38,74,42)>;
G:=Group( (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)(58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76), (1,41)(2,42)(3,43)(4,44)(5,45)(6,46)(7,47)(8,48)(9,49)(10,50)(11,51)(12,52)(13,53)(14,54)(15,55)(16,56)(17,57)(18,39)(19,40)(20,71)(21,72)(22,73)(23,74)(24,75)(25,76)(26,58)(27,59)(28,60)(29,61)(30,62)(31,63)(32,64)(33,65)(34,66)(35,67)(36,68)(37,69)(38,70), (1,31)(2,32)(3,33)(4,34)(5,35)(6,36)(7,37)(8,38)(9,20)(10,21)(11,22)(12,23)(13,24)(14,25)(15,26)(16,27)(17,28)(18,29)(19,30)(39,61)(40,62)(41,63)(42,64)(43,65)(44,66)(45,67)(46,68)(47,69)(48,70)(49,71)(50,72)(51,73)(52,74)(53,75)(54,76)(55,58)(56,59)(57,60), (2,8,12)(3,15,4)(5,10,7)(6,17,18)(9,19,13)(11,14,16)(20,62,53)(21,69,45)(22,76,56)(23,64,48)(24,71,40)(25,59,51)(26,66,43)(27,73,54)(28,61,46)(29,68,57)(30,75,49)(31,63,41)(32,70,52)(33,58,44)(34,65,55)(35,72,47)(36,60,39)(37,67,50)(38,74,42) );
G=PermutationGroup([[(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),(58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76)], [(1,41),(2,42),(3,43),(4,44),(5,45),(6,46),(7,47),(8,48),(9,49),(10,50),(11,51),(12,52),(13,53),(14,54),(15,55),(16,56),(17,57),(18,39),(19,40),(20,71),(21,72),(22,73),(23,74),(24,75),(25,76),(26,58),(27,59),(28,60),(29,61),(30,62),(31,63),(32,64),(33,65),(34,66),(35,67),(36,68),(37,69),(38,70)], [(1,31),(2,32),(3,33),(4,34),(5,35),(6,36),(7,37),(8,38),(9,20),(10,21),(11,22),(12,23),(13,24),(14,25),(15,26),(16,27),(17,28),(18,29),(19,30),(39,61),(40,62),(41,63),(42,64),(43,65),(44,66),(45,67),(46,68),(47,69),(48,70),(49,71),(50,72),(51,73),(52,74),(53,75),(54,76),(55,58),(56,59),(57,60)], [(2,8,12),(3,15,4),(5,10,7),(6,17,18),(9,19,13),(11,14,16),(20,62,53),(21,69,45),(22,76,56),(23,64,48),(24,71,40),(25,59,51),(26,66,43),(27,73,54),(28,61,46),(29,68,57),(30,75,49),(31,63,41),(32,70,52),(33,58,44),(34,65,55),(35,72,47),(36,60,39),(37,67,50),(38,74,42)]])
C19⋊A4 is a maximal subgroup of
D19⋊A4
C19⋊A4 is a maximal quotient of C38.A4
Matrix representation of C19⋊A4 ►in GL3(𝔽229) generated by
| 0 | 1 | 0 |
| 0 | 0 | 1 |
| 1 | 167 | 46 |
| 50 | 92 | 183 |
| 183 | 154 | 37 |
| 37 | 179 | 24 |
| 222 | 6 | 90 |
| 90 | 138 | 24 |
| 24 | 205 | 97 |
| 1 | 0 | 0 |
| 109 | 182 | 142 |
| 1 | 167 | 46 |
G:=sub<GL(3,GF(229))| [0,0,1,1,0,167,0,1,46],[50,183,37,92,154,179,183,37,24],[222,90,24,6,138,205,90,24,97],[1,109,1,0,182,167,0,142,46] >;
C19⋊A4 in GAP, Magma, Sage, TeX
C_{19}\rtimes A_4 % in TeX
G:=Group("C19:A4"); // GroupNames label
G:=SmallGroup(228,11);
// by ID
G=gap.SmallGroup(228,11);
# by ID
G:=PCGroup([4,-3,-2,2,-19,49,110,1347]);
// Polycyclic
G:=Group<a,b,c,d|a^19=b^2=c^2=d^3=1,a*b=b*a,a*c=c*a,d*a*d^-1=a^11,d*b*d^-1=b*c=c*b,d*c*d^-1=b>;
// generators/relations
Export
Subgroup lattice of C19⋊A4 in TeX
Character table of C19⋊A4 in TeX