metacyclic, supersoluble, monomial
Aliases: D76⋊C3, C76⋊1C6, D38⋊1C6, C4⋊(C19⋊C6), C19⋊C3⋊1D4, C19⋊1(C3×D4), C38.3(C2×C6), (C2×C19⋊C6)⋊1C2, (C4×C19⋊C3)⋊1C2, C2.4(C2×C19⋊C6), (C2×C19⋊C3).3C22, SmallGroup(456,9)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D76⋊C3
G = < a,b,c | a76=b2=c3=1, bab=a-1, cac-1=a49, cbc-1=a48b >
Character table of D76⋊C3
| class | 1 | 2A | 2B | 2C | 3A | 3B | 4 | 6A | 6B | 6C | 6D | 6E | 6F | 12A | 12B | 19A | 19B | 19C | 38A | 38B | 38C | 76A | 76B | 76C | 76D | 76E | 76F | |
| size | 1 | 1 | 38 | 38 | 19 | 19 | 2 | 19 | 19 | 38 | 38 | 38 | 38 | 38 | 38 | 6 | 6 | 6 | 6 | 6 | 6 | 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 | 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 |
| ρ4 | 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 |
| ρ5 | 1 | 1 | 1 | -1 | ζ32 | ζ3 | -1 | ζ32 | ζ3 | ζ32 | ζ65 | ζ3 | ζ6 | ζ65 | ζ6 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | linear of order 6 |
| ρ6 | 1 | 1 | -1 | -1 | ζ3 | ζ32 | 1 | ζ3 | ζ32 | ζ65 | ζ6 | ζ6 | ζ65 | ζ32 | ζ3 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 6 |
| ρ7 | 1 | 1 | -1 | 1 | ζ32 | ζ3 | -1 | ζ32 | ζ3 | ζ6 | ζ3 | ζ65 | ζ32 | ζ65 | ζ6 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | linear of order 6 |
| ρ8 | 1 | 1 | 1 | -1 | ζ3 | ζ32 | -1 | ζ3 | ζ32 | ζ3 | ζ6 | ζ32 | ζ65 | ζ6 | ζ65 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | linear of order 6 |
| ρ9 | 1 | 1 | 1 | 1 | ζ32 | ζ3 | 1 | ζ32 | ζ3 | ζ32 | ζ3 | ζ3 | ζ32 | ζ3 | ζ32 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 3 |
| ρ10 | 1 | 1 | 1 | 1 | ζ3 | ζ32 | 1 | ζ3 | ζ32 | ζ3 | ζ32 | ζ32 | ζ3 | ζ32 | ζ3 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 3 |
| ρ11 | 1 | 1 | -1 | -1 | ζ32 | ζ3 | 1 | ζ32 | ζ3 | ζ6 | ζ65 | ζ65 | ζ6 | ζ3 | ζ32 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 6 |
| ρ12 | 1 | 1 | -1 | 1 | ζ3 | ζ32 | -1 | ζ3 | ζ32 | ζ65 | ζ32 | ζ6 | ζ3 | ζ6 | ζ65 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | linear of order 6 |
| ρ13 | 2 | -2 | 0 | 0 | 2 | 2 | 0 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 2 | 2 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ14 | 2 | -2 | 0 | 0 | -1-√-3 | -1+√-3 | 0 | 1+√-3 | 1-√-3 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 2 | 2 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C3×D4 |
| ρ15 | 2 | -2 | 0 | 0 | -1+√-3 | -1-√-3 | 0 | 1-√-3 | 1+√-3 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 2 | 2 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C3×D4 |
| ρ16 | 6 | 6 | 0 | 0 | 0 | 0 | 6 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ζ1917+ζ1916+ζ1914+ζ195+ζ193+ζ192 | ζ1915+ζ1913+ζ1910+ζ199+ζ196+ζ194 | ζ1918+ζ1912+ζ1911+ζ198+ζ197+ζ19 | ζ1915+ζ1913+ζ1910+ζ199+ζ196+ζ194 | ζ1918+ζ1912+ζ1911+ζ198+ζ197+ζ19 | ζ1917+ζ1916+ζ1914+ζ195+ζ193+ζ192 | ζ1915+ζ1913+ζ1910+ζ199+ζ196+ζ194 | ζ1917+ζ1916+ζ1914+ζ195+ζ193+ζ192 | ζ1918+ζ1912+ζ1911+ζ198+ζ197+ζ19 | ζ1915+ζ1913+ζ1910+ζ199+ζ196+ζ194 | ζ1918+ζ1912+ζ1911+ζ198+ζ197+ζ19 | ζ1917+ζ1916+ζ1914+ζ195+ζ193+ζ192 | orthogonal lifted from C19⋊C6 |
| ρ17 | 6 | 6 | 0 | 0 | 0 | 0 | -6 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ζ1917+ζ1916+ζ1914+ζ195+ζ193+ζ192 | ζ1915+ζ1913+ζ1910+ζ199+ζ196+ζ194 | ζ1918+ζ1912+ζ1911+ζ198+ζ197+ζ19 | ζ1915+ζ1913+ζ1910+ζ199+ζ196+ζ194 | ζ1918+ζ1912+ζ1911+ζ198+ζ197+ζ19 | ζ1917+ζ1916+ζ1914+ζ195+ζ193+ζ192 | -ζ1915-ζ1913-ζ1910-ζ199-ζ196-ζ194 | -ζ1917-ζ1916-ζ1914-ζ195-ζ193-ζ192 | -ζ1918-ζ1912-ζ1911-ζ198-ζ197-ζ19 | -ζ1915-ζ1913-ζ1910-ζ199-ζ196-ζ194 | -ζ1918-ζ1912-ζ1911-ζ198-ζ197-ζ19 | -ζ1917-ζ1916-ζ1914-ζ195-ζ193-ζ192 | orthogonal lifted from C2×C19⋊C6 |
| ρ18 | 6 | 6 | 0 | 0 | 0 | 0 | -6 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ζ1915+ζ1913+ζ1910+ζ199+ζ196+ζ194 | ζ1918+ζ1912+ζ1911+ζ198+ζ197+ζ19 | ζ1917+ζ1916+ζ1914+ζ195+ζ193+ζ192 | ζ1918+ζ1912+ζ1911+ζ198+ζ197+ζ19 | ζ1917+ζ1916+ζ1914+ζ195+ζ193+ζ192 | ζ1915+ζ1913+ζ1910+ζ199+ζ196+ζ194 | -ζ1918-ζ1912-ζ1911-ζ198-ζ197-ζ19 | -ζ1915-ζ1913-ζ1910-ζ199-ζ196-ζ194 | -ζ1917-ζ1916-ζ1914-ζ195-ζ193-ζ192 | -ζ1918-ζ1912-ζ1911-ζ198-ζ197-ζ19 | -ζ1917-ζ1916-ζ1914-ζ195-ζ193-ζ192 | -ζ1915-ζ1913-ζ1910-ζ199-ζ196-ζ194 | orthogonal lifted from C2×C19⋊C6 |
| ρ19 | 6 | 6 | 0 | 0 | 0 | 0 | -6 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ζ1918+ζ1912+ζ1911+ζ198+ζ197+ζ19 | ζ1917+ζ1916+ζ1914+ζ195+ζ193+ζ192 | ζ1915+ζ1913+ζ1910+ζ199+ζ196+ζ194 | ζ1917+ζ1916+ζ1914+ζ195+ζ193+ζ192 | ζ1915+ζ1913+ζ1910+ζ199+ζ196+ζ194 | ζ1918+ζ1912+ζ1911+ζ198+ζ197+ζ19 | -ζ1917-ζ1916-ζ1914-ζ195-ζ193-ζ192 | -ζ1918-ζ1912-ζ1911-ζ198-ζ197-ζ19 | -ζ1915-ζ1913-ζ1910-ζ199-ζ196-ζ194 | -ζ1917-ζ1916-ζ1914-ζ195-ζ193-ζ192 | -ζ1915-ζ1913-ζ1910-ζ199-ζ196-ζ194 | -ζ1918-ζ1912-ζ1911-ζ198-ζ197-ζ19 | orthogonal lifted from C2×C19⋊C6 |
| ρ20 | 6 | 6 | 0 | 0 | 0 | 0 | 6 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ζ1918+ζ1912+ζ1911+ζ198+ζ197+ζ19 | ζ1917+ζ1916+ζ1914+ζ195+ζ193+ζ192 | ζ1915+ζ1913+ζ1910+ζ199+ζ196+ζ194 | ζ1917+ζ1916+ζ1914+ζ195+ζ193+ζ192 | ζ1915+ζ1913+ζ1910+ζ199+ζ196+ζ194 | ζ1918+ζ1912+ζ1911+ζ198+ζ197+ζ19 | ζ1917+ζ1916+ζ1914+ζ195+ζ193+ζ192 | ζ1918+ζ1912+ζ1911+ζ198+ζ197+ζ19 | ζ1915+ζ1913+ζ1910+ζ199+ζ196+ζ194 | ζ1917+ζ1916+ζ1914+ζ195+ζ193+ζ192 | ζ1915+ζ1913+ζ1910+ζ199+ζ196+ζ194 | ζ1918+ζ1912+ζ1911+ζ198+ζ197+ζ19 | orthogonal lifted from C19⋊C6 |
| ρ21 | 6 | 6 | 0 | 0 | 0 | 0 | 6 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ζ1915+ζ1913+ζ1910+ζ199+ζ196+ζ194 | ζ1918+ζ1912+ζ1911+ζ198+ζ197+ζ19 | ζ1917+ζ1916+ζ1914+ζ195+ζ193+ζ192 | ζ1918+ζ1912+ζ1911+ζ198+ζ197+ζ19 | ζ1917+ζ1916+ζ1914+ζ195+ζ193+ζ192 | ζ1915+ζ1913+ζ1910+ζ199+ζ196+ζ194 | ζ1918+ζ1912+ζ1911+ζ198+ζ197+ζ19 | ζ1915+ζ1913+ζ1910+ζ199+ζ196+ζ194 | ζ1917+ζ1916+ζ1914+ζ195+ζ193+ζ192 | ζ1918+ζ1912+ζ1911+ζ198+ζ197+ζ19 | ζ1917+ζ1916+ζ1914+ζ195+ζ193+ζ192 | ζ1915+ζ1913+ζ1910+ζ199+ζ196+ζ194 | orthogonal lifted from C19⋊C6 |
| ρ22 | 6 | -6 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ζ1918+ζ1912+ζ1911+ζ198+ζ197+ζ19 | ζ1917+ζ1916+ζ1914+ζ195+ζ193+ζ192 | ζ1915+ζ1913+ζ1910+ζ199+ζ196+ζ194 | -ζ1917-ζ1916-ζ1914-ζ195-ζ193-ζ192 | -ζ1915-ζ1913-ζ1910-ζ199-ζ196-ζ194 | -ζ1918-ζ1912-ζ1911-ζ198-ζ197-ζ19 | ζ4ζ1917+ζ4ζ1916-ζ4ζ1914+ζ4ζ195-ζ4ζ193-ζ4ζ192 | ζ4ζ1918+ζ4ζ1912-ζ4ζ1911+ζ4ζ198-ζ4ζ197-ζ4ζ19 | ζ4ζ1915+ζ4ζ1913+ζ4ζ1910-ζ4ζ199-ζ4ζ196-ζ4ζ194 | ζ43ζ1917+ζ43ζ1916-ζ43ζ1914+ζ43ζ195-ζ43ζ193-ζ43ζ192 | ζ43ζ1915+ζ43ζ1913+ζ43ζ1910-ζ43ζ199-ζ43ζ196-ζ43ζ194 | -ζ4ζ1918-ζ4ζ1912+ζ4ζ1911-ζ4ζ198+ζ4ζ197+ζ4ζ19 | orthogonal faithful |
| ρ23 | 6 | -6 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ζ1915+ζ1913+ζ1910+ζ199+ζ196+ζ194 | ζ1918+ζ1912+ζ1911+ζ198+ζ197+ζ19 | ζ1917+ζ1916+ζ1914+ζ195+ζ193+ζ192 | -ζ1918-ζ1912-ζ1911-ζ198-ζ197-ζ19 | -ζ1917-ζ1916-ζ1914-ζ195-ζ193-ζ192 | -ζ1915-ζ1913-ζ1910-ζ199-ζ196-ζ194 | ζ4ζ1918+ζ4ζ1912-ζ4ζ1911+ζ4ζ198-ζ4ζ197-ζ4ζ19 | ζ43ζ1915+ζ43ζ1913+ζ43ζ1910-ζ43ζ199-ζ43ζ196-ζ43ζ194 | ζ4ζ1917+ζ4ζ1916-ζ4ζ1914+ζ4ζ195-ζ4ζ193-ζ4ζ192 | -ζ4ζ1918-ζ4ζ1912+ζ4ζ1911-ζ4ζ198+ζ4ζ197+ζ4ζ19 | ζ43ζ1917+ζ43ζ1916-ζ43ζ1914+ζ43ζ195-ζ43ζ193-ζ43ζ192 | ζ4ζ1915+ζ4ζ1913+ζ4ζ1910-ζ4ζ199-ζ4ζ196-ζ4ζ194 | orthogonal faithful |
| ρ24 | 6 | -6 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ζ1917+ζ1916+ζ1914+ζ195+ζ193+ζ192 | ζ1915+ζ1913+ζ1910+ζ199+ζ196+ζ194 | ζ1918+ζ1912+ζ1911+ζ198+ζ197+ζ19 | -ζ1915-ζ1913-ζ1910-ζ199-ζ196-ζ194 | -ζ1918-ζ1912-ζ1911-ζ198-ζ197-ζ19 | -ζ1917-ζ1916-ζ1914-ζ195-ζ193-ζ192 | ζ43ζ1915+ζ43ζ1913+ζ43ζ1910-ζ43ζ199-ζ43ζ196-ζ43ζ194 | ζ43ζ1917+ζ43ζ1916-ζ43ζ1914+ζ43ζ195-ζ43ζ193-ζ43ζ192 | ζ4ζ1918+ζ4ζ1912-ζ4ζ1911+ζ4ζ198-ζ4ζ197-ζ4ζ19 | ζ4ζ1915+ζ4ζ1913+ζ4ζ1910-ζ4ζ199-ζ4ζ196-ζ4ζ194 | -ζ4ζ1918-ζ4ζ1912+ζ4ζ1911-ζ4ζ198+ζ4ζ197+ζ4ζ19 | ζ4ζ1917+ζ4ζ1916-ζ4ζ1914+ζ4ζ195-ζ4ζ193-ζ4ζ192 | orthogonal faithful |
| ρ25 | 6 | -6 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ζ1918+ζ1912+ζ1911+ζ198+ζ197+ζ19 | ζ1917+ζ1916+ζ1914+ζ195+ζ193+ζ192 | ζ1915+ζ1913+ζ1910+ζ199+ζ196+ζ194 | -ζ1917-ζ1916-ζ1914-ζ195-ζ193-ζ192 | -ζ1915-ζ1913-ζ1910-ζ199-ζ196-ζ194 | -ζ1918-ζ1912-ζ1911-ζ198-ζ197-ζ19 | ζ43ζ1917+ζ43ζ1916-ζ43ζ1914+ζ43ζ195-ζ43ζ193-ζ43ζ192 | -ζ4ζ1918-ζ4ζ1912+ζ4ζ1911-ζ4ζ198+ζ4ζ197+ζ4ζ19 | ζ43ζ1915+ζ43ζ1913+ζ43ζ1910-ζ43ζ199-ζ43ζ196-ζ43ζ194 | ζ4ζ1917+ζ4ζ1916-ζ4ζ1914+ζ4ζ195-ζ4ζ193-ζ4ζ192 | ζ4ζ1915+ζ4ζ1913+ζ4ζ1910-ζ4ζ199-ζ4ζ196-ζ4ζ194 | ζ4ζ1918+ζ4ζ1912-ζ4ζ1911+ζ4ζ198-ζ4ζ197-ζ4ζ19 | orthogonal faithful |
| ρ26 | 6 | -6 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ζ1917+ζ1916+ζ1914+ζ195+ζ193+ζ192 | ζ1915+ζ1913+ζ1910+ζ199+ζ196+ζ194 | ζ1918+ζ1912+ζ1911+ζ198+ζ197+ζ19 | -ζ1915-ζ1913-ζ1910-ζ199-ζ196-ζ194 | -ζ1918-ζ1912-ζ1911-ζ198-ζ197-ζ19 | -ζ1917-ζ1916-ζ1914-ζ195-ζ193-ζ192 | ζ4ζ1915+ζ4ζ1913+ζ4ζ1910-ζ4ζ199-ζ4ζ196-ζ4ζ194 | ζ4ζ1917+ζ4ζ1916-ζ4ζ1914+ζ4ζ195-ζ4ζ193-ζ4ζ192 | -ζ4ζ1918-ζ4ζ1912+ζ4ζ1911-ζ4ζ198+ζ4ζ197+ζ4ζ19 | ζ43ζ1915+ζ43ζ1913+ζ43ζ1910-ζ43ζ199-ζ43ζ196-ζ43ζ194 | ζ4ζ1918+ζ4ζ1912-ζ4ζ1911+ζ4ζ198-ζ4ζ197-ζ4ζ19 | ζ43ζ1917+ζ43ζ1916-ζ43ζ1914+ζ43ζ195-ζ43ζ193-ζ43ζ192 | orthogonal faithful |
| ρ27 | 6 | -6 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ζ1915+ζ1913+ζ1910+ζ199+ζ196+ζ194 | ζ1918+ζ1912+ζ1911+ζ198+ζ197+ζ19 | ζ1917+ζ1916+ζ1914+ζ195+ζ193+ζ192 | -ζ1918-ζ1912-ζ1911-ζ198-ζ197-ζ19 | -ζ1917-ζ1916-ζ1914-ζ195-ζ193-ζ192 | -ζ1915-ζ1913-ζ1910-ζ199-ζ196-ζ194 | -ζ4ζ1918-ζ4ζ1912+ζ4ζ1911-ζ4ζ198+ζ4ζ197+ζ4ζ19 | ζ4ζ1915+ζ4ζ1913+ζ4ζ1910-ζ4ζ199-ζ4ζ196-ζ4ζ194 | ζ43ζ1917+ζ43ζ1916-ζ43ζ1914+ζ43ζ195-ζ43ζ193-ζ43ζ192 | ζ4ζ1918+ζ4ζ1912-ζ4ζ1911+ζ4ζ198-ζ4ζ197-ζ4ζ19 | ζ4ζ1917+ζ4ζ1916-ζ4ζ1914+ζ4ζ195-ζ4ζ193-ζ4ζ192 | ζ43ζ1915+ζ43ζ1913+ζ43ζ1910-ζ43ζ199-ζ43ζ196-ζ43ζ194 | orthogonal 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 19)(2 18)(3 17)(4 16)(5 15)(6 14)(7 13)(8 12)(9 11)(20 76)(21 75)(22 74)(23 73)(24 72)(25 71)(26 70)(27 69)(28 68)(29 67)(30 66)(31 65)(32 64)(33 63)(34 62)(35 61)(36 60)(37 59)(38 58)(39 57)(40 56)(41 55)(42 54)(43 53)(44 52)(45 51)(46 50)(47 49)
(2 46 50)(3 15 23)(4 60 72)(5 29 45)(6 74 18)(7 43 67)(8 12 40)(9 57 13)(10 26 62)(11 71 35)(14 54 30)(16 68 52)(17 37 25)(19 51 47)(21 65 69)(22 34 42)(24 48 64)(27 31 59)(28 76 32)(33 73 49)(36 56 44)(38 70 66)(41 53 61)(55 75 63)
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), (1,19)(2,18)(3,17)(4,16)(5,15)(6,14)(7,13)(8,12)(9,11)(20,76)(21,75)(22,74)(23,73)(24,72)(25,71)(26,70)(27,69)(28,68)(29,67)(30,66)(31,65)(32,64)(33,63)(34,62)(35,61)(36,60)(37,59)(38,58)(39,57)(40,56)(41,55)(42,54)(43,53)(44,52)(45,51)(46,50)(47,49), (2,46,50)(3,15,23)(4,60,72)(5,29,45)(6,74,18)(7,43,67)(8,12,40)(9,57,13)(10,26,62)(11,71,35)(14,54,30)(16,68,52)(17,37,25)(19,51,47)(21,65,69)(22,34,42)(24,48,64)(27,31,59)(28,76,32)(33,73,49)(36,56,44)(38,70,66)(41,53,61)(55,75,63)>;
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), (1,19)(2,18)(3,17)(4,16)(5,15)(6,14)(7,13)(8,12)(9,11)(20,76)(21,75)(22,74)(23,73)(24,72)(25,71)(26,70)(27,69)(28,68)(29,67)(30,66)(31,65)(32,64)(33,63)(34,62)(35,61)(36,60)(37,59)(38,58)(39,57)(40,56)(41,55)(42,54)(43,53)(44,52)(45,51)(46,50)(47,49), (2,46,50)(3,15,23)(4,60,72)(5,29,45)(6,74,18)(7,43,67)(8,12,40)(9,57,13)(10,26,62)(11,71,35)(14,54,30)(16,68,52)(17,37,25)(19,51,47)(21,65,69)(22,34,42)(24,48,64)(27,31,59)(28,76,32)(33,73,49)(36,56,44)(38,70,66)(41,53,61)(55,75,63) );
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)], [(1,19),(2,18),(3,17),(4,16),(5,15),(6,14),(7,13),(8,12),(9,11),(20,76),(21,75),(22,74),(23,73),(24,72),(25,71),(26,70),(27,69),(28,68),(29,67),(30,66),(31,65),(32,64),(33,63),(34,62),(35,61),(36,60),(37,59),(38,58),(39,57),(40,56),(41,55),(42,54),(43,53),(44,52),(45,51),(46,50),(47,49)], [(2,46,50),(3,15,23),(4,60,72),(5,29,45),(6,74,18),(7,43,67),(8,12,40),(9,57,13),(10,26,62),(11,71,35),(14,54,30),(16,68,52),(17,37,25),(19,51,47),(21,65,69),(22,34,42),(24,48,64),(27,31,59),(28,76,32),(33,73,49),(36,56,44),(38,70,66),(41,53,61),(55,75,63)]])
Matrix representation of D76⋊C3 ►in GL8(𝔽229)
| 1 | 83 | 0 | 0 | 0 | 0 | 0 | 0 |
| 160 | 228 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 100 | 40 | 77 | 60 | 79 | 20 |
| 0 | 0 | 209 | 22 | 188 | 149 | 208 | 1 |
| 0 | 0 | 228 | 228 | 121 | 100 | 19 | 227 |
| 0 | 0 | 2 | 190 | 30 | 68 | 131 | 210 |
| 0 | 0 | 19 | 99 | 141 | 99 | 19 | 228 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 228 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 69 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 100 | 40 | 77 | 60 | 79 | 20 |
| 0 | 0 | 108 | 130 | 191 | 132 | 88 | 129 |
| 0 | 0 | 121 | 120 | 17 | 17 | 120 | 121 |
| 0 | 0 | 129 | 88 | 132 | 191 | 130 | 108 |
| 0 | 0 | 20 | 79 | 60 | 77 | 40 | 100 |
| 0 | 0 | 1 | 208 | 149 | 188 | 22 | 209 |
| 134 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 134 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 129 | 88 | 132 | 191 | 130 | 108 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 100 | 40 | 77 | 60 | 79 | 20 |
| 0 | 0 | 228 | 19 | 99 | 141 | 99 | 19 |
| 0 | 0 | 2 | 190 | 30 | 68 | 131 | 210 |
G:=sub<GL(8,GF(229))| [1,160,0,0,0,0,0,0,83,228,0,0,0,0,0,0,0,0,100,209,228,2,19,1,0,0,40,22,228,190,99,0,0,0,77,188,121,30,141,0,0,0,60,149,100,68,99,0,0,0,79,208,19,131,19,0,0,0,20,1,227,210,228,0],[228,69,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,100,108,121,129,20,1,0,0,40,130,120,88,79,208,0,0,77,191,17,132,60,149,0,0,60,132,17,191,77,188,0,0,79,88,120,130,40,22,0,0,20,129,121,108,100,209],[134,0,0,0,0,0,0,0,0,134,0,0,0,0,0,0,0,0,1,129,0,100,228,2,0,0,0,88,0,40,19,190,0,0,0,132,0,77,99,30,0,0,0,191,1,60,141,68,0,0,0,130,0,79,99,131,0,0,0,108,0,20,19,210] >;
D76⋊C3 in GAP, Magma, Sage, TeX
D_{76}\rtimes C_3 % in TeX
G:=Group("D76:C3"); // GroupNames label
G:=SmallGroup(456,9);
// by ID
G=gap.SmallGroup(456,9);
# by ID
G:=PCGroup([5,-2,-2,-3,-2,-19,141,66,10804,1064]);
// Polycyclic
G:=Group<a,b,c|a^76=b^2=c^3=1,b*a*b=a^-1,c*a*c^-1=a^49,c*b*c^-1=a^48*b>;
// generators/relations
Export
Subgroup lattice of D76⋊C3 in TeX
Character table of D76⋊C3 in TeX