metabelian, supersoluble, monomial
Aliases: D38⋊2C6, Dic19⋊C6, C19⋊C12⋊C2, C19⋊C3⋊2D4, C19⋊D4⋊C3, C19⋊2(C3×D4), (C2×C38)⋊3C6, C38.5(C2×C6), C22⋊2(C19⋊C6), (C2×C19⋊C6)⋊2C2, C2.5(C2×C19⋊C6), (C22×C19⋊C3)⋊1C2, (C2×C19⋊C3).5C22, SmallGroup(456,11)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D38⋊C6
G = < a,b,c | a38=b2=c6=1, bab=a-1, cac-1=a7, cbc-1=a25b >
Character table of D38⋊C6
| class | 1 | 2A | 2B | 2C | 3A | 3B | 4 | 6A | 6B | 6C | 6D | 6E | 6F | 12A | 12B | 19A | 19B | 19C | 38A | 38B | 38C | 38D | 38E | 38F | 38G | 38H | 38I | |
| size | 1 | 1 | 2 | 38 | 19 | 19 | 38 | 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 | ζ6 | ζ65 | ζ3 | ζ65 | ζ6 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 6 |
| ρ6 | 1 | 1 | -1 | -1 | ζ32 | ζ3 | 1 | ζ32 | ζ3 | ζ6 | ζ6 | ζ65 | ζ65 | ζ3 | ζ32 | 1 | 1 | 1 | -1 | 1 | -1 | -1 | -1 | 1 | 1 | -1 | -1 | linear of order 6 |
| ρ7 | 1 | 1 | 1 | -1 | ζ3 | ζ32 | -1 | ζ3 | ζ32 | ζ3 | ζ65 | ζ6 | ζ32 | ζ6 | ζ65 | 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 | ζ65 | ζ3 | ζ32 | ζ6 | 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 | ζ32 | ζ3 | ζ3 | 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 | ζ3 | ζ32 | ζ32 | 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 | ζ32 | ζ3 | ζ65 | ζ65 | ζ6 | 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 | ζ65 | ζ6 | ζ6 | ζ32 | ζ3 | 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 | 0 | -2 | 0 | 0 | 0 | -2 | -2 | 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 | 0 | -2 | 0 | 0 | 0 | -2 | -2 | 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 | 0 | -2 | 0 | 0 | 0 | -2 | -2 | 0 | 0 | complex lifted from C3×D4 |
| ρ16 | 6 | 6 | -6 | 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 | -ζ1917-ζ1916-ζ1914-ζ195-ζ193-ζ192 | 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 | -ζ1915-ζ1913-ζ1910-ζ199-ζ196-ζ194 | -ζ1918-ζ1912-ζ1911-ζ198-ζ197-ζ19 | orthogonal lifted from C2×C19⋊C6 |
| ρ17 | 6 | 6 | 6 | 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 | ζ1917+ζ1916+ζ1914+ζ195+ζ193+ζ192 | 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 | ζ1915+ζ1913+ζ1910+ζ199+ζ196+ζ194 | ζ1918+ζ1912+ζ1911+ζ198+ζ197+ζ19 | orthogonal lifted from C19⋊C6 |
| ρ18 | 6 | 6 | 6 | 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 | ζ1915+ζ1913+ζ1910+ζ199+ζ196+ζ194 | 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 | ζ1918+ζ1912+ζ1911+ζ198+ζ197+ζ19 | ζ1917+ζ1916+ζ1914+ζ195+ζ193+ζ192 | orthogonal lifted from C19⋊C6 |
| ρ19 | 6 | 6 | -6 | 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 | -ζ1915-ζ1913-ζ1910-ζ199-ζ196-ζ194 | 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 | -ζ1918-ζ1912-ζ1911-ζ198-ζ197-ζ19 | -ζ1917-ζ1916-ζ1914-ζ195-ζ193-ζ192 | orthogonal lifted from C2×C19⋊C6 |
| ρ20 | 6 | 6 | -6 | 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 | -ζ1918-ζ1912-ζ1911-ζ198-ζ197-ζ19 | 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 | -ζ1917-ζ1916-ζ1914-ζ195-ζ193-ζ192 | -ζ1915-ζ1913-ζ1910-ζ199-ζ196-ζ194 | orthogonal lifted from C2×C19⋊C6 |
| ρ21 | 6 | 6 | 6 | 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 | ζ1918+ζ1912+ζ1911+ζ198+ζ197+ζ19 | 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 | 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 | ζ1917+ζ1916+ζ1914+ζ195+ζ193+ζ192 | ζ1915+ζ1913+ζ1910+ζ199+ζ196+ζ194 | ζ1918+ζ1912+ζ1911+ζ198+ζ197+ζ19 | ζ1918+ζ1912-ζ1911+ζ198-ζ197-ζ19 | -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 | -ζ1917-ζ1916+ζ1914-ζ195+ζ193+ζ192 | -ζ1915-ζ1913-ζ1910+ζ199+ζ196+ζ194 | complex 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 | -ζ1917-ζ1916+ζ1914-ζ195+ζ193+ζ192 | -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 | ζ1915+ζ1913+ζ1910-ζ199-ζ196-ζ194 | -ζ1918-ζ1912+ζ1911-ζ198+ζ197+ζ19 | complex faithful |
| ρ24 | 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 | ζ1917+ζ1916-ζ1914+ζ195-ζ193-ζ192 | -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 | -ζ1915-ζ1913-ζ1910+ζ199+ζ196+ζ194 | ζ1918+ζ1912-ζ1911+ζ198-ζ197-ζ19 | complex faithful |
| ρ25 | 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 | -ζ1918-ζ1912+ζ1911-ζ198+ζ197+ζ19 | -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 | ζ1917+ζ1916-ζ1914+ζ195-ζ193-ζ192 | ζ1915+ζ1913+ζ1910-ζ199-ζ196-ζ194 | complex faithful |
| ρ26 | 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 | ζ1915+ζ1913+ζ1910-ζ199-ζ196-ζ194 | -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 | ζ1918+ζ1912-ζ1911+ζ198-ζ197-ζ19 | ζ1917+ζ1916-ζ1914+ζ195-ζ193-ζ192 | complex faithful |
| ρ27 | 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 | -ζ1915-ζ1913-ζ1910+ζ199+ζ196+ζ194 | -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 | -ζ1918-ζ1912+ζ1911-ζ198+ζ197+ζ19 | -ζ1917-ζ1916+ζ1914-ζ195+ζ193+ζ192 | 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 38)(2 37)(3 36)(4 35)(5 34)(6 33)(7 32)(8 31)(9 30)(10 29)(11 28)(12 27)(13 26)(14 25)(15 24)(16 23)(17 22)(18 21)(19 20)(39 57)(40 56)(41 55)(42 54)(43 53)(44 52)(45 51)(46 50)(47 49)(58 76)(59 75)(60 74)(61 73)(62 72)(63 71)(64 70)(65 69)(66 68)
(1 39)(2 50 8 40 12 46)(3 61 15 41 23 53)(4 72 22 42 34 60)(5 45 29 43 7 67)(6 56 36 44 18 74)(9 51 19 47 13 57)(10 62 26 48 24 64)(11 73 33 49 35 71)(14 68 16 52 30 54)(17 63 37 55 25 75)(20 58)(21 69 27 59 31 65)(28 70 38 66 32 76)
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), (1,38)(2,37)(3,36)(4,35)(5,34)(6,33)(7,32)(8,31)(9,30)(10,29)(11,28)(12,27)(13,26)(14,25)(15,24)(16,23)(17,22)(18,21)(19,20)(39,57)(40,56)(41,55)(42,54)(43,53)(44,52)(45,51)(46,50)(47,49)(58,76)(59,75)(60,74)(61,73)(62,72)(63,71)(64,70)(65,69)(66,68), (1,39)(2,50,8,40,12,46)(3,61,15,41,23,53)(4,72,22,42,34,60)(5,45,29,43,7,67)(6,56,36,44,18,74)(9,51,19,47,13,57)(10,62,26,48,24,64)(11,73,33,49,35,71)(14,68,16,52,30,54)(17,63,37,55,25,75)(20,58)(21,69,27,59,31,65)(28,70,38,66,32,76)>;
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), (1,38)(2,37)(3,36)(4,35)(5,34)(6,33)(7,32)(8,31)(9,30)(10,29)(11,28)(12,27)(13,26)(14,25)(15,24)(16,23)(17,22)(18,21)(19,20)(39,57)(40,56)(41,55)(42,54)(43,53)(44,52)(45,51)(46,50)(47,49)(58,76)(59,75)(60,74)(61,73)(62,72)(63,71)(64,70)(65,69)(66,68), (1,39)(2,50,8,40,12,46)(3,61,15,41,23,53)(4,72,22,42,34,60)(5,45,29,43,7,67)(6,56,36,44,18,74)(9,51,19,47,13,57)(10,62,26,48,24,64)(11,73,33,49,35,71)(14,68,16,52,30,54)(17,63,37,55,25,75)(20,58)(21,69,27,59,31,65)(28,70,38,66,32,76) );
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)], [(1,38),(2,37),(3,36),(4,35),(5,34),(6,33),(7,32),(8,31),(9,30),(10,29),(11,28),(12,27),(13,26),(14,25),(15,24),(16,23),(17,22),(18,21),(19,20),(39,57),(40,56),(41,55),(42,54),(43,53),(44,52),(45,51),(46,50),(47,49),(58,76),(59,75),(60,74),(61,73),(62,72),(63,71),(64,70),(65,69),(66,68)], [(1,39),(2,50,8,40,12,46),(3,61,15,41,23,53),(4,72,22,42,34,60),(5,45,29,43,7,67),(6,56,36,44,18,74),(9,51,19,47,13,57),(10,62,26,48,24,64),(11,73,33,49,35,71),(14,68,16,52,30,54),(17,63,37,55,25,75),(20,58),(21,69,27,59,31,65),(28,70,38,66,32,76)]])
Matrix representation of D38⋊C6 ►in GL8(𝔽229)
| 228 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 228 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 20 | 79 | 60 | 77 | 40 | 100 |
| 0 | 0 | 129 | 88 | 132 | 191 | 130 | 108 |
| 0 | 0 | 121 | 120 | 17 | 17 | 120 | 121 |
| 0 | 0 | 108 | 130 | 191 | 132 | 88 | 129 |
| 0 | 0 | 100 | 40 | 77 | 60 | 79 | 20 |
| 0 | 0 | 209 | 22 | 188 | 149 | 208 | 1 |
| 228 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 202 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 20 | 79 | 60 | 77 | 40 | 100 |
| 0 | 0 | 1 | 208 | 149 | 188 | 22 | 209 |
| 0 | 0 | 227 | 19 | 100 | 121 | 228 | 228 |
| 0 | 0 | 210 | 131 | 68 | 30 | 190 | 2 |
| 0 | 0 | 228 | 19 | 99 | 141 | 99 | 19 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 135 | 219 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 94 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 228 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 19 | 98 | 161 | 199 | 39 | 227 |
| 0 | 0 | 129 | 189 | 152 | 169 | 150 | 209 |
| 0 | 0 | 0 | 0 | 228 | 0 | 0 | 0 |
| 0 | 0 | 228 | 21 | 80 | 41 | 207 | 20 |
| 0 | 0 | 1 | 1 | 108 | 129 | 210 | 2 |
G:=sub<GL(8,GF(229))| [228,0,0,0,0,0,0,0,0,228,0,0,0,0,0,0,0,0,20,129,121,108,100,209,0,0,79,88,120,130,40,22,0,0,60,132,17,191,77,188,0,0,77,191,17,132,60,149,0,0,40,130,120,88,79,208,0,0,100,108,121,129,20,1],[228,202,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,20,1,227,210,228,0,0,0,79,208,19,131,19,0,0,0,60,149,100,68,99,0,0,0,77,188,121,30,141,0,0,0,40,22,228,190,99,0,0,0,100,209,228,2,19,1],[135,0,0,0,0,0,0,0,219,94,0,0,0,0,0,0,0,0,228,19,129,0,228,1,0,0,0,98,189,0,21,1,0,0,0,161,152,228,80,108,0,0,0,199,169,0,41,129,0,0,0,39,150,0,207,210,0,0,0,227,209,0,20,2] >;
D38⋊C6 in GAP, Magma, Sage, TeX
D_{38}\rtimes C_6 % in TeX
G:=Group("D38:C6"); // GroupNames label
G:=SmallGroup(456,11);
// by ID
G=gap.SmallGroup(456,11);
# by ID
G:=PCGroup([5,-2,-2,-3,-2,-19,141,10804,1064]);
// Polycyclic
G:=Group<a,b,c|a^38=b^2=c^6=1,b*a*b=a^-1,c*a*c^-1=a^7,c*b*c^-1=a^25*b>;
// generators/relations
Export
Subgroup lattice of D38⋊C6 in TeX
Character table of D38⋊C6 in TeX