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## G = D76⋊C3order 456 = 23·3·19

### The semidirect product of D76 and C3 acting faithfully

Aliases: D76⋊C3, C761C6, D381C6, C4⋊(C19⋊C6), C19⋊C31D4, C191(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

 Derived series C1 — C38 — D76⋊C3
 Chief series C1 — C19 — C38 — C2×C19⋊C3 — C2×C19⋊C6 — D76⋊C3
 Lower central C19 — C38 — D76⋊C3
 Upper central C1 — C2 — C4

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

Smallest permutation representation of D76⋊C3
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`

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