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## G = Dic38⋊C3order 456 = 23·3·19

### The semidirect product of Dic38 and C3 acting faithfully

Aliases: Dic38⋊C3, C76.1C6, Dic19.C6, C19⋊C3⋊Q8, C19⋊(C3×Q8), C19⋊C12.C2, C4.(C19⋊C6), C38.1(C2×C6), C2.3(C2×C19⋊C6), (C4×C19⋊C3).1C2, (C2×C19⋊C3).1C22, SmallGroup(456,7)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C38 — Dic38⋊C3
 Chief series C1 — C19 — C38 — C2×C19⋊C3 — C19⋊C12 — Dic38⋊C3
 Lower central C19 — C38 — Dic38⋊C3
 Upper central C1 — C2 — C4

Generators and relations for Dic38⋊C3
G = < a,b,c | a76=c3=1, b2=a38, bab-1=a-1, cac-1=a49, bc=cb >

Character table of Dic38⋊C3

 class 1 2 3A 3B 4A 4B 4C 6A 6B 12A 12B 12C 12D 12E 12F 19A 19B 19C 38A 38B 38C 76A 76B 76C 76D 76E 76F size 1 1 19 19 2 38 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 ζ3 ζ32 1 -1 -1 ζ3 ζ32 ζ65 ζ6 ζ65 ζ3 ζ6 ζ32 1 1 1 1 1 1 1 1 1 1 1 1 linear of order 6 ρ6 1 1 ζ32 ζ3 1 -1 -1 ζ32 ζ3 ζ6 ζ65 ζ6 ζ32 ζ65 ζ3 1 1 1 1 1 1 1 1 1 1 1 1 linear of order 6 ρ7 1 1 ζ3 ζ32 -1 -1 1 ζ3 ζ32 ζ3 ζ6 ζ65 ζ65 ζ32 ζ6 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 linear of order 6 ρ8 1 1 ζ32 ζ3 1 1 1 ζ32 ζ3 ζ32 ζ3 ζ32 ζ32 ζ3 ζ3 1 1 1 1 1 1 1 1 1 1 1 1 linear of order 3 ρ9 1 1 ζ32 ζ3 -1 -1 1 ζ32 ζ3 ζ32 ζ65 ζ6 ζ6 ζ3 ζ65 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 linear of order 6 ρ10 1 1 ζ32 ζ3 -1 1 -1 ζ32 ζ3 ζ6 ζ3 ζ32 ζ6 ζ65 ζ65 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 linear of order 6 ρ11 1 1 ζ3 ζ32 1 1 1 ζ3 ζ32 ζ3 ζ32 ζ3 ζ3 ζ32 ζ32 1 1 1 1 1 1 1 1 1 1 1 1 linear of order 3 ρ12 1 1 ζ3 ζ32 -1 1 -1 ζ3 ζ32 ζ65 ζ32 ζ3 ζ65 ζ6 ζ6 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 linear of order 6 ρ13 2 -2 2 2 0 0 0 -2 -2 0 0 0 0 0 0 2 2 2 -2 -2 -2 0 0 0 0 0 0 symplectic lifted from Q8, Schur index 2 ρ14 2 -2 -1+√-3 -1-√-3 0 0 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×Q8 ρ15 2 -2 -1-√-3 -1+√-3 0 0 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×Q8 ρ16 6 6 0 0 6 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 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 ρ17 6 6 0 0 -6 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 -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 ρ18 6 6 0 0 6 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 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 ρ19 6 6 0 0 6 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 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 ρ20 6 6 0 0 -6 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 -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 ρ21 6 6 0 0 -6 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 -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 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 -ζ1915-ζ1913-ζ1910-ζ199-ζ196-ζ194 -ζ1918-ζ1912-ζ1911-ζ198-ζ197-ζ19 -ζ1917-ζ1916-ζ1914-ζ195-ζ193-ζ192 ζ4ζ1918+ζ4ζ1912-ζ4ζ1911+ζ4ζ198-ζ4ζ197-ζ4ζ19 ζ4ζ1915+ζ4ζ1913+ζ4ζ1910-ζ4ζ199-ζ4ζ196-ζ4ζ194 ζ4ζ1917+ζ4ζ1916-ζ4ζ1914+ζ4ζ195-ζ4ζ193-ζ4ζ192 -ζ4ζ1918-ζ4ζ1912+ζ4ζ1911-ζ4ζ198+ζ4ζ197+ζ4ζ19 ζ43ζ1917+ζ43ζ1916-ζ43ζ1914+ζ43ζ195-ζ43ζ193-ζ43ζ192 ζ43ζ1915+ζ43ζ1913+ζ43ζ1910-ζ43ζ199-ζ43ζ196-ζ43ζ194 symplectic faithful, Schur index 2 ρ23 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 ζ4ζ1915+ζ4ζ1913+ζ4ζ1910-ζ4ζ199-ζ4ζ196-ζ4ζ194 ζ43ζ1917+ζ43ζ1916-ζ43ζ1914+ζ43ζ195-ζ43ζ193-ζ43ζ192 -ζ4ζ1918-ζ4ζ1912+ζ4ζ1911-ζ4ζ198+ζ4ζ197+ζ4ζ19 ζ43ζ1915+ζ43ζ1913+ζ43ζ1910-ζ43ζ199-ζ43ζ196-ζ43ζ194 ζ4ζ1918+ζ4ζ1912-ζ4ζ1911+ζ4ζ198-ζ4ζ197-ζ4ζ19 ζ4ζ1917+ζ4ζ1916-ζ4ζ1914+ζ4ζ195-ζ4ζ193-ζ4ζ192 symplectic faithful, Schur index 2 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 -ζ4ζ1918-ζ4ζ1912+ζ4ζ1911-ζ4ζ198+ζ4ζ197+ζ4ζ19 ζ43ζ1915+ζ43ζ1913+ζ43ζ1910-ζ43ζ199-ζ43ζ196-ζ43ζ194 ζ43ζ1917+ζ43ζ1916-ζ43ζ1914+ζ43ζ195-ζ43ζ193-ζ43ζ192 ζ4ζ1918+ζ4ζ1912-ζ4ζ1911+ζ4ζ198-ζ4ζ197-ζ4ζ19 ζ4ζ1917+ζ4ζ1916-ζ4ζ1914+ζ4ζ195-ζ4ζ193-ζ4ζ192 ζ4ζ1915+ζ4ζ1913+ζ4ζ1910-ζ4ζ199-ζ4ζ196-ζ4ζ194 symplectic faithful, Schur index 2 ρ25 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 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 symplectic faithful, Schur index 2 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 -ζ1917-ζ1916-ζ1914-ζ195-ζ193-ζ192 -ζ1915-ζ1913-ζ1910-ζ199-ζ196-ζ194 -ζ1918-ζ1912-ζ1911-ζ198-ζ197-ζ19 ζ43ζ1915+ζ43ζ1913+ζ43ζ1910-ζ43ζ199-ζ43ζ196-ζ43ζ194 ζ4ζ1917+ζ4ζ1916-ζ4ζ1914+ζ4ζ195-ζ4ζ193-ζ4ζ192 ζ4ζ1918+ζ4ζ1912-ζ4ζ1911+ζ4ζ198-ζ4ζ197-ζ4ζ19 ζ4ζ1915+ζ4ζ1913+ζ4ζ1910-ζ4ζ199-ζ4ζ196-ζ4ζ194 -ζ4ζ1918-ζ4ζ1912+ζ4ζ1911-ζ4ζ198+ζ4ζ197+ζ4ζ19 ζ43ζ1917+ζ43ζ1916-ζ43ζ1914+ζ43ζ195-ζ43ζ193-ζ43ζ192 symplectic faithful, Schur index 2 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 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 symplectic faithful, Schur index 2

Smallest permutation representation of Dic38⋊C3
On 152 points
Generators in S152
(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)(77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152)
(1 96 39 134)(2 95 40 133)(3 94 41 132)(4 93 42 131)(5 92 43 130)(6 91 44 129)(7 90 45 128)(8 89 46 127)(9 88 47 126)(10 87 48 125)(11 86 49 124)(12 85 50 123)(13 84 51 122)(14 83 52 121)(15 82 53 120)(16 81 54 119)(17 80 55 118)(18 79 56 117)(19 78 57 116)(20 77 58 115)(21 152 59 114)(22 151 60 113)(23 150 61 112)(24 149 62 111)(25 148 63 110)(26 147 64 109)(27 146 65 108)(28 145 66 107)(29 144 67 106)(30 143 68 105)(31 142 69 104)(32 141 70 103)(33 140 71 102)(34 139 72 101)(35 138 73 100)(36 137 74 99)(37 136 75 98)(38 135 76 97)
(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)(78 122 126)(79 91 99)(80 136 148)(81 105 121)(82 150 94)(83 119 143)(84 88 116)(85 133 89)(86 102 138)(87 147 111)(90 130 106)(92 144 128)(93 113 101)(95 127 123)(97 141 145)(98 110 118)(100 124 140)(103 107 135)(104 152 108)(109 149 125)(112 132 120)(114 146 142)(117 129 137)(131 151 139)

G:=sub<Sym(152)| (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)(77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152), (1,96,39,134)(2,95,40,133)(3,94,41,132)(4,93,42,131)(5,92,43,130)(6,91,44,129)(7,90,45,128)(8,89,46,127)(9,88,47,126)(10,87,48,125)(11,86,49,124)(12,85,50,123)(13,84,51,122)(14,83,52,121)(15,82,53,120)(16,81,54,119)(17,80,55,118)(18,79,56,117)(19,78,57,116)(20,77,58,115)(21,152,59,114)(22,151,60,113)(23,150,61,112)(24,149,62,111)(25,148,63,110)(26,147,64,109)(27,146,65,108)(28,145,66,107)(29,144,67,106)(30,143,68,105)(31,142,69,104)(32,141,70,103)(33,140,71,102)(34,139,72,101)(35,138,73,100)(36,137,74,99)(37,136,75,98)(38,135,76,97), (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)(78,122,126)(79,91,99)(80,136,148)(81,105,121)(82,150,94)(83,119,143)(84,88,116)(85,133,89)(86,102,138)(87,147,111)(90,130,106)(92,144,128)(93,113,101)(95,127,123)(97,141,145)(98,110,118)(100,124,140)(103,107,135)(104,152,108)(109,149,125)(112,132,120)(114,146,142)(117,129,137)(131,151,139)>;

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)(77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152), (1,96,39,134)(2,95,40,133)(3,94,41,132)(4,93,42,131)(5,92,43,130)(6,91,44,129)(7,90,45,128)(8,89,46,127)(9,88,47,126)(10,87,48,125)(11,86,49,124)(12,85,50,123)(13,84,51,122)(14,83,52,121)(15,82,53,120)(16,81,54,119)(17,80,55,118)(18,79,56,117)(19,78,57,116)(20,77,58,115)(21,152,59,114)(22,151,60,113)(23,150,61,112)(24,149,62,111)(25,148,63,110)(26,147,64,109)(27,146,65,108)(28,145,66,107)(29,144,67,106)(30,143,68,105)(31,142,69,104)(32,141,70,103)(33,140,71,102)(34,139,72,101)(35,138,73,100)(36,137,74,99)(37,136,75,98)(38,135,76,97), (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)(78,122,126)(79,91,99)(80,136,148)(81,105,121)(82,150,94)(83,119,143)(84,88,116)(85,133,89)(86,102,138)(87,147,111)(90,130,106)(92,144,128)(93,113,101)(95,127,123)(97,141,145)(98,110,118)(100,124,140)(103,107,135)(104,152,108)(109,149,125)(112,132,120)(114,146,142)(117,129,137)(131,151,139) );

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),(77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152)], [(1,96,39,134),(2,95,40,133),(3,94,41,132),(4,93,42,131),(5,92,43,130),(6,91,44,129),(7,90,45,128),(8,89,46,127),(9,88,47,126),(10,87,48,125),(11,86,49,124),(12,85,50,123),(13,84,51,122),(14,83,52,121),(15,82,53,120),(16,81,54,119),(17,80,55,118),(18,79,56,117),(19,78,57,116),(20,77,58,115),(21,152,59,114),(22,151,60,113),(23,150,61,112),(24,149,62,111),(25,148,63,110),(26,147,64,109),(27,146,65,108),(28,145,66,107),(29,144,67,106),(30,143,68,105),(31,142,69,104),(32,141,70,103),(33,140,71,102),(34,139,72,101),(35,138,73,100),(36,137,74,99),(37,136,75,98),(38,135,76,97)], [(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),(78,122,126),(79,91,99),(80,136,148),(81,105,121),(82,150,94),(83,119,143),(84,88,116),(85,133,89),(86,102,138),(87,147,111),(90,130,106),(92,144,128),(93,113,101),(95,127,123),(97,141,145),(98,110,118),(100,124,140),(103,107,135),(104,152,108),(109,149,125),(112,132,120),(114,146,142),(117,129,137),(131,151,139)])

Matrix representation of Dic38⋊C3 in GL8(𝔽229)

 181 31 0 0 0 0 0 0 66 48 0 0 0 0 0 0 0 0 121 1 0 0 0 0 0 0 227 0 1 0 0 0 0 0 1 0 0 1 0 0 0 0 109 0 0 0 1 0 0 0 211 0 0 0 0 1 0 0 1 121 228 126 125 127
,
 209 80 0 0 0 0 0 0 78 20 0 0 0 0 0 0 0 0 104 141 143 184 215 79 0 0 97 195 82 128 29 226 0 0 12 19 145 12 45 42 0 0 192 208 42 9 25 125 0 0 29 202 145 0 13 48 0 0 148 108 12 20 38 221
,
 94 0 0 0 0 0 0 0 0 94 0 0 0 0 0 0 0 0 21 192 155 98 144 142 0 0 72 207 40 9 43 159 0 0 174 90 45 211 121 154 0 0 167 225 103 102 47 6 0 0 214 182 115 83 142 166 0 0 132 124 179 104 207 170

G:=sub<GL(8,GF(229))| [181,66,0,0,0,0,0,0,31,48,0,0,0,0,0,0,0,0,121,227,1,109,211,1,0,0,1,0,0,0,0,121,0,0,0,1,0,0,0,228,0,0,0,0,1,0,0,126,0,0,0,0,0,1,0,125,0,0,0,0,0,0,1,127],[209,78,0,0,0,0,0,0,80,20,0,0,0,0,0,0,0,0,104,97,12,192,29,148,0,0,141,195,19,208,202,108,0,0,143,82,145,42,145,12,0,0,184,128,12,9,0,20,0,0,215,29,45,25,13,38,0,0,79,226,42,125,48,221],[94,0,0,0,0,0,0,0,0,94,0,0,0,0,0,0,0,0,21,72,174,167,214,132,0,0,192,207,90,225,182,124,0,0,155,40,45,103,115,179,0,0,98,9,211,102,83,104,0,0,144,43,121,47,142,207,0,0,142,159,154,6,166,170] >;

Dic38⋊C3 in GAP, Magma, Sage, TeX

{\rm Dic}_{38}\rtimes C_3
% in TeX

G:=Group("Dic38:C3");
// GroupNames label

G:=SmallGroup(456,7);
// by ID

G=gap.SmallGroup(456,7);
# by ID

G:=PCGroup([5,-2,-2,-3,-2,-19,60,141,66,10804,1064]);
// Polycyclic

G:=Group<a,b,c|a^76=c^3=1,b^2=a^38,b*a*b^-1=a^-1,c*a*c^-1=a^49,b*c=c*b>;
// generators/relations

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