direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: D38, C2×D19, C38⋊C2, C19⋊C22, sometimes denoted D76 or Dih38 or Dih76, SmallGroup(76,3)
Series: Derived ►Chief ►Lower central ►Upper central
| C19 — D38 |
Generators and relations for D38
G = < a,b | a38=b2=1, bab=a-1 >
Character table of D38
| class | 1 | 2A | 2B | 2C | 19A | 19B | 19C | 19D | 19E | 19F | 19G | 19H | 19I | 38A | 38B | 38C | 38D | 38E | 38F | 38G | 38H | 38I | |
| size | 1 | 1 | 19 | 19 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | |
| ρ1 | 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 | 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 | 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 | linear of order 2 |
| ρ5 | 2 | -2 | 0 | 0 | ζ1914+ζ195 | ζ1911+ζ198 | ζ1917+ζ192 | ζ1915+ζ194 | ζ1910+ζ199 | ζ1916+ζ193 | ζ1913+ζ196 | ζ1912+ζ197 | ζ1918+ζ19 | -ζ1918-ζ19 | -ζ1914-ζ195 | -ζ1911-ζ198 | -ζ1917-ζ192 | -ζ1915-ζ194 | -ζ1910-ζ199 | -ζ1916-ζ193 | -ζ1913-ζ196 | -ζ1912-ζ197 | orthogonal faithful |
| ρ6 | 2 | 2 | 0 | 0 | ζ1913+ζ196 | ζ1917+ζ192 | ζ1910+ζ199 | ζ1918+ζ19 | ζ1912+ζ197 | ζ1915+ζ194 | ζ1911+ζ198 | ζ1916+ζ193 | ζ1914+ζ195 | ζ1914+ζ195 | ζ1913+ζ196 | ζ1917+ζ192 | ζ1910+ζ199 | ζ1918+ζ19 | ζ1912+ζ197 | ζ1915+ζ194 | ζ1911+ζ198 | ζ1916+ζ193 | orthogonal lifted from D19 |
| ρ7 | 2 | 2 | 0 | 0 | ζ1914+ζ195 | ζ1911+ζ198 | ζ1917+ζ192 | ζ1915+ζ194 | ζ1910+ζ199 | ζ1916+ζ193 | ζ1913+ζ196 | ζ1912+ζ197 | ζ1918+ζ19 | ζ1918+ζ19 | ζ1914+ζ195 | ζ1911+ζ198 | ζ1917+ζ192 | ζ1915+ζ194 | ζ1910+ζ199 | ζ1916+ζ193 | ζ1913+ζ196 | ζ1912+ζ197 | orthogonal lifted from D19 |
| ρ8 | 2 | 2 | 0 | 0 | ζ1911+ζ198 | ζ1910+ζ199 | ζ1912+ζ197 | ζ1914+ζ195 | ζ1916+ζ193 | ζ1918+ζ19 | ζ1917+ζ192 | ζ1915+ζ194 | ζ1913+ζ196 | ζ1913+ζ196 | ζ1911+ζ198 | ζ1910+ζ199 | ζ1912+ζ197 | ζ1914+ζ195 | ζ1916+ζ193 | ζ1918+ζ19 | ζ1917+ζ192 | ζ1915+ζ194 | orthogonal lifted from D19 |
| ρ9 | 2 | -2 | 0 | 0 | ζ1913+ζ196 | ζ1917+ζ192 | ζ1910+ζ199 | ζ1918+ζ19 | ζ1912+ζ197 | ζ1915+ζ194 | ζ1911+ζ198 | ζ1916+ζ193 | ζ1914+ζ195 | -ζ1914-ζ195 | -ζ1913-ζ196 | -ζ1917-ζ192 | -ζ1910-ζ199 | -ζ1918-ζ19 | -ζ1912-ζ197 | -ζ1915-ζ194 | -ζ1911-ζ198 | -ζ1916-ζ193 | orthogonal faithful |
| ρ10 | 2 | -2 | 0 | 0 | ζ1910+ζ199 | ζ1916+ζ193 | ζ1915+ζ194 | ζ1911+ζ198 | ζ1918+ζ19 | ζ1913+ζ196 | ζ1912+ζ197 | ζ1914+ζ195 | ζ1917+ζ192 | -ζ1917-ζ192 | -ζ1910-ζ199 | -ζ1916-ζ193 | -ζ1915-ζ194 | -ζ1911-ζ198 | -ζ1918-ζ19 | -ζ1913-ζ196 | -ζ1912-ζ197 | -ζ1914-ζ195 | orthogonal faithful |
| ρ11 | 2 | 2 | 0 | 0 | ζ1918+ζ19 | ζ1913+ζ196 | ζ1911+ζ198 | ζ1916+ζ193 | ζ1917+ζ192 | ζ1912+ζ197 | ζ1914+ζ195 | ζ1910+ζ199 | ζ1915+ζ194 | ζ1915+ζ194 | ζ1918+ζ19 | ζ1913+ζ196 | ζ1911+ζ198 | ζ1916+ζ193 | ζ1917+ζ192 | ζ1912+ζ197 | ζ1914+ζ195 | ζ1910+ζ199 | orthogonal lifted from D19 |
| ρ12 | 2 | 2 | 0 | 0 | ζ1916+ζ193 | ζ1918+ζ19 | ζ1914+ζ195 | ζ1910+ζ199 | ζ1913+ζ196 | ζ1917+ζ192 | ζ1915+ζ194 | ζ1911+ζ198 | ζ1912+ζ197 | ζ1912+ζ197 | ζ1916+ζ193 | ζ1918+ζ19 | ζ1914+ζ195 | ζ1910+ζ199 | ζ1913+ζ196 | ζ1917+ζ192 | ζ1915+ζ194 | ζ1911+ζ198 | orthogonal lifted from D19 |
| ρ13 | 2 | -2 | 0 | 0 | ζ1911+ζ198 | ζ1910+ζ199 | ζ1912+ζ197 | ζ1914+ζ195 | ζ1916+ζ193 | ζ1918+ζ19 | ζ1917+ζ192 | ζ1915+ζ194 | ζ1913+ζ196 | -ζ1913-ζ196 | -ζ1911-ζ198 | -ζ1910-ζ199 | -ζ1912-ζ197 | -ζ1914-ζ195 | -ζ1916-ζ193 | -ζ1918-ζ19 | -ζ1917-ζ192 | -ζ1915-ζ194 | orthogonal faithful |
| ρ14 | 2 | -2 | 0 | 0 | ζ1918+ζ19 | ζ1913+ζ196 | ζ1911+ζ198 | ζ1916+ζ193 | ζ1917+ζ192 | ζ1912+ζ197 | ζ1914+ζ195 | ζ1910+ζ199 | ζ1915+ζ194 | -ζ1915-ζ194 | -ζ1918-ζ19 | -ζ1913-ζ196 | -ζ1911-ζ198 | -ζ1916-ζ193 | -ζ1917-ζ192 | -ζ1912-ζ197 | -ζ1914-ζ195 | -ζ1910-ζ199 | orthogonal faithful |
| ρ15 | 2 | -2 | 0 | 0 | ζ1912+ζ197 | ζ1915+ζ194 | ζ1918+ζ19 | ζ1917+ζ192 | ζ1914+ζ195 | ζ1911+ζ198 | ζ1916+ζ193 | ζ1913+ζ196 | ζ1910+ζ199 | -ζ1910-ζ199 | -ζ1912-ζ197 | -ζ1915-ζ194 | -ζ1918-ζ19 | -ζ1917-ζ192 | -ζ1914-ζ195 | -ζ1911-ζ198 | -ζ1916-ζ193 | -ζ1913-ζ196 | orthogonal faithful |
| ρ16 | 2 | 2 | 0 | 0 | ζ1915+ζ194 | ζ1914+ζ195 | ζ1913+ζ196 | ζ1912+ζ197 | ζ1911+ζ198 | ζ1910+ζ199 | ζ1918+ζ19 | ζ1917+ζ192 | ζ1916+ζ193 | ζ1916+ζ193 | ζ1915+ζ194 | ζ1914+ζ195 | ζ1913+ζ196 | ζ1912+ζ197 | ζ1911+ζ198 | ζ1910+ζ199 | ζ1918+ζ19 | ζ1917+ζ192 | orthogonal lifted from D19 |
| ρ17 | 2 | -2 | 0 | 0 | ζ1915+ζ194 | ζ1914+ζ195 | ζ1913+ζ196 | ζ1912+ζ197 | ζ1911+ζ198 | ζ1910+ζ199 | ζ1918+ζ19 | ζ1917+ζ192 | ζ1916+ζ193 | -ζ1916-ζ193 | -ζ1915-ζ194 | -ζ1914-ζ195 | -ζ1913-ζ196 | -ζ1912-ζ197 | -ζ1911-ζ198 | -ζ1910-ζ199 | -ζ1918-ζ19 | -ζ1917-ζ192 | orthogonal faithful |
| ρ18 | 2 | -2 | 0 | 0 | ζ1917+ζ192 | ζ1912+ζ197 | ζ1916+ζ193 | ζ1913+ζ196 | ζ1915+ζ194 | ζ1914+ζ195 | ζ1910+ζ199 | ζ1918+ζ19 | ζ1911+ζ198 | -ζ1911-ζ198 | -ζ1917-ζ192 | -ζ1912-ζ197 | -ζ1916-ζ193 | -ζ1913-ζ196 | -ζ1915-ζ194 | -ζ1914-ζ195 | -ζ1910-ζ199 | -ζ1918-ζ19 | orthogonal faithful |
| ρ19 | 2 | -2 | 0 | 0 | ζ1916+ζ193 | ζ1918+ζ19 | ζ1914+ζ195 | ζ1910+ζ199 | ζ1913+ζ196 | ζ1917+ζ192 | ζ1915+ζ194 | ζ1911+ζ198 | ζ1912+ζ197 | -ζ1912-ζ197 | -ζ1916-ζ193 | -ζ1918-ζ19 | -ζ1914-ζ195 | -ζ1910-ζ199 | -ζ1913-ζ196 | -ζ1917-ζ192 | -ζ1915-ζ194 | -ζ1911-ζ198 | orthogonal faithful |
| ρ20 | 2 | 2 | 0 | 0 | ζ1917+ζ192 | ζ1912+ζ197 | ζ1916+ζ193 | ζ1913+ζ196 | ζ1915+ζ194 | ζ1914+ζ195 | ζ1910+ζ199 | ζ1918+ζ19 | ζ1911+ζ198 | ζ1911+ζ198 | ζ1917+ζ192 | ζ1912+ζ197 | ζ1916+ζ193 | ζ1913+ζ196 | ζ1915+ζ194 | ζ1914+ζ195 | ζ1910+ζ199 | ζ1918+ζ19 | orthogonal lifted from D19 |
| ρ21 | 2 | 2 | 0 | 0 | ζ1910+ζ199 | ζ1916+ζ193 | ζ1915+ζ194 | ζ1911+ζ198 | ζ1918+ζ19 | ζ1913+ζ196 | ζ1912+ζ197 | ζ1914+ζ195 | ζ1917+ζ192 | ζ1917+ζ192 | ζ1910+ζ199 | ζ1916+ζ193 | ζ1915+ζ194 | ζ1911+ζ198 | ζ1918+ζ19 | ζ1913+ζ196 | ζ1912+ζ197 | ζ1914+ζ195 | orthogonal lifted from D19 |
| ρ22 | 2 | 2 | 0 | 0 | ζ1912+ζ197 | ζ1915+ζ194 | ζ1918+ζ19 | ζ1917+ζ192 | ζ1914+ζ195 | ζ1911+ζ198 | ζ1916+ζ193 | ζ1913+ζ196 | ζ1910+ζ199 | ζ1910+ζ199 | ζ1912+ζ197 | ζ1915+ζ194 | ζ1918+ζ19 | ζ1917+ζ192 | ζ1914+ζ195 | ζ1911+ζ198 | ζ1916+ζ193 | ζ1913+ζ196 | orthogonal lifted from D19 |
►On 38 points
Generators in S38
(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)
(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)
G:=sub<Sym(38)| (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), (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)>;
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), (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) );
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)], [(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)]])
D38 is a maximal subgroup of
D76 C19⋊D4
D38 is a maximal quotient of Dic38 D76 C19⋊D4
Matrix representation of D38 ►in GL2(𝔽37) generated by
| 0 | 36 |
| 1 | 11 |
| 11 | 9 |
| 36 | 26 |
G:=sub<GL(2,GF(37))| [0,1,36,11],[11,36,9,26] >;
D38 in GAP, Magma, Sage, TeX
D_{38} % in TeX
G:=Group("D38"); // GroupNames label
G:=SmallGroup(76,3);
// by ID
G=gap.SmallGroup(76,3);
# by ID
G:=PCGroup([3,-2,-2,-19,650]);
// Polycyclic
G:=Group<a,b|a^38=b^2=1,b*a*b=a^-1>;
// generators/relations
Export
Subgroup lattice of D38 in TeX
Character table of D38 in TeX