direct product, metacyclic, supersoluble, monomial, Z-group, 3-hyperelementary
Aliases: C2×C19⋊C3, C38⋊C3, C19⋊2C6, SmallGroup(114,2)
Series: Derived ►Chief ►Lower central ►Upper central
| C19 — C2×C19⋊C3 |
Generators and relations for C2×C19⋊C3
G = < a,b,c | a2=b19=c3=1, ab=ba, ac=ca, cbc-1=b11 >
Character table of C2×C19⋊C3
| class | 1 | 2 | 3A | 3B | 6A | 6B | 19A | 19B | 19C | 19D | 19E | 19F | 38A | 38B | 38C | 38D | 38E | 38F | |
| size | 1 | 1 | 19 | 19 | 19 | 19 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | |
| ρ1 | 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 | linear of order 2 |
| ρ3 | 1 | -1 | ζ3 | ζ32 | ζ6 | ζ65 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | linear of order 6 |
| ρ4 | 1 | 1 | ζ32 | ζ3 | ζ3 | ζ32 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 3 |
| ρ5 | 1 | -1 | ζ32 | ζ3 | ζ65 | ζ6 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | linear of order 6 |
| ρ6 | 1 | 1 | ζ3 | ζ32 | ζ32 | ζ3 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 3 |
| ρ7 | 3 | 3 | 0 | 0 | 0 | 0 | ζ199+ζ196+ζ194 | ζ1918+ζ1912+ζ198 | ζ1915+ζ1913+ζ1910 | ζ1914+ζ193+ζ192 | ζ1917+ζ1916+ζ195 | ζ1911+ζ197+ζ19 | ζ1918+ζ1912+ζ198 | ζ1915+ζ1913+ζ1910 | ζ1917+ζ1916+ζ195 | ζ1911+ζ197+ζ19 | ζ1914+ζ193+ζ192 | ζ199+ζ196+ζ194 | complex lifted from C19⋊C3 |
| ρ8 | 3 | 3 | 0 | 0 | 0 | 0 | ζ1911+ζ197+ζ19 | ζ1914+ζ193+ζ192 | ζ1918+ζ1912+ζ198 | ζ1915+ζ1913+ζ1910 | ζ199+ζ196+ζ194 | ζ1917+ζ1916+ζ195 | ζ1914+ζ193+ζ192 | ζ1918+ζ1912+ζ198 | ζ199+ζ196+ζ194 | ζ1917+ζ1916+ζ195 | ζ1915+ζ1913+ζ1910 | ζ1911+ζ197+ζ19 | complex lifted from C19⋊C3 |
| ρ9 | 3 | -3 | 0 | 0 | 0 | 0 | ζ1917+ζ1916+ζ195 | ζ1915+ζ1913+ζ1910 | ζ1914+ζ193+ζ192 | ζ1918+ζ1912+ζ198 | ζ1911+ζ197+ζ19 | ζ199+ζ196+ζ194 | -ζ1915-ζ1913-ζ1910 | -ζ1914-ζ193-ζ192 | -ζ1911-ζ197-ζ19 | -ζ199-ζ196-ζ194 | -ζ1918-ζ1912-ζ198 | -ζ1917-ζ1916-ζ195 | complex faithful |
| ρ10 | 3 | -3 | 0 | 0 | 0 | 0 | ζ1915+ζ1913+ζ1910 | ζ1911+ζ197+ζ19 | ζ199+ζ196+ζ194 | ζ1917+ζ1916+ζ195 | ζ1914+ζ193+ζ192 | ζ1918+ζ1912+ζ198 | -ζ1911-ζ197-ζ19 | -ζ199-ζ196-ζ194 | -ζ1914-ζ193-ζ192 | -ζ1918-ζ1912-ζ198 | -ζ1917-ζ1916-ζ195 | -ζ1915-ζ1913-ζ1910 | complex faithful |
| ρ11 | 3 | -3 | 0 | 0 | 0 | 0 | ζ1918+ζ1912+ζ198 | ζ1917+ζ1916+ζ195 | ζ1911+ζ197+ζ19 | ζ199+ζ196+ζ194 | ζ1915+ζ1913+ζ1910 | ζ1914+ζ193+ζ192 | -ζ1917-ζ1916-ζ195 | -ζ1911-ζ197-ζ19 | -ζ1915-ζ1913-ζ1910 | -ζ1914-ζ193-ζ192 | -ζ199-ζ196-ζ194 | -ζ1918-ζ1912-ζ198 | complex faithful |
| ρ12 | 3 | 3 | 0 | 0 | 0 | 0 | ζ1918+ζ1912+ζ198 | ζ1917+ζ1916+ζ195 | ζ1911+ζ197+ζ19 | ζ199+ζ196+ζ194 | ζ1915+ζ1913+ζ1910 | ζ1914+ζ193+ζ192 | ζ1917+ζ1916+ζ195 | ζ1911+ζ197+ζ19 | ζ1915+ζ1913+ζ1910 | ζ1914+ζ193+ζ192 | ζ199+ζ196+ζ194 | ζ1918+ζ1912+ζ198 | complex lifted from C19⋊C3 |
| ρ13 | 3 | -3 | 0 | 0 | 0 | 0 | ζ1911+ζ197+ζ19 | ζ1914+ζ193+ζ192 | ζ1918+ζ1912+ζ198 | ζ1915+ζ1913+ζ1910 | ζ199+ζ196+ζ194 | ζ1917+ζ1916+ζ195 | -ζ1914-ζ193-ζ192 | -ζ1918-ζ1912-ζ198 | -ζ199-ζ196-ζ194 | -ζ1917-ζ1916-ζ195 | -ζ1915-ζ1913-ζ1910 | -ζ1911-ζ197-ζ19 | complex faithful |
| ρ14 | 3 | -3 | 0 | 0 | 0 | 0 | ζ199+ζ196+ζ194 | ζ1918+ζ1912+ζ198 | ζ1915+ζ1913+ζ1910 | ζ1914+ζ193+ζ192 | ζ1917+ζ1916+ζ195 | ζ1911+ζ197+ζ19 | -ζ1918-ζ1912-ζ198 | -ζ1915-ζ1913-ζ1910 | -ζ1917-ζ1916-ζ195 | -ζ1911-ζ197-ζ19 | -ζ1914-ζ193-ζ192 | -ζ199-ζ196-ζ194 | complex faithful |
| ρ15 | 3 | 3 | 0 | 0 | 0 | 0 | ζ1917+ζ1916+ζ195 | ζ1915+ζ1913+ζ1910 | ζ1914+ζ193+ζ192 | ζ1918+ζ1912+ζ198 | ζ1911+ζ197+ζ19 | ζ199+ζ196+ζ194 | ζ1915+ζ1913+ζ1910 | ζ1914+ζ193+ζ192 | ζ1911+ζ197+ζ19 | ζ199+ζ196+ζ194 | ζ1918+ζ1912+ζ198 | ζ1917+ζ1916+ζ195 | complex lifted from C19⋊C3 |
| ρ16 | 3 | 3 | 0 | 0 | 0 | 0 | ζ1914+ζ193+ζ192 | ζ199+ζ196+ζ194 | ζ1917+ζ1916+ζ195 | ζ1911+ζ197+ζ19 | ζ1918+ζ1912+ζ198 | ζ1915+ζ1913+ζ1910 | ζ199+ζ196+ζ194 | ζ1917+ζ1916+ζ195 | ζ1918+ζ1912+ζ198 | ζ1915+ζ1913+ζ1910 | ζ1911+ζ197+ζ19 | ζ1914+ζ193+ζ192 | complex lifted from C19⋊C3 |
| ρ17 | 3 | 3 | 0 | 0 | 0 | 0 | ζ1915+ζ1913+ζ1910 | ζ1911+ζ197+ζ19 | ζ199+ζ196+ζ194 | ζ1917+ζ1916+ζ195 | ζ1914+ζ193+ζ192 | ζ1918+ζ1912+ζ198 | ζ1911+ζ197+ζ19 | ζ199+ζ196+ζ194 | ζ1914+ζ193+ζ192 | ζ1918+ζ1912+ζ198 | ζ1917+ζ1916+ζ195 | ζ1915+ζ1913+ζ1910 | complex lifted from C19⋊C3 |
| ρ18 | 3 | -3 | 0 | 0 | 0 | 0 | ζ1914+ζ193+ζ192 | ζ199+ζ196+ζ194 | ζ1917+ζ1916+ζ195 | ζ1911+ζ197+ζ19 | ζ1918+ζ1912+ζ198 | ζ1915+ζ1913+ζ1910 | -ζ199-ζ196-ζ194 | -ζ1917-ζ1916-ζ195 | -ζ1918-ζ1912-ζ198 | -ζ1915-ζ1913-ζ1910 | -ζ1911-ζ197-ζ19 | -ζ1914-ζ193-ζ192 | complex faithful |
►On 38 points
Generators in S38
(1 20)(2 21)(3 22)(4 23)(5 24)(6 25)(7 26)(8 27)(9 28)(10 29)(11 30)(12 31)(13 32)(14 33)(15 34)(16 35)(17 36)(18 37)(19 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)
(2 8 12)(3 15 4)(5 10 7)(6 17 18)(9 19 13)(11 14 16)(21 27 31)(22 34 23)(24 29 26)(25 36 37)(28 38 32)(30 33 35)
G:=sub<Sym(38)| (1,20)(2,21)(3,22)(4,23)(5,24)(6,25)(7,26)(8,27)(9,28)(10,29)(11,30)(12,31)(13,32)(14,33)(15,34)(16,35)(17,36)(18,37)(19,38), (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), (2,8,12)(3,15,4)(5,10,7)(6,17,18)(9,19,13)(11,14,16)(21,27,31)(22,34,23)(24,29,26)(25,36,37)(28,38,32)(30,33,35)>;
G:=Group( (1,20)(2,21)(3,22)(4,23)(5,24)(6,25)(7,26)(8,27)(9,28)(10,29)(11,30)(12,31)(13,32)(14,33)(15,34)(16,35)(17,36)(18,37)(19,38), (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), (2,8,12)(3,15,4)(5,10,7)(6,17,18)(9,19,13)(11,14,16)(21,27,31)(22,34,23)(24,29,26)(25,36,37)(28,38,32)(30,33,35) );
G=PermutationGroup([[(1,20),(2,21),(3,22),(4,23),(5,24),(6,25),(7,26),(8,27),(9,28),(10,29),(11,30),(12,31),(13,32),(14,33),(15,34),(16,35),(17,36),(18,37),(19,38)], [(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)], [(2,8,12),(3,15,4),(5,10,7),(6,17,18),(9,19,13),(11,14,16),(21,27,31),(22,34,23),(24,29,26),(25,36,37),(28,38,32),(30,33,35)]])
C2×C19⋊C3 is a maximal subgroup of
C19⋊C12 C38.A4
Matrix representation of C2×C19⋊C3 ►in GL3(𝔽7) generated by
| 6 | 0 | 0 |
| 0 | 6 | 0 |
| 0 | 0 | 6 |
| 6 | 3 | 4 |
| 5 | 6 | 4 |
| 3 | 3 | 5 |
| 1 | 1 | 4 |
| 0 | 2 | 0 |
| 0 | 2 | 4 |
G:=sub<GL(3,GF(7))| [6,0,0,0,6,0,0,0,6],[6,5,3,3,6,3,4,4,5],[1,0,0,1,2,2,4,0,4] >;
C2×C19⋊C3 in GAP, Magma, Sage, TeX
C_2\times C_{19}\rtimes C_3 % in TeX
G:=Group("C2xC19:C3"); // GroupNames label
G:=SmallGroup(114,2);
// by ID
G=gap.SmallGroup(114,2);
# by ID
G:=PCGroup([3,-2,-3,-19,194]);
// Polycyclic
G:=Group<a,b,c|a^2=b^19=c^3=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^11>;
// generators/relations
Export
Subgroup lattice of C2×C19⋊C3 in TeX
Character table of C2×C19⋊C3 in TeX