direct product, metacyclic, supersoluble, monomial, A-group
Aliases: C2×C19⋊C6, C38⋊C6, D38⋊C3, D19⋊C6, C19⋊(C2×C6), C19⋊C3⋊C22, (C2×C19⋊C3)⋊C2, SmallGroup(228,7)
Series: Derived ►Chief ►Lower central ►Upper central
| C19 — C2×C19⋊C6 |
Generators and relations for C2×C19⋊C6
G = < a,b,c | a2=b19=c6=1, ab=ba, ac=ca, cbc-1=b12 >
Character table of C2×C19⋊C6
| class | 1 | 2A | 2B | 2C | 3A | 3B | 6A | 6B | 6C | 6D | 6E | 6F | 19A | 19B | 19C | 38A | 38B | 38C | |
| size | 1 | 1 | 19 | 19 | 19 | 19 | 19 | 19 | 19 | 19 | 19 | 19 | 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 | 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 | 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 | linear of order 2 |
| ρ5 | 1 | -1 | 1 | -1 | ζ3 | ζ32 | ζ65 | ζ32 | ζ3 | ζ65 | ζ6 | ζ6 | 1 | 1 | 1 | -1 | -1 | -1 | linear of order 6 |
| ρ6 | 1 | -1 | -1 | 1 | ζ3 | ζ32 | ζ3 | ζ6 | ζ65 | ζ65 | ζ32 | ζ6 | 1 | 1 | 1 | -1 | -1 | -1 | linear of order 6 |
| ρ7 | 1 | 1 | 1 | 1 | ζ32 | ζ3 | ζ32 | ζ3 | ζ32 | ζ32 | ζ3 | ζ3 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 3 |
| ρ8 | 1 | -1 | -1 | 1 | ζ32 | ζ3 | ζ32 | ζ65 | ζ6 | ζ6 | ζ3 | ζ65 | 1 | 1 | 1 | -1 | -1 | -1 | linear of order 6 |
| ρ9 | 1 | 1 | -1 | -1 | ζ3 | ζ32 | ζ65 | ζ6 | ζ65 | ζ3 | ζ6 | ζ32 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 6 |
| ρ10 | 1 | 1 | 1 | 1 | ζ3 | ζ32 | ζ3 | ζ32 | ζ3 | ζ3 | ζ32 | ζ32 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 3 |
| ρ11 | 1 | -1 | 1 | -1 | ζ32 | ζ3 | ζ6 | ζ3 | ζ32 | ζ6 | ζ65 | ζ65 | 1 | 1 | 1 | -1 | -1 | -1 | linear of order 6 |
| ρ12 | 1 | 1 | -1 | -1 | ζ32 | ζ3 | ζ6 | ζ65 | ζ6 | ζ32 | ζ65 | ζ3 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 6 |
| ρ13 | 6 | -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 | -ζ1915-ζ1913-ζ1910-ζ199-ζ196-ζ194 | -ζ1918-ζ1912-ζ1911-ζ198-ζ197-ζ19 | -ζ1917-ζ1916-ζ1914-ζ195-ζ193-ζ192 | orthogonal faithful |
| ρ14 | 6 | 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 | ζ1915+ζ1913+ζ1910+ζ199+ζ196+ζ194 | ζ1918+ζ1912+ζ1911+ζ198+ζ197+ζ19 | ζ1917+ζ1916+ζ1914+ζ195+ζ193+ζ192 | orthogonal lifted from C19⋊C6 |
| ρ15 | 6 | 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 | ζ1917+ζ1916+ζ1914+ζ195+ζ193+ζ192 | ζ1915+ζ1913+ζ1910+ζ199+ζ196+ζ194 | ζ1918+ζ1912+ζ1911+ζ198+ζ197+ζ19 | orthogonal lifted from C19⋊C6 |
| ρ16 | 6 | -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 | -ζ1917-ζ1916-ζ1914-ζ195-ζ193-ζ192 | -ζ1915-ζ1913-ζ1910-ζ199-ζ196-ζ194 | -ζ1918-ζ1912-ζ1911-ζ198-ζ197-ζ19 | orthogonal faithful |
| ρ17 | 6 | 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 | ζ1918+ζ1912+ζ1911+ζ198+ζ197+ζ19 | ζ1917+ζ1916+ζ1914+ζ195+ζ193+ζ192 | ζ1915+ζ1913+ζ1910+ζ199+ζ196+ζ194 | orthogonal lifted from C19⋊C6 |
| ρ18 | 6 | -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 | -ζ1918-ζ1912-ζ1911-ζ198-ζ197-ζ19 | -ζ1917-ζ1916-ζ1914-ζ195-ζ193-ζ192 | -ζ1915-ζ1913-ζ1910-ζ199-ζ196-ζ194 | orthogonal 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 9 8 19 12 13)(3 17 15 18 4 6)(5 14 10 16 7 11)(21 28 27 38 31 32)(22 36 34 37 23 25)(24 33 29 35 26 30)
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), (2,9,8,19,12,13)(3,17,15,18,4,6)(5,14,10,16,7,11)(21,28,27,38,31,32)(22,36,34,37,23,25)(24,33,29,35,26,30)>;
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), (2,9,8,19,12,13)(3,17,15,18,4,6)(5,14,10,16,7,11)(21,28,27,38,31,32)(22,36,34,37,23,25)(24,33,29,35,26,30) );
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)], [(2,9,8,19,12,13),(3,17,15,18,4,6),(5,14,10,16,7,11),(21,28,27,38,31,32),(22,36,34,37,23,25),(24,33,29,35,26,30)]])
C2×C19⋊C6 is a maximal subgroup of
D76⋊C3 D38⋊C6
C2×C19⋊C6 is a maximal quotient of Dic38⋊C3 D76⋊C3 D38⋊C6
Matrix representation of C2×C19⋊C6 ►in GL6(𝔽229)
| 228 | 0 | 0 | 0 | 0 | 0 |
| 0 | 228 | 0 | 0 | 0 | 0 |
| 0 | 0 | 228 | 0 | 0 | 0 |
| 0 | 0 | 0 | 228 | 0 | 0 |
| 0 | 0 | 0 | 0 | 228 | 0 |
| 0 | 0 | 0 | 0 | 0 | 228 |
| 101 | 106 | 83 | 106 | 101 | 228 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 210 | 209 | 99 | 99 | 209 | 210 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 2 | 26 | 16 | 44 | 124 | 128 |
| 102 | 4 | 77 | 2 | 204 | 107 |
| 0 | 0 | 0 | 1 | 0 | 0 |
G:=sub<GL(6,GF(229))| [228,0,0,0,0,0,0,228,0,0,0,0,0,0,228,0,0,0,0,0,0,228,0,0,0,0,0,0,228,0,0,0,0,0,0,228],[101,1,0,0,0,0,106,0,1,0,0,0,83,0,0,1,0,0,106,0,0,0,1,0,101,0,0,0,0,1,228,0,0,0,0,0],[1,210,0,2,102,0,0,209,0,26,4,0,0,99,0,16,77,0,0,99,0,44,2,1,0,209,0,124,204,0,0,210,1,128,107,0] >;
C2×C19⋊C6 in GAP, Magma, Sage, TeX
C_2\times C_{19}\rtimes C_6 % in TeX
G:=Group("C2xC19:C6"); // GroupNames label
G:=SmallGroup(228,7);
// by ID
G=gap.SmallGroup(228,7);
# by ID
G:=PCGroup([4,-2,-2,-3,-19,3459,347]);
// Polycyclic
G:=Group<a,b,c|a^2=b^19=c^6=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^12>;
// generators/relations
Export
Subgroup lattice of C2×C19⋊C6 in TeX
Character table of C2×C19⋊C6 in TeX