metacyclic, supersoluble, monomial, Z-group
Aliases: C19⋊C12, C38.C6, Dic19⋊C3, C19⋊C3⋊C4, C2.(C19⋊C6), (C2×C19⋊C3).C2, SmallGroup(228,1)
Series: Derived ►Chief ►Lower central ►Upper central
| C19 — C19⋊C12 |
Generators and relations for C19⋊C12
G = < a,b | a19=b12=1, bab-1=a8 >
Character table of C19⋊C12
| class | 1 | 2 | 3A | 3B | 4A | 4B | 6A | 6B | 12A | 12B | 12C | 12D | 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 | ζ3 | ζ32 | 1 | 1 | ζ3 | ζ32 | ζ32 | ζ3 | ζ3 | ζ32 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 3 |
| ρ4 | 1 | 1 | ζ32 | ζ3 | 1 | 1 | ζ32 | ζ3 | ζ3 | ζ32 | ζ32 | ζ3 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 3 |
| ρ5 | 1 | 1 | ζ32 | ζ3 | -1 | -1 | ζ32 | ζ3 | ζ65 | ζ6 | ζ6 | ζ65 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 6 |
| ρ6 | 1 | 1 | ζ3 | ζ32 | -1 | -1 | ζ3 | ζ32 | ζ6 | ζ65 | ζ65 | ζ6 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 6 |
| ρ7 | 1 | -1 | 1 | 1 | -i | i | -1 | -1 | -i | i | -i | i | 1 | 1 | 1 | -1 | -1 | -1 | linear of order 4 |
| ρ8 | 1 | -1 | 1 | 1 | i | -i | -1 | -1 | i | -i | i | -i | 1 | 1 | 1 | -1 | -1 | -1 | linear of order 4 |
| ρ9 | 1 | -1 | ζ32 | ζ3 | -i | i | ζ6 | ζ65 | ζ43ζ3 | ζ4ζ32 | ζ43ζ32 | ζ4ζ3 | 1 | 1 | 1 | -1 | -1 | -1 | linear of order 12 |
| ρ10 | 1 | -1 | ζ3 | ζ32 | -i | i | ζ65 | ζ6 | ζ43ζ32 | ζ4ζ3 | ζ43ζ3 | ζ4ζ32 | 1 | 1 | 1 | -1 | -1 | -1 | linear of order 12 |
| ρ11 | 1 | -1 | ζ3 | ζ32 | i | -i | ζ65 | ζ6 | ζ4ζ32 | ζ43ζ3 | ζ4ζ3 | ζ43ζ32 | 1 | 1 | 1 | -1 | -1 | -1 | linear of order 12 |
| ρ12 | 1 | -1 | ζ32 | ζ3 | i | -i | ζ6 | ζ65 | ζ4ζ3 | ζ43ζ32 | ζ4ζ32 | ζ43ζ3 | 1 | 1 | 1 | -1 | -1 | -1 | linear of order 12 |
| ρ13 | 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 |
| 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 | symplectic faithful, Schur index 2 |
| ρ17 | 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 | symplectic faithful, Schur index 2 |
| 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 | symplectic faithful, Schur index 2 |
►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 58 21 39)(2 70 32 57 8 66 22 51 12 76 28 47)(3 63 24 56 15 74 23 44 4 75 35 55)(5 68 27 54 10 71 25 49 7 73 30 52)(6 61 38 53 17 60 26 42 18 72 37 41)(9 59 33 50 19 65 29 40 13 69 20 46)(11 64 36 48 14 62 31 45 16 67 34 43)
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), (1,58,21,39)(2,70,32,57,8,66,22,51,12,76,28,47)(3,63,24,56,15,74,23,44,4,75,35,55)(5,68,27,54,10,71,25,49,7,73,30,52)(6,61,38,53,17,60,26,42,18,72,37,41)(9,59,33,50,19,65,29,40,13,69,20,46)(11,64,36,48,14,62,31,45,16,67,34,43)>;
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), (1,58,21,39)(2,70,32,57,8,66,22,51,12,76,28,47)(3,63,24,56,15,74,23,44,4,75,35,55)(5,68,27,54,10,71,25,49,7,73,30,52)(6,61,38,53,17,60,26,42,18,72,37,41)(9,59,33,50,19,65,29,40,13,69,20,46)(11,64,36,48,14,62,31,45,16,67,34,43) );
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)], [(1,58,21,39),(2,70,32,57,8,66,22,51,12,76,28,47),(3,63,24,56,15,74,23,44,4,75,35,55),(5,68,27,54,10,71,25,49,7,73,30,52),(6,61,38,53,17,60,26,42,18,72,37,41),(9,59,33,50,19,65,29,40,13,69,20,46),(11,64,36,48,14,62,31,45,16,67,34,43)]])
C19⋊C12 is a maximal subgroup of
Dic38⋊C3 C4×C19⋊C6 D38⋊C6
C19⋊C12 is a maximal quotient of C19⋊C24
Matrix representation of C19⋊C12 ►in GL6(𝔽229)
| 0 | 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 |
| 228 | 108 | 17 | 8 | 17 | 108 |
| 101 | 158 | 170 | 181 | 114 | 79 |
| 14 | 109 | 200 | 101 | 1 | 13 |
| 207 | 219 | 135 | 191 | 176 | 197 |
| 150 | 115 | 48 | 59 | 71 | 137 |
| 117 | 83 | 185 | 92 | 227 | 73 |
| 56 | 56 | 124 | 137 | 79 | 56 |
G:=sub<GL(6,GF(229))| [0,0,0,0,0,228,1,0,0,0,0,108,0,1,0,0,0,17,0,0,1,0,0,8,0,0,0,1,0,17,0,0,0,0,1,108],[101,14,207,150,117,56,158,109,219,115,83,56,170,200,135,48,185,124,181,101,191,59,92,137,114,1,176,71,227,79,79,13,197,137,73,56] >;
C19⋊C12 in GAP, Magma, Sage, TeX
C_{19}\rtimes C_{12} % in TeX
G:=Group("C19:C12"); // GroupNames label
G:=SmallGroup(228,1);
// by ID
G=gap.SmallGroup(228,1);
# by ID
G:=PCGroup([4,-2,-3,-2,-19,24,3459,679]);
// Polycyclic
G:=Group<a,b|a^19=b^12=1,b*a*b^-1=a^8>;
// generators/relations
Export
Subgroup lattice of C19⋊C12 in TeX
Character table of C19⋊C12 in TeX