direct product, metacyclic, supersoluble, monomial, Z-group, 3-hyperelementary
Aliases: C2×C19⋊C9, C38⋊C9, C19⋊2C18, C19⋊C3.2C6, (C2×C19⋊C3).C3, SmallGroup(342,8)
Series: Derived ►Chief ►Lower central ►Upper central
| C19 — C2×C19⋊C9 |
Generators and relations for C2×C19⋊C9
G = < a,b,c | a2=b19=c9=1, ab=ba, ac=ca, cbc-1=b5 >
Character table of C2×C19⋊C9
| class | 1 | 2 | 3A | 3B | 6A | 6B | 9A | 9B | 9C | 9D | 9E | 9F | 18A | 18B | 18C | 18D | 18E | 18F | 19A | 19B | 38A | 38B | |
| size | 1 | 1 | 19 | 19 | 19 | 19 | 19 | 19 | 19 | 19 | 19 | 19 | 19 | 19 | 19 | 19 | 19 | 19 | 9 | 9 | 9 | 9 | |
| 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 | ζ32 | ζ32 | ζ3 | ζ3 | ζ3 | ζ32 | ζ3 | ζ32 | ζ32 | ζ32 | ζ3 | ζ3 | 1 | 1 | 1 | 1 | linear of order 3 |
| ρ4 | 1 | -1 | 1 | 1 | -1 | -1 | ζ3 | ζ3 | ζ32 | ζ32 | ζ32 | ζ3 | ζ6 | ζ65 | ζ65 | ζ65 | ζ6 | ζ6 | 1 | 1 | -1 | -1 | linear of order 6 |
| ρ5 | 1 | -1 | 1 | 1 | -1 | -1 | ζ32 | ζ32 | ζ3 | ζ3 | ζ3 | ζ32 | ζ65 | ζ6 | ζ6 | ζ6 | ζ65 | ζ65 | 1 | 1 | -1 | -1 | linear of order 6 |
| ρ6 | 1 | 1 | 1 | 1 | 1 | 1 | ζ3 | ζ3 | ζ32 | ζ32 | ζ32 | ζ3 | ζ32 | ζ3 | ζ3 | ζ3 | ζ32 | ζ32 | 1 | 1 | 1 | 1 | linear of order 3 |
| ρ7 | 1 | -1 | ζ32 | ζ3 | ζ6 | ζ65 | ζ94 | ζ97 | ζ92 | ζ95 | ζ98 | ζ9 | -ζ98 | -ζ9 | -ζ94 | -ζ97 | -ζ92 | -ζ95 | 1 | 1 | -1 | -1 | linear of order 18 |
| ρ8 | 1 | -1 | ζ32 | ζ3 | ζ6 | ζ65 | ζ9 | ζ94 | ζ95 | ζ98 | ζ92 | ζ97 | -ζ92 | -ζ97 | -ζ9 | -ζ94 | -ζ95 | -ζ98 | 1 | 1 | -1 | -1 | linear of order 18 |
| ρ9 | 1 | 1 | ζ3 | ζ32 | ζ3 | ζ32 | ζ95 | ζ92 | ζ97 | ζ94 | ζ9 | ζ98 | ζ9 | ζ98 | ζ95 | ζ92 | ζ97 | ζ94 | 1 | 1 | 1 | 1 | linear of order 9 |
| ρ10 | 1 | -1 | ζ3 | ζ32 | ζ65 | ζ6 | ζ95 | ζ92 | ζ97 | ζ94 | ζ9 | ζ98 | -ζ9 | -ζ98 | -ζ95 | -ζ92 | -ζ97 | -ζ94 | 1 | 1 | -1 | -1 | linear of order 18 |
| ρ11 | 1 | 1 | ζ32 | ζ3 | ζ32 | ζ3 | ζ94 | ζ97 | ζ92 | ζ95 | ζ98 | ζ9 | ζ98 | ζ9 | ζ94 | ζ97 | ζ92 | ζ95 | 1 | 1 | 1 | 1 | linear of order 9 |
| ρ12 | 1 | 1 | ζ3 | ζ32 | ζ3 | ζ32 | ζ98 | ζ95 | ζ94 | ζ9 | ζ97 | ζ92 | ζ97 | ζ92 | ζ98 | ζ95 | ζ94 | ζ9 | 1 | 1 | 1 | 1 | linear of order 9 |
| ρ13 | 1 | -1 | ζ32 | ζ3 | ζ6 | ζ65 | ζ97 | ζ9 | ζ98 | ζ92 | ζ95 | ζ94 | -ζ95 | -ζ94 | -ζ97 | -ζ9 | -ζ98 | -ζ92 | 1 | 1 | -1 | -1 | linear of order 18 |
| ρ14 | 1 | 1 | ζ32 | ζ3 | ζ32 | ζ3 | ζ9 | ζ94 | ζ95 | ζ98 | ζ92 | ζ97 | ζ92 | ζ97 | ζ9 | ζ94 | ζ95 | ζ98 | 1 | 1 | 1 | 1 | linear of order 9 |
| ρ15 | 1 | 1 | ζ3 | ζ32 | ζ3 | ζ32 | ζ92 | ζ98 | ζ9 | ζ97 | ζ94 | ζ95 | ζ94 | ζ95 | ζ92 | ζ98 | ζ9 | ζ97 | 1 | 1 | 1 | 1 | linear of order 9 |
| ρ16 | 1 | -1 | ζ3 | ζ32 | ζ65 | ζ6 | ζ92 | ζ98 | ζ9 | ζ97 | ζ94 | ζ95 | -ζ94 | -ζ95 | -ζ92 | -ζ98 | -ζ9 | -ζ97 | 1 | 1 | -1 | -1 | linear of order 18 |
| ρ17 | 1 | 1 | ζ32 | ζ3 | ζ32 | ζ3 | ζ97 | ζ9 | ζ98 | ζ92 | ζ95 | ζ94 | ζ95 | ζ94 | ζ97 | ζ9 | ζ98 | ζ92 | 1 | 1 | 1 | 1 | linear of order 9 |
| ρ18 | 1 | -1 | ζ3 | ζ32 | ζ65 | ζ6 | ζ98 | ζ95 | ζ94 | ζ9 | ζ97 | ζ92 | -ζ97 | -ζ92 | -ζ98 | -ζ95 | -ζ94 | -ζ9 | 1 | 1 | -1 | -1 | linear of order 18 |
| ρ19 | 9 | 9 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1+√-19/2 | -1-√-19/2 | -1-√-19/2 | -1+√-19/2 | complex lifted from C19⋊C9 |
| ρ20 | 9 | -9 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1-√-19/2 | -1+√-19/2 | 1-√-19/2 | 1+√-19/2 | complex faithful |
| ρ21 | 9 | -9 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1+√-19/2 | -1-√-19/2 | 1+√-19/2 | 1-√-19/2 | complex faithful |
| ρ22 | 9 | 9 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1-√-19/2 | -1+√-19/2 | -1+√-19/2 | -1-√-19/2 | complex lifted from C19⋊C9 |
►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 5 17 8 10 18 12 7 6)(3 9 14 15 19 16 4 13 11)(21 24 36 27 29 37 31 26 25)(22 28 33 34 38 35 23 32 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), (2,5,17,8,10,18,12,7,6)(3,9,14,15,19,16,4,13,11)(21,24,36,27,29,37,31,26,25)(22,28,33,34,38,35,23,32,30)>;
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), (2,5,17,8,10,18,12,7,6)(3,9,14,15,19,16,4,13,11)(21,24,36,27,29,37,31,26,25)(22,28,33,34,38,35,23,32,30) );
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)], [(2,5,17,8,10,18,12,7,6),(3,9,14,15,19,16,4,13,11),(21,24,36,27,29,37,31,26,25),(22,28,33,34,38,35,23,32,30)]])
Matrix representation of C2×C19⋊C9 ►in GL9(𝔽2053)
| 2052 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 2052 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 2052 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 2052 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 2052 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 2052 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 2052 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2052 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2052 |
| 858 | 2 | 1193 | 856 | 861 | 1196 | 2051 | 859 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 1194 | 2049 | 860 | 338 | 1190 | 1716 | 3 | 1192 | 856 |
| 1718 | 5 | 1191 | 519 | 865 | 1532 | 2049 | 1721 | 1198 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 856 | 1720 | 337 | 2046 | 523 | 1201 | 334 | 1715 | 862 |
| 1198 | 334 | 1715 | 865 | 1531 | 2046 | 1718 | 339 | 1192 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
G:=sub<GL(9,GF(2053))| [2052,0,0,0,0,0,0,0,0,0,2052,0,0,0,0,0,0,0,0,0,2052,0,0,0,0,0,0,0,0,0,2052,0,0,0,0,0,0,0,0,0,2052,0,0,0,0,0,0,0,0,0,2052,0,0,0,0,0,0,0,0,0,2052,0,0,0,0,0,0,0,0,0,2052,0,0,0,0,0,0,0,0,0,2052],[858,1,0,0,0,0,0,0,0,2,0,1,0,0,0,0,0,0,1193,0,0,1,0,0,0,0,0,856,0,0,0,1,0,0,0,0,861,0,0,0,0,1,0,0,0,1196,0,0,0,0,0,1,0,0,2051,0,0,0,0,0,0,1,0,859,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0],[1,0,1194,1718,0,0,856,1198,0,0,0,2049,5,1,0,1720,334,0,0,0,860,1191,0,0,337,1715,1,0,0,338,519,0,0,2046,865,0,0,0,1190,865,0,0,523,1531,0,0,1,1716,1532,0,0,1201,2046,0,0,0,3,2049,0,1,334,1718,0,0,0,1192,1721,0,0,1715,339,0,0,0,856,1198,0,0,862,1192,0] >;
C2×C19⋊C9 in GAP, Magma, Sage, TeX
C_2\times C_{19}\rtimes C_9 % in TeX
G:=Group("C2xC19:C9"); // GroupNames label
G:=SmallGroup(342,8);
// by ID
G=gap.SmallGroup(342,8);
# by ID
G:=PCGroup([4,-2,-3,-3,-19,29,583,347]);
// Polycyclic
G:=Group<a,b,c|a^2=b^19=c^9=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^5>;
// generators/relations
Export
Subgroup lattice of C2×C19⋊C9 in TeX
Character table of C2×C19⋊C9 in TeX