metacyclic, supersoluble, monomial
Aliases: Dic26⋊C3, C52.1C6, Dic13.2C6, C13⋊C3⋊Q8, C13⋊(C3×Q8), C4.(C13⋊C6), C26.1(C2×C6), C26.C6.2C2, C2.3(C2×C13⋊C6), (C4×C13⋊C3).1C2, (C2×C13⋊C3).1C22, SmallGroup(312,8)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic26⋊C3
G = < a,b,c | a52=c3=1, b2=a26, bab-1=a-1, cac-1=a9, bc=cb >
Character table of Dic26⋊C3
| class | 1 | 2 | 3A | 3B | 4A | 4B | 4C | 6A | 6B | 12A | 12B | 12C | 12D | 12E | 12F | 13A | 13B | 26A | 26B | 52A | 52B | 52C | 52D | |
| size | 1 | 1 | 13 | 13 | 2 | 26 | 26 | 13 | 13 | 26 | 26 | 26 | 26 | 26 | 26 | 6 | 6 | 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 | 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 | 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 | -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 | -1 | linear of order 2 |
| ρ5 | 1 | 1 | ζ3 | ζ32 | 1 | -1 | -1 | ζ3 | ζ32 | ζ65 | ζ6 | ζ65 | ζ3 | ζ6 | ζ32 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 6 |
| ρ6 | 1 | 1 | ζ3 | ζ32 | -1 | 1 | -1 | ζ3 | ζ32 | ζ65 | ζ32 | ζ3 | ζ65 | ζ6 | ζ6 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | linear of order 6 |
| ρ7 | 1 | 1 | ζ3 | ζ32 | -1 | -1 | 1 | ζ3 | ζ32 | ζ3 | ζ6 | ζ65 | ζ65 | ζ32 | ζ6 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | linear of order 6 |
| ρ8 | 1 | 1 | ζ32 | ζ3 | -1 | -1 | 1 | ζ32 | ζ3 | ζ32 | ζ65 | ζ6 | ζ6 | ζ3 | ζ65 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | linear of order 6 |
| ρ9 | 1 | 1 | ζ32 | ζ3 | 1 | -1 | -1 | ζ32 | ζ3 | ζ6 | ζ65 | ζ6 | ζ32 | ζ65 | ζ3 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 6 |
| ρ10 | 1 | 1 | ζ32 | ζ3 | 1 | 1 | 1 | ζ32 | ζ3 | ζ32 | ζ3 | ζ32 | ζ32 | ζ3 | ζ3 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 3 |
| ρ11 | 1 | 1 | ζ32 | ζ3 | -1 | 1 | -1 | ζ32 | ζ3 | ζ6 | ζ3 | ζ32 | ζ6 | ζ65 | ζ65 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | linear of order 6 |
| ρ12 | 1 | 1 | ζ3 | ζ32 | 1 | 1 | 1 | ζ3 | ζ32 | ζ3 | ζ32 | ζ3 | ζ3 | ζ32 | ζ32 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 3 |
| ρ13 | 2 | -2 | 2 | 2 | 0 | 0 | 0 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | symplectic lifted from Q8, Schur index 2 |
| ρ14 | 2 | -2 | -1-√-3 | -1+√-3 | 0 | 0 | 0 | 1+√-3 | 1-√-3 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | complex lifted from C3×Q8 |
| ρ15 | 2 | -2 | -1+√-3 | -1-√-3 | 0 | 0 | 0 | 1-√-3 | 1+√-3 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | complex lifted from C3×Q8 |
| ρ16 | 6 | 6 | 0 | 0 | -6 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1-√13/2 | -1+√13/2 | -1+√13/2 | -1-√13/2 | 1-√13/2 | 1-√13/2 | 1+√13/2 | 1+√13/2 | orthogonal lifted from C2×C13⋊C6 |
| ρ17 | 6 | 6 | 0 | 0 | 6 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1+√13/2 | -1-√13/2 | -1-√13/2 | -1+√13/2 | -1-√13/2 | -1-√13/2 | -1+√13/2 | -1+√13/2 | orthogonal lifted from C13⋊C6 |
| ρ18 | 6 | 6 | 0 | 0 | -6 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1+√13/2 | -1-√13/2 | -1-√13/2 | -1+√13/2 | 1+√13/2 | 1+√13/2 | 1-√13/2 | 1-√13/2 | orthogonal lifted from C2×C13⋊C6 |
| ρ19 | 6 | 6 | 0 | 0 | 6 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1-√13/2 | -1+√13/2 | -1+√13/2 | -1-√13/2 | -1+√13/2 | -1+√13/2 | -1-√13/2 | -1-√13/2 | orthogonal lifted from C13⋊C6 |
| ρ20 | 6 | -6 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1-√13/2 | -1+√13/2 | 1-√13/2 | 1+√13/2 | ζ43ζ1312+ζ43ζ1310-ζ43ζ139+ζ43ζ134-ζ43ζ133-ζ43ζ13 | ζ4ζ1312+ζ4ζ1310-ζ4ζ139+ζ4ζ134-ζ4ζ133-ζ4ζ13 | -ζ43ζ1311-ζ43ζ138-ζ43ζ137+ζ43ζ136+ζ43ζ135+ζ43ζ132 | -ζ4ζ1311-ζ4ζ138-ζ4ζ137+ζ4ζ136+ζ4ζ135+ζ4ζ132 | symplectic faithful, Schur index 2 |
| ρ21 | 6 | -6 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1+√13/2 | -1-√13/2 | 1+√13/2 | 1-√13/2 | -ζ43ζ1311-ζ43ζ138-ζ43ζ137+ζ43ζ136+ζ43ζ135+ζ43ζ132 | -ζ4ζ1311-ζ4ζ138-ζ4ζ137+ζ4ζ136+ζ4ζ135+ζ4ζ132 | ζ4ζ1312+ζ4ζ1310-ζ4ζ139+ζ4ζ134-ζ4ζ133-ζ4ζ13 | ζ43ζ1312+ζ43ζ1310-ζ43ζ139+ζ43ζ134-ζ43ζ133-ζ43ζ13 | symplectic faithful, Schur index 2 |
| ρ22 | 6 | -6 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1+√13/2 | -1-√13/2 | 1+√13/2 | 1-√13/2 | -ζ4ζ1311-ζ4ζ138-ζ4ζ137+ζ4ζ136+ζ4ζ135+ζ4ζ132 | -ζ43ζ1311-ζ43ζ138-ζ43ζ137+ζ43ζ136+ζ43ζ135+ζ43ζ132 | ζ43ζ1312+ζ43ζ1310-ζ43ζ139+ζ43ζ134-ζ43ζ133-ζ43ζ13 | ζ4ζ1312+ζ4ζ1310-ζ4ζ139+ζ4ζ134-ζ4ζ133-ζ4ζ13 | symplectic faithful, Schur index 2 |
| ρ23 | 6 | -6 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1-√13/2 | -1+√13/2 | 1-√13/2 | 1+√13/2 | ζ4ζ1312+ζ4ζ1310-ζ4ζ139+ζ4ζ134-ζ4ζ133-ζ4ζ13 | ζ43ζ1312+ζ43ζ1310-ζ43ζ139+ζ43ζ134-ζ43ζ133-ζ43ζ13 | -ζ4ζ1311-ζ4ζ138-ζ4ζ137+ζ4ζ136+ζ4ζ135+ζ4ζ132 | -ζ43ζ1311-ζ43ζ138-ζ43ζ137+ζ43ζ136+ζ43ζ135+ζ43ζ132 | symplectic faithful, Schur index 2 |
►On 104 points
Generators in S104
(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 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104)
(1 53 27 79)(2 104 28 78)(3 103 29 77)(4 102 30 76)(5 101 31 75)(6 100 32 74)(7 99 33 73)(8 98 34 72)(9 97 35 71)(10 96 36 70)(11 95 37 69)(12 94 38 68)(13 93 39 67)(14 92 40 66)(15 91 41 65)(16 90 42 64)(17 89 43 63)(18 88 44 62)(19 87 45 61)(20 86 46 60)(21 85 47 59)(22 84 48 58)(23 83 49 57)(24 82 50 56)(25 81 51 55)(26 80 52 54)
(2 30 10)(3 7 19)(4 36 28)(5 13 37)(6 42 46)(8 48 12)(9 25 21)(11 31 39)(15 43 23)(16 20 32)(17 49 41)(18 26 50)(22 38 34)(24 44 52)(29 33 45)(35 51 47)(54 82 62)(55 59 71)(56 88 80)(57 65 89)(58 94 98)(60 100 64)(61 77 73)(63 83 91)(67 95 75)(68 72 84)(69 101 93)(70 78 102)(74 90 86)(76 96 104)(81 85 97)(87 103 99)
G:=sub<Sym(104)| (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)(53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104), (1,53,27,79)(2,104,28,78)(3,103,29,77)(4,102,30,76)(5,101,31,75)(6,100,32,74)(7,99,33,73)(8,98,34,72)(9,97,35,71)(10,96,36,70)(11,95,37,69)(12,94,38,68)(13,93,39,67)(14,92,40,66)(15,91,41,65)(16,90,42,64)(17,89,43,63)(18,88,44,62)(19,87,45,61)(20,86,46,60)(21,85,47,59)(22,84,48,58)(23,83,49,57)(24,82,50,56)(25,81,51,55)(26,80,52,54), (2,30,10)(3,7,19)(4,36,28)(5,13,37)(6,42,46)(8,48,12)(9,25,21)(11,31,39)(15,43,23)(16,20,32)(17,49,41)(18,26,50)(22,38,34)(24,44,52)(29,33,45)(35,51,47)(54,82,62)(55,59,71)(56,88,80)(57,65,89)(58,94,98)(60,100,64)(61,77,73)(63,83,91)(67,95,75)(68,72,84)(69,101,93)(70,78,102)(74,90,86)(76,96,104)(81,85,97)(87,103,99)>;
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)(53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104), (1,53,27,79)(2,104,28,78)(3,103,29,77)(4,102,30,76)(5,101,31,75)(6,100,32,74)(7,99,33,73)(8,98,34,72)(9,97,35,71)(10,96,36,70)(11,95,37,69)(12,94,38,68)(13,93,39,67)(14,92,40,66)(15,91,41,65)(16,90,42,64)(17,89,43,63)(18,88,44,62)(19,87,45,61)(20,86,46,60)(21,85,47,59)(22,84,48,58)(23,83,49,57)(24,82,50,56)(25,81,51,55)(26,80,52,54), (2,30,10)(3,7,19)(4,36,28)(5,13,37)(6,42,46)(8,48,12)(9,25,21)(11,31,39)(15,43,23)(16,20,32)(17,49,41)(18,26,50)(22,38,34)(24,44,52)(29,33,45)(35,51,47)(54,82,62)(55,59,71)(56,88,80)(57,65,89)(58,94,98)(60,100,64)(61,77,73)(63,83,91)(67,95,75)(68,72,84)(69,101,93)(70,78,102)(74,90,86)(76,96,104)(81,85,97)(87,103,99) );
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),(53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104)], [(1,53,27,79),(2,104,28,78),(3,103,29,77),(4,102,30,76),(5,101,31,75),(6,100,32,74),(7,99,33,73),(8,98,34,72),(9,97,35,71),(10,96,36,70),(11,95,37,69),(12,94,38,68),(13,93,39,67),(14,92,40,66),(15,91,41,65),(16,90,42,64),(17,89,43,63),(18,88,44,62),(19,87,45,61),(20,86,46,60),(21,85,47,59),(22,84,48,58),(23,83,49,57),(24,82,50,56),(25,81,51,55),(26,80,52,54)], [(2,30,10),(3,7,19),(4,36,28),(5,13,37),(6,42,46),(8,48,12),(9,25,21),(11,31,39),(15,43,23),(16,20,32),(17,49,41),(18,26,50),(22,38,34),(24,44,52),(29,33,45),(35,51,47),(54,82,62),(55,59,71),(56,88,80),(57,65,89),(58,94,98),(60,100,64),(61,77,73),(63,83,91),(67,95,75),(68,72,84),(69,101,93),(70,78,102),(74,90,86),(76,96,104),(81,85,97),(87,103,99)]])
Matrix representation of Dic26⋊C3 ►in GL6(𝔽157)
| 138 | 69 | 115 | 149 | 4 | 50 |
| 138 | 46 | 138 | 153 | 0 | 0 |
| 134 | 50 | 4 | 130 | 0 | 50 |
| 138 | 42 | 119 | 42 | 138 | 153 |
| 92 | 88 | 123 | 145 | 8 | 73 |
| 140 | 17 | 117 | 44 | 136 | 155 |
| 53 | 98 | 106 | 10 | 94 | 41 |
| 100 | 10 | 153 | 155 | 153 | 110 |
| 109 | 152 | 46 | 117 | 40 | 145 |
| 150 | 61 | 149 | 105 | 46 | 60 |
| 106 | 155 | 146 | 64 | 46 | 98 |
| 96 | 55 | 45 | 11 | 93 | 54 |
| 89 | 68 | 88 | 2 | 156 | 67 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 156 | 2 | 88 | 68 | 89 | 2 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 | 0 | 0 |
G:=sub<GL(6,GF(157))| [138,138,134,138,92,140,69,46,50,42,88,17,115,138,4,119,123,117,149,153,130,42,145,44,4,0,0,138,8,136,50,0,50,153,73,155],[53,100,109,150,106,96,98,10,152,61,155,55,106,153,46,149,146,45,10,155,117,105,64,11,94,153,40,46,46,93,41,110,145,60,98,54],[89,0,1,156,0,0,68,0,0,2,0,1,88,0,0,88,0,0,2,0,0,68,0,0,156,1,0,89,0,0,67,0,0,2,1,0] >;
Dic26⋊C3 in GAP, Magma, Sage, TeX
{\rm Dic}_{26}\rtimes C_3 % in TeX
G:=Group("Dic26:C3"); // GroupNames label
G:=SmallGroup(312,8);
// by ID
G=gap.SmallGroup(312,8);
# by ID
G:=PCGroup([5,-2,-2,-3,-2,-13,60,141,66,7204,464]);
// Polycyclic
G:=Group<a,b,c|a^52=c^3=1,b^2=a^26,b*a*b^-1=a^-1,c*a*c^-1=a^9,b*c=c*b>;
// generators/relations
Export
Subgroup lattice of Dic26⋊C3 in TeX
Character table of Dic26⋊C3 in TeX