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## G = Dic26⋊C3order 312 = 23·3·13

### The semidirect product of Dic26 and C3 acting faithfully

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

 Derived series C1 — C26 — Dic26⋊C3
 Chief series C1 — C13 — C26 — C2×C13⋊C3 — C26.C6 — Dic26⋊C3
 Lower central C13 — C26 — Dic26⋊C3
 Upper central C1 — C2 — C4

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

Smallest permutation representation of Dic26⋊C3
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

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