metacyclic, supersoluble, monomial
Aliases: D52⋊C3, C52⋊1C6, D26⋊1C6, C4⋊(C13⋊C6), C13⋊C3⋊1D4, C13⋊1(C3×D4), C26.3(C2×C6), (C2×C13⋊C6)⋊1C2, (C4×C13⋊C3)⋊1C2, C2.4(C2×C13⋊C6), (C2×C13⋊C3).3C22, SmallGroup(312,10)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D52⋊C3
G = < a,b,c | a52=b2=c3=1, bab=a-1, cac-1=a9, cbc-1=a8b >
Character table of D52⋊C3
| class | 1 | 2A | 2B | 2C | 3A | 3B | 4 | 6A | 6B | 6C | 6D | 6E | 6F | 12A | 12B | 13A | 13B | 26A | 26B | 52A | 52B | 52C | 52D | |
| size | 1 | 1 | 26 | 26 | 13 | 13 | 2 | 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 | 1 | -1 | ζ32 | ζ3 | -1 | ζ32 | ζ3 | ζ32 | ζ65 | ζ3 | ζ6 | ζ6 | ζ65 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | linear of order 6 |
| ρ6 | 1 | 1 | 1 | 1 | ζ3 | ζ32 | 1 | ζ3 | ζ32 | ζ3 | ζ32 | ζ32 | ζ3 | ζ3 | ζ32 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 3 |
| ρ7 | 1 | 1 | -1 | 1 | ζ32 | ζ3 | -1 | ζ32 | ζ3 | ζ6 | ζ3 | ζ65 | ζ32 | ζ6 | ζ65 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | linear of order 6 |
| ρ8 | 1 | 1 | -1 | 1 | ζ3 | ζ32 | -1 | ζ3 | ζ32 | ζ65 | ζ32 | ζ6 | ζ3 | ζ65 | ζ6 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | linear of order 6 |
| ρ9 | 1 | 1 | 1 | -1 | ζ3 | ζ32 | -1 | ζ3 | ζ32 | ζ3 | ζ6 | ζ32 | ζ65 | ζ65 | ζ6 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | linear of order 6 |
| ρ10 | 1 | 1 | 1 | 1 | ζ32 | ζ3 | 1 | ζ32 | ζ3 | ζ32 | ζ3 | ζ3 | ζ32 | ζ32 | ζ3 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 3 |
| ρ11 | 1 | 1 | -1 | -1 | ζ3 | ζ32 | 1 | ζ3 | ζ32 | ζ65 | ζ6 | ζ6 | ζ65 | ζ3 | ζ32 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 6 |
| ρ12 | 1 | 1 | -1 | -1 | ζ32 | ζ3 | 1 | ζ32 | ζ3 | ζ6 | ζ65 | ζ65 | ζ6 | ζ32 | ζ3 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 6 |
| ρ13 | 2 | -2 | 0 | 0 | 2 | 2 | 0 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ14 | 2 | -2 | 0 | 0 | -1+√-3 | -1-√-3 | 0 | 1-√-3 | 1+√-3 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | complex lifted from C3×D4 |
| ρ15 | 2 | -2 | 0 | 0 | -1-√-3 | -1+√-3 | 0 | 1+√-3 | 1-√-3 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | complex lifted from C3×D4 |
| ρ16 | 6 | 6 | 0 | 0 | 0 | 0 | -6 | 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 | 0 | 0 | 6 | 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 | 0 | 0 | -6 | 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 | 0 | 0 | 6 | 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 | ζ4ζ1311+ζ4ζ138+ζ4ζ137-ζ4ζ136-ζ4ζ135-ζ4ζ132 | -ζ4ζ1312-ζ4ζ1310+ζ4ζ139-ζ4ζ134+ζ4ζ133+ζ4ζ13 | ζ43ζ1311+ζ43ζ138+ζ43ζ137-ζ43ζ136-ζ43ζ135-ζ43ζ132 | orthogonal faithful |
| 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 | -ζ4ζ1312-ζ4ζ1310+ζ4ζ139-ζ4ζ134+ζ4ζ133+ζ4ζ13 | ζ43ζ1311+ζ43ζ138+ζ43ζ137-ζ43ζ136-ζ43ζ135-ζ43ζ132 | -ζ43ζ1312-ζ43ζ1310+ζ43ζ139-ζ43ζ134+ζ43ζ133+ζ43ζ13 | ζ4ζ1311+ζ4ζ138+ζ4ζ137-ζ4ζ136-ζ4ζ135-ζ4ζ132 | orthogonal faithful |
| 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 | -ζ4ζ1312-ζ4ζ1310+ζ4ζ139-ζ4ζ134+ζ4ζ133+ζ4ζ13 | ζ43ζ1311+ζ43ζ138+ζ43ζ137-ζ43ζ136-ζ43ζ135-ζ43ζ132 | -ζ43ζ1312-ζ43ζ1310+ζ43ζ139-ζ43ζ134+ζ43ζ133+ζ43ζ13 | orthogonal faithful |
| 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 | ζ43ζ1311+ζ43ζ138+ζ43ζ137-ζ43ζ136-ζ43ζ135-ζ43ζ132 | -ζ43ζ1312-ζ43ζ1310+ζ43ζ139-ζ43ζ134+ζ43ζ133+ζ43ζ13 | ζ4ζ1311+ζ4ζ138+ζ4ζ137-ζ4ζ136-ζ4ζ135-ζ4ζ132 | -ζ4ζ1312-ζ4ζ1310+ζ4ζ139-ζ4ζ134+ζ4ζ133+ζ4ζ13 | orthogonal faithful |
►On 52 points
Generators in S52
(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)
(1 52)(2 51)(3 50)(4 49)(5 48)(6 47)(7 46)(8 45)(9 44)(10 43)(11 42)(12 41)(13 40)(14 39)(15 38)(16 37)(17 36)(18 35)(19 34)(20 33)(21 32)(22 31)(23 30)(24 29)(25 28)(26 27)
(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)
G:=sub<Sym(52)| (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), (1,52)(2,51)(3,50)(4,49)(5,48)(6,47)(7,46)(8,45)(9,44)(10,43)(11,42)(12,41)(13,40)(14,39)(15,38)(16,37)(17,36)(18,35)(19,34)(20,33)(21,32)(22,31)(23,30)(24,29)(25,28)(26,27), (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)>;
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), (1,52)(2,51)(3,50)(4,49)(5,48)(6,47)(7,46)(8,45)(9,44)(10,43)(11,42)(12,41)(13,40)(14,39)(15,38)(16,37)(17,36)(18,35)(19,34)(20,33)(21,32)(22,31)(23,30)(24,29)(25,28)(26,27), (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) );
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)], [(1,52),(2,51),(3,50),(4,49),(5,48),(6,47),(7,46),(8,45),(9,44),(10,43),(11,42),(12,41),(13,40),(14,39),(15,38),(16,37),(17,36),(18,35),(19,34),(20,33),(21,32),(22,31),(23,30),(24,29),(25,28),(26,27)], [(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)]])
Matrix representation of D52⋊C3 ►in GL6(𝔽157)
| 4 | 111 | 4 | 134 | 4 | 4 |
| 153 | 46 | 103 | 50 | 126 | 46 |
| 111 | 8 | 111 | 4 | 115 | 138 |
| 19 | 147 | 46 | 128 | 42 | 151 |
| 6 | 113 | 2 | 134 | 140 | 136 |
| 21 | 21 | 155 | 153 | 19 | 155 |
| 4 | 111 | 4 | 134 | 4 | 4 |
| 153 | 153 | 23 | 153 | 46 | 153 |
| 111 | 31 | 107 | 54 | 111 | 4 |
| 19 | 42 | 153 | 46 | 149 | 46 |
| 6 | 115 | 29 | 111 | 10 | 138 |
| 21 | 17 | 23 | 155 | 44 | 151 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 70 | 138 | 69 | 69 | 138 | 70 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 87 | 20 | 155 | 89 | 86 | 88 |
| 156 | 89 | 155 | 90 | 155 | 89 |
G:=sub<GL(6,GF(157))| [4,153,111,19,6,21,111,46,8,147,113,21,4,103,111,46,2,155,134,50,4,128,134,153,4,126,115,42,140,19,4,46,138,151,136,155],[4,153,111,19,6,21,111,153,31,42,115,17,4,23,107,153,29,23,134,153,54,46,111,155,4,46,111,149,10,44,4,153,4,46,138,151],[1,70,0,0,87,156,0,138,0,1,20,89,0,69,0,0,155,155,0,69,0,0,89,90,0,138,0,0,86,155,0,70,1,0,88,89] >;
D52⋊C3 in GAP, Magma, Sage, TeX
D_{52}\rtimes C_3 % in TeX
G:=Group("D52:C3"); // GroupNames label
G:=SmallGroup(312,10);
// by ID
G=gap.SmallGroup(312,10);
# by ID
G:=PCGroup([5,-2,-2,-3,-2,-13,141,66,7204,464]);
// Polycyclic
G:=Group<a,b,c|a^52=b^2=c^3=1,b*a*b=a^-1,c*a*c^-1=a^9,c*b*c^-1=a^8*b>;
// generators/relations
Export
Subgroup lattice of D52⋊C3 in TeX
Character table of D52⋊C3 in TeX