metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: C17⋊2C8, C34.C4, Dic17.2C2, C2.(C17⋊C4), SmallGroup(136,3)
Series: Derived ►Chief ►Lower central ►Upper central
| C17 — C17⋊2C8 |
Generators and relations for C17⋊2C8
G = < a,b | a17=b8=1, bab-1=a4 >
Character table of C17⋊2C8
| class | 1 | 2 | 4A | 4B | 8A | 8B | 8C | 8D | 17A | 17B | 17C | 17D | 34A | 34B | 34C | 34D | |
| size | 1 | 1 | 17 | 17 | 17 | 17 | 17 | 17 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | |
| ρ1 | 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 | linear of order 2 |
| ρ3 | 1 | 1 | -1 | -1 | -i | i | -i | i | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 4 |
| ρ4 | 1 | 1 | -1 | -1 | i | -i | i | -i | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 4 |
| ρ5 | 1 | -1 | -i | i | ζ8 | ζ83 | ζ85 | ζ87 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | linear of order 8 |
| ρ6 | 1 | -1 | i | -i | ζ87 | ζ85 | ζ83 | ζ8 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | linear of order 8 |
| ρ7 | 1 | -1 | i | -i | ζ83 | ζ8 | ζ87 | ζ85 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | linear of order 8 |
| ρ8 | 1 | -1 | -i | i | ζ85 | ζ87 | ζ8 | ζ83 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | linear of order 8 |
| ρ9 | 4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | ζ1711+ζ1710+ζ177+ζ176 | ζ1714+ζ1712+ζ175+ζ173 | ζ1716+ζ1713+ζ174+ζ17 | ζ1715+ζ179+ζ178+ζ172 | ζ1716+ζ1713+ζ174+ζ17 | ζ1715+ζ179+ζ178+ζ172 | ζ1711+ζ1710+ζ177+ζ176 | ζ1714+ζ1712+ζ175+ζ173 | orthogonal lifted from C17⋊C4 |
| ρ10 | 4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | ζ1715+ζ179+ζ178+ζ172 | ζ1716+ζ1713+ζ174+ζ17 | ζ1711+ζ1710+ζ177+ζ176 | ζ1714+ζ1712+ζ175+ζ173 | ζ1711+ζ1710+ζ177+ζ176 | ζ1714+ζ1712+ζ175+ζ173 | ζ1715+ζ179+ζ178+ζ172 | ζ1716+ζ1713+ζ174+ζ17 | orthogonal lifted from C17⋊C4 |
| ρ11 | 4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | ζ1714+ζ1712+ζ175+ζ173 | ζ1711+ζ1710+ζ177+ζ176 | ζ1715+ζ179+ζ178+ζ172 | ζ1716+ζ1713+ζ174+ζ17 | ζ1715+ζ179+ζ178+ζ172 | ζ1716+ζ1713+ζ174+ζ17 | ζ1714+ζ1712+ζ175+ζ173 | ζ1711+ζ1710+ζ177+ζ176 | orthogonal lifted from C17⋊C4 |
| ρ12 | 4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | ζ1716+ζ1713+ζ174+ζ17 | ζ1715+ζ179+ζ178+ζ172 | ζ1714+ζ1712+ζ175+ζ173 | ζ1711+ζ1710+ζ177+ζ176 | ζ1714+ζ1712+ζ175+ζ173 | ζ1711+ζ1710+ζ177+ζ176 | ζ1716+ζ1713+ζ174+ζ17 | ζ1715+ζ179+ζ178+ζ172 | orthogonal lifted from C17⋊C4 |
| ρ13 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | ζ1714+ζ1712+ζ175+ζ173 | ζ1711+ζ1710+ζ177+ζ176 | ζ1715+ζ179+ζ178+ζ172 | ζ1716+ζ1713+ζ174+ζ17 | -ζ1715-ζ179-ζ178-ζ172 | -ζ1716-ζ1713-ζ174-ζ17 | -ζ1714-ζ1712-ζ175-ζ173 | -ζ1711-ζ1710-ζ177-ζ176 | symplectic faithful, Schur index 2 |
| ρ14 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | ζ1711+ζ1710+ζ177+ζ176 | ζ1714+ζ1712+ζ175+ζ173 | ζ1716+ζ1713+ζ174+ζ17 | ζ1715+ζ179+ζ178+ζ172 | -ζ1716-ζ1713-ζ174-ζ17 | -ζ1715-ζ179-ζ178-ζ172 | -ζ1711-ζ1710-ζ177-ζ176 | -ζ1714-ζ1712-ζ175-ζ173 | symplectic faithful, Schur index 2 |
| ρ15 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | ζ1715+ζ179+ζ178+ζ172 | ζ1716+ζ1713+ζ174+ζ17 | ζ1711+ζ1710+ζ177+ζ176 | ζ1714+ζ1712+ζ175+ζ173 | -ζ1711-ζ1710-ζ177-ζ176 | -ζ1714-ζ1712-ζ175-ζ173 | -ζ1715-ζ179-ζ178-ζ172 | -ζ1716-ζ1713-ζ174-ζ17 | symplectic faithful, Schur index 2 |
| ρ16 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | ζ1716+ζ1713+ζ174+ζ17 | ζ1715+ζ179+ζ178+ζ172 | ζ1714+ζ1712+ζ175+ζ173 | ζ1711+ζ1710+ζ177+ζ176 | -ζ1714-ζ1712-ζ175-ζ173 | -ζ1711-ζ1710-ζ177-ζ176 | -ζ1716-ζ1713-ζ174-ζ17 | -ζ1715-ζ179-ζ178-ζ172 | symplectic faithful, Schur index 2 |
►Regular action on 136 points
Generators in S136
(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)(103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119)(120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136)
(1 125 52 86 30 103 35 83)(2 121 68 90 31 116 51 70)(3 134 67 94 32 112 50 74)(4 130 66 98 33 108 49 78)(5 126 65 102 34 104 48 82)(6 122 64 89 18 117 47 69)(7 135 63 93 19 113 46 73)(8 131 62 97 20 109 45 77)(9 127 61 101 21 105 44 81)(10 123 60 88 22 118 43 85)(11 136 59 92 23 114 42 72)(12 132 58 96 24 110 41 76)(13 128 57 100 25 106 40 80)(14 124 56 87 26 119 39 84)(15 120 55 91 27 115 38 71)(16 133 54 95 28 111 37 75)(17 129 53 99 29 107 36 79)
G:=sub<Sym(136)| (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)(103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119)(120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136), (1,125,52,86,30,103,35,83)(2,121,68,90,31,116,51,70)(3,134,67,94,32,112,50,74)(4,130,66,98,33,108,49,78)(5,126,65,102,34,104,48,82)(6,122,64,89,18,117,47,69)(7,135,63,93,19,113,46,73)(8,131,62,97,20,109,45,77)(9,127,61,101,21,105,44,81)(10,123,60,88,22,118,43,85)(11,136,59,92,23,114,42,72)(12,132,58,96,24,110,41,76)(13,128,57,100,25,106,40,80)(14,124,56,87,26,119,39,84)(15,120,55,91,27,115,38,71)(16,133,54,95,28,111,37,75)(17,129,53,99,29,107,36,79)>;
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)(103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119)(120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136), (1,125,52,86,30,103,35,83)(2,121,68,90,31,116,51,70)(3,134,67,94,32,112,50,74)(4,130,66,98,33,108,49,78)(5,126,65,102,34,104,48,82)(6,122,64,89,18,117,47,69)(7,135,63,93,19,113,46,73)(8,131,62,97,20,109,45,77)(9,127,61,101,21,105,44,81)(10,123,60,88,22,118,43,85)(11,136,59,92,23,114,42,72)(12,132,58,96,24,110,41,76)(13,128,57,100,25,106,40,80)(14,124,56,87,26,119,39,84)(15,120,55,91,27,115,38,71)(16,133,54,95,28,111,37,75)(17,129,53,99,29,107,36,79) );
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),(103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119),(120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136)], [(1,125,52,86,30,103,35,83),(2,121,68,90,31,116,51,70),(3,134,67,94,32,112,50,74),(4,130,66,98,33,108,49,78),(5,126,65,102,34,104,48,82),(6,122,64,89,18,117,47,69),(7,135,63,93,19,113,46,73),(8,131,62,97,20,109,45,77),(9,127,61,101,21,105,44,81),(10,123,60,88,22,118,43,85),(11,136,59,92,23,114,42,72),(12,132,58,96,24,110,41,76),(13,128,57,100,25,106,40,80),(14,124,56,87,26,119,39,84),(15,120,55,91,27,115,38,71),(16,133,54,95,28,111,37,75),(17,129,53,99,29,107,36,79)]])
C17⋊2C8 is a maximal subgroup of
C34.C8 C68.C4 D34.4C4 C17⋊M4(2) C51⋊3C8
C17⋊2C8 is a maximal quotient of C17⋊3C16 C51⋊3C8
Matrix representation of C17⋊2C8 ►in GL5(𝔽137)
| 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 0 | 136 | 52 | 72 | 52 |
| 10 | 0 | 0 | 0 | 0 |
| 0 | 117 | 73 | 22 | 106 |
| 0 | 27 | 2 | 103 | 76 |
| 0 | 94 | 33 | 24 | 92 |
| 0 | 32 | 45 | 57 | 131 |
G:=sub<GL(5,GF(137))| [1,0,0,0,0,0,0,0,0,136,0,1,0,0,52,0,0,1,0,72,0,0,0,1,52],[10,0,0,0,0,0,117,27,94,32,0,73,2,33,45,0,22,103,24,57,0,106,76,92,131] >;
C17⋊2C8 in GAP, Magma, Sage, TeX
C_{17}\rtimes_2C_8 % in TeX
G:=Group("C17:2C8"); // GroupNames label
G:=SmallGroup(136,3);
// by ID
G=gap.SmallGroup(136,3);
# by ID
G:=PCGroup([4,-2,-2,-2,-17,8,21,1667,1031]);
// Polycyclic
G:=Group<a,b|a^17=b^8=1,b*a*b^-1=a^4>;
// generators/relations
Export
Subgroup lattice of C17⋊2C8 in TeX
Character table of C17⋊2C8 in TeX