metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: C17⋊3C8, C34.2C4, C68.2C2, C4.2D17, C2.Dic17, SmallGroup(136,1)
Series: Derived ►Chief ►Lower central ►Upper central
| C17 — C17⋊3C8 |
Generators and relations for C17⋊3C8
G = < a,b | a17=b8=1, bab-1=a-1 >
Smallest permutation representation of C17⋊3C8
►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)
(1 120 65 86 28 103 36 69)(2 136 66 102 29 119 37 85)(3 135 67 101 30 118 38 84)(4 134 68 100 31 117 39 83)(5 133 52 99 32 116 40 82)(6 132 53 98 33 115 41 81)(7 131 54 97 34 114 42 80)(8 130 55 96 18 113 43 79)(9 129 56 95 19 112 44 78)(10 128 57 94 20 111 45 77)(11 127 58 93 21 110 46 76)(12 126 59 92 22 109 47 75)(13 125 60 91 23 108 48 74)(14 124 61 90 24 107 49 73)(15 123 62 89 25 106 50 72)(16 122 63 88 26 105 51 71)(17 121 64 87 27 104 35 70)
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), (1,120,65,86,28,103,36,69)(2,136,66,102,29,119,37,85)(3,135,67,101,30,118,38,84)(4,134,68,100,31,117,39,83)(5,133,52,99,32,116,40,82)(6,132,53,98,33,115,41,81)(7,131,54,97,34,114,42,80)(8,130,55,96,18,113,43,79)(9,129,56,95,19,112,44,78)(10,128,57,94,20,111,45,77)(11,127,58,93,21,110,46,76)(12,126,59,92,22,109,47,75)(13,125,60,91,23,108,48,74)(14,124,61,90,24,107,49,73)(15,123,62,89,25,106,50,72)(16,122,63,88,26,105,51,71)(17,121,64,87,27,104,35,70)>;
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), (1,120,65,86,28,103,36,69)(2,136,66,102,29,119,37,85)(3,135,67,101,30,118,38,84)(4,134,68,100,31,117,39,83)(5,133,52,99,32,116,40,82)(6,132,53,98,33,115,41,81)(7,131,54,97,34,114,42,80)(8,130,55,96,18,113,43,79)(9,129,56,95,19,112,44,78)(10,128,57,94,20,111,45,77)(11,127,58,93,21,110,46,76)(12,126,59,92,22,109,47,75)(13,125,60,91,23,108,48,74)(14,124,61,90,24,107,49,73)(15,123,62,89,25,106,50,72)(16,122,63,88,26,105,51,71)(17,121,64,87,27,104,35,70) );
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)], [(1,120,65,86,28,103,36,69),(2,136,66,102,29,119,37,85),(3,135,67,101,30,118,38,84),(4,134,68,100,31,117,39,83),(5,133,52,99,32,116,40,82),(6,132,53,98,33,115,41,81),(7,131,54,97,34,114,42,80),(8,130,55,96,18,113,43,79),(9,129,56,95,19,112,44,78),(10,128,57,94,20,111,45,77),(11,127,58,93,21,110,46,76),(12,126,59,92,22,109,47,75),(13,125,60,91,23,108,48,74),(14,124,61,90,24,107,49,73),(15,123,62,89,25,106,50,72),(16,122,63,88,26,105,51,71),(17,121,64,87,27,104,35,70)]])
C17⋊3C8 is a maximal subgroup of
C17⋊3C16 C8×D17 C8⋊D17 C68.4C4 D4⋊D17 D4.D17 Q8⋊D17 C17⋊Q16 C51⋊5C8
C17⋊3C8 is a maximal quotient of
C17⋊4C16 C51⋊5C8
40 conjugacy classes
| class | 1 | 2 | 4A | 4B | 8A | 8B | 8C | 8D | 17A | ··· | 17H | 34A | ··· | 34H | 68A | ··· | 68P |
| order | 1 | 2 | 4 | 4 | 8 | 8 | 8 | 8 | 17 | ··· | 17 | 34 | ··· | 34 | 68 | ··· | 68 |
| size | 1 | 1 | 1 | 1 | 17 | 17 | 17 | 17 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
40 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 |
| type | + | + | + | - | |||
| image | C1 | C2 | C4 | C8 | D17 | Dic17 | C17⋊3C8 |
| kernel | C17⋊3C8 | C68 | C34 | C17 | C4 | C2 | C1 |
| # reps | 1 | 1 | 2 | 4 | 8 | 8 | 16 |
Matrix representation of C17⋊3C8 ►in GL2(𝔽137) generated by
| 0 | 1 |
| 136 | 30 |
| 99 | 16 |
| 109 | 38 |
G:=sub<GL(2,GF(137))| [0,136,1,30],[99,109,16,38] >;
C17⋊3C8 in GAP, Magma, Sage, TeX
C_{17}\rtimes_3C_8 % in TeX
G:=Group("C17:3C8"); // GroupNames label
G:=SmallGroup(136,1);
// by ID
G=gap.SmallGroup(136,1);
# by ID
G:=PCGroup([4,-2,-2,-2,-17,8,21,2051]);
// Polycyclic
G:=Group<a,b|a^17=b^8=1,b*a*b^-1=a^-1>;
// generators/relations
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