direct product, metacyclic, nilpotent (class 2), monomial, 2-elementary
Aliases: Q8×C17, C4.C34, C68.3C2, C34.7C22, C2.2(C2×C34), SmallGroup(136,11)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Q8×C17
G = < a,b,c | a17=b4=1, c2=b2, ab=ba, ac=ca, cbc-1=b-1 >
Smallest permutation representation of Q8×C17
►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 53 26 97)(2 54 27 98)(3 55 28 99)(4 56 29 100)(5 57 30 101)(6 58 31 102)(7 59 32 86)(8 60 33 87)(9 61 34 88)(10 62 18 89)(11 63 19 90)(12 64 20 91)(13 65 21 92)(14 66 22 93)(15 67 23 94)(16 68 24 95)(17 52 25 96)(35 135 106 70)(36 136 107 71)(37 120 108 72)(38 121 109 73)(39 122 110 74)(40 123 111 75)(41 124 112 76)(42 125 113 77)(43 126 114 78)(44 127 115 79)(45 128 116 80)(46 129 117 81)(47 130 118 82)(48 131 119 83)(49 132 103 84)(50 133 104 85)(51 134 105 69)
(1 85 26 133)(2 69 27 134)(3 70 28 135)(4 71 29 136)(5 72 30 120)(6 73 31 121)(7 74 32 122)(8 75 33 123)(9 76 34 124)(10 77 18 125)(11 78 19 126)(12 79 20 127)(13 80 21 128)(14 81 22 129)(15 82 23 130)(16 83 24 131)(17 84 25 132)(35 55 106 99)(36 56 107 100)(37 57 108 101)(38 58 109 102)(39 59 110 86)(40 60 111 87)(41 61 112 88)(42 62 113 89)(43 63 114 90)(44 64 115 91)(45 65 116 92)(46 66 117 93)(47 67 118 94)(48 68 119 95)(49 52 103 96)(50 53 104 97)(51 54 105 98)
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), (1,53,26,97)(2,54,27,98)(3,55,28,99)(4,56,29,100)(5,57,30,101)(6,58,31,102)(7,59,32,86)(8,60,33,87)(9,61,34,88)(10,62,18,89)(11,63,19,90)(12,64,20,91)(13,65,21,92)(14,66,22,93)(15,67,23,94)(16,68,24,95)(17,52,25,96)(35,135,106,70)(36,136,107,71)(37,120,108,72)(38,121,109,73)(39,122,110,74)(40,123,111,75)(41,124,112,76)(42,125,113,77)(43,126,114,78)(44,127,115,79)(45,128,116,80)(46,129,117,81)(47,130,118,82)(48,131,119,83)(49,132,103,84)(50,133,104,85)(51,134,105,69), (1,85,26,133)(2,69,27,134)(3,70,28,135)(4,71,29,136)(5,72,30,120)(6,73,31,121)(7,74,32,122)(8,75,33,123)(9,76,34,124)(10,77,18,125)(11,78,19,126)(12,79,20,127)(13,80,21,128)(14,81,22,129)(15,82,23,130)(16,83,24,131)(17,84,25,132)(35,55,106,99)(36,56,107,100)(37,57,108,101)(38,58,109,102)(39,59,110,86)(40,60,111,87)(41,61,112,88)(42,62,113,89)(43,63,114,90)(44,64,115,91)(45,65,116,92)(46,66,117,93)(47,67,118,94)(48,68,119,95)(49,52,103,96)(50,53,104,97)(51,54,105,98)>;
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), (1,53,26,97)(2,54,27,98)(3,55,28,99)(4,56,29,100)(5,57,30,101)(6,58,31,102)(7,59,32,86)(8,60,33,87)(9,61,34,88)(10,62,18,89)(11,63,19,90)(12,64,20,91)(13,65,21,92)(14,66,22,93)(15,67,23,94)(16,68,24,95)(17,52,25,96)(35,135,106,70)(36,136,107,71)(37,120,108,72)(38,121,109,73)(39,122,110,74)(40,123,111,75)(41,124,112,76)(42,125,113,77)(43,126,114,78)(44,127,115,79)(45,128,116,80)(46,129,117,81)(47,130,118,82)(48,131,119,83)(49,132,103,84)(50,133,104,85)(51,134,105,69), (1,85,26,133)(2,69,27,134)(3,70,28,135)(4,71,29,136)(5,72,30,120)(6,73,31,121)(7,74,32,122)(8,75,33,123)(9,76,34,124)(10,77,18,125)(11,78,19,126)(12,79,20,127)(13,80,21,128)(14,81,22,129)(15,82,23,130)(16,83,24,131)(17,84,25,132)(35,55,106,99)(36,56,107,100)(37,57,108,101)(38,58,109,102)(39,59,110,86)(40,60,111,87)(41,61,112,88)(42,62,113,89)(43,63,114,90)(44,64,115,91)(45,65,116,92)(46,66,117,93)(47,67,118,94)(48,68,119,95)(49,52,103,96)(50,53,104,97)(51,54,105,98) );
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)], [(1,53,26,97),(2,54,27,98),(3,55,28,99),(4,56,29,100),(5,57,30,101),(6,58,31,102),(7,59,32,86),(8,60,33,87),(9,61,34,88),(10,62,18,89),(11,63,19,90),(12,64,20,91),(13,65,21,92),(14,66,22,93),(15,67,23,94),(16,68,24,95),(17,52,25,96),(35,135,106,70),(36,136,107,71),(37,120,108,72),(38,121,109,73),(39,122,110,74),(40,123,111,75),(41,124,112,76),(42,125,113,77),(43,126,114,78),(44,127,115,79),(45,128,116,80),(46,129,117,81),(47,130,118,82),(48,131,119,83),(49,132,103,84),(50,133,104,85),(51,134,105,69)], [(1,85,26,133),(2,69,27,134),(3,70,28,135),(4,71,29,136),(5,72,30,120),(6,73,31,121),(7,74,32,122),(8,75,33,123),(9,76,34,124),(10,77,18,125),(11,78,19,126),(12,79,20,127),(13,80,21,128),(14,81,22,129),(15,82,23,130),(16,83,24,131),(17,84,25,132),(35,55,106,99),(36,56,107,100),(37,57,108,101),(38,58,109,102),(39,59,110,86),(40,60,111,87),(41,61,112,88),(42,62,113,89),(43,63,114,90),(44,64,115,91),(45,65,116,92),(46,66,117,93),(47,67,118,94),(48,68,119,95),(49,52,103,96),(50,53,104,97),(51,54,105,98)]])
Q8×C17 is a maximal subgroup of
Q8⋊D17 C17⋊Q16 D68⋊C2
85 conjugacy classes
| class | 1 | 2 | 4A | 4B | 4C | 17A | ··· | 17P | 34A | ··· | 34P | 68A | ··· | 68AV |
| order | 1 | 2 | 4 | 4 | 4 | 17 | ··· | 17 | 34 | ··· | 34 | 68 | ··· | 68 |
| size | 1 | 1 | 2 | 2 | 2 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 |
85 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | - | |||
| image | C1 | C2 | C17 | C34 | Q8 | Q8×C17 |
| kernel | Q8×C17 | C68 | Q8 | C4 | C17 | C1 |
| # reps | 1 | 3 | 16 | 48 | 1 | 16 |
Matrix representation of Q8×C17 ►in GL2(𝔽137) generated by
| 38 | 0 |
| 0 | 38 |
| 0 | 1 |
| 136 | 0 |
| 39 | 56 |
| 56 | 98 |
G:=sub<GL(2,GF(137))| [38,0,0,38],[0,136,1,0],[39,56,56,98] >;
Q8×C17 in GAP, Magma, Sage, TeX
Q_8\times C_{17} % in TeX
G:=Group("Q8xC17"); // GroupNames label
G:=SmallGroup(136,11);
// by ID
G=gap.SmallGroup(136,11);
# by ID
G:=PCGroup([4,-2,-2,-17,-2,272,561,277]);
// Polycyclic
G:=Group<a,b,c|a^17=b^4=1,c^2=b^2,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
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