metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C8⋊3D17, C136⋊4C2, D34.2C4, C4.13D34, C17⋊3M4(2), C68.13C22, Dic17.2C4, C17⋊3C8⋊4C2, C34.9(C2×C4), C2.3(C4×D17), (C4×D17).2C2, SmallGroup(272,5)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C8⋊D17
G = < a,b,c | a8=b17=c2=1, ab=ba, cac=a5, cbc=b-1 >
Smallest permutation representation of C8⋊D17
►On 136 points
Generators in S136
(1 130 57 99 23 115 41 73)(2 131 58 100 24 116 42 74)(3 132 59 101 25 117 43 75)(4 133 60 102 26 118 44 76)(5 134 61 86 27 119 45 77)(6 135 62 87 28 103 46 78)(7 136 63 88 29 104 47 79)(8 120 64 89 30 105 48 80)(9 121 65 90 31 106 49 81)(10 122 66 91 32 107 50 82)(11 123 67 92 33 108 51 83)(12 124 68 93 34 109 35 84)(13 125 52 94 18 110 36 85)(14 126 53 95 19 111 37 69)(15 127 54 96 20 112 38 70)(16 128 55 97 21 113 39 71)(17 129 56 98 22 114 40 72)
(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 17)(2 16)(3 15)(4 14)(5 13)(6 12)(7 11)(8 10)(18 27)(19 26)(20 25)(21 24)(22 23)(28 34)(29 33)(30 32)(35 46)(36 45)(37 44)(38 43)(39 42)(40 41)(47 51)(48 50)(52 61)(53 60)(54 59)(55 58)(56 57)(62 68)(63 67)(64 66)(69 102)(70 101)(71 100)(72 99)(73 98)(74 97)(75 96)(76 95)(77 94)(78 93)(79 92)(80 91)(81 90)(82 89)(83 88)(84 87)(85 86)(103 124)(104 123)(105 122)(106 121)(107 120)(108 136)(109 135)(110 134)(111 133)(112 132)(113 131)(114 130)(115 129)(116 128)(117 127)(118 126)(119 125)
G:=sub<Sym(136)| (1,130,57,99,23,115,41,73)(2,131,58,100,24,116,42,74)(3,132,59,101,25,117,43,75)(4,133,60,102,26,118,44,76)(5,134,61,86,27,119,45,77)(6,135,62,87,28,103,46,78)(7,136,63,88,29,104,47,79)(8,120,64,89,30,105,48,80)(9,121,65,90,31,106,49,81)(10,122,66,91,32,107,50,82)(11,123,67,92,33,108,51,83)(12,124,68,93,34,109,35,84)(13,125,52,94,18,110,36,85)(14,126,53,95,19,111,37,69)(15,127,54,96,20,112,38,70)(16,128,55,97,21,113,39,71)(17,129,56,98,22,114,40,72), (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), (1,17)(2,16)(3,15)(4,14)(5,13)(6,12)(7,11)(8,10)(18,27)(19,26)(20,25)(21,24)(22,23)(28,34)(29,33)(30,32)(35,46)(36,45)(37,44)(38,43)(39,42)(40,41)(47,51)(48,50)(52,61)(53,60)(54,59)(55,58)(56,57)(62,68)(63,67)(64,66)(69,102)(70,101)(71,100)(72,99)(73,98)(74,97)(75,96)(76,95)(77,94)(78,93)(79,92)(80,91)(81,90)(82,89)(83,88)(84,87)(85,86)(103,124)(104,123)(105,122)(106,121)(107,120)(108,136)(109,135)(110,134)(111,133)(112,132)(113,131)(114,130)(115,129)(116,128)(117,127)(118,126)(119,125)>;
G:=Group( (1,130,57,99,23,115,41,73)(2,131,58,100,24,116,42,74)(3,132,59,101,25,117,43,75)(4,133,60,102,26,118,44,76)(5,134,61,86,27,119,45,77)(6,135,62,87,28,103,46,78)(7,136,63,88,29,104,47,79)(8,120,64,89,30,105,48,80)(9,121,65,90,31,106,49,81)(10,122,66,91,32,107,50,82)(11,123,67,92,33,108,51,83)(12,124,68,93,34,109,35,84)(13,125,52,94,18,110,36,85)(14,126,53,95,19,111,37,69)(15,127,54,96,20,112,38,70)(16,128,55,97,21,113,39,71)(17,129,56,98,22,114,40,72), (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), (1,17)(2,16)(3,15)(4,14)(5,13)(6,12)(7,11)(8,10)(18,27)(19,26)(20,25)(21,24)(22,23)(28,34)(29,33)(30,32)(35,46)(36,45)(37,44)(38,43)(39,42)(40,41)(47,51)(48,50)(52,61)(53,60)(54,59)(55,58)(56,57)(62,68)(63,67)(64,66)(69,102)(70,101)(71,100)(72,99)(73,98)(74,97)(75,96)(76,95)(77,94)(78,93)(79,92)(80,91)(81,90)(82,89)(83,88)(84,87)(85,86)(103,124)(104,123)(105,122)(106,121)(107,120)(108,136)(109,135)(110,134)(111,133)(112,132)(113,131)(114,130)(115,129)(116,128)(117,127)(118,126)(119,125) );
G=PermutationGroup([[(1,130,57,99,23,115,41,73),(2,131,58,100,24,116,42,74),(3,132,59,101,25,117,43,75),(4,133,60,102,26,118,44,76),(5,134,61,86,27,119,45,77),(6,135,62,87,28,103,46,78),(7,136,63,88,29,104,47,79),(8,120,64,89,30,105,48,80),(9,121,65,90,31,106,49,81),(10,122,66,91,32,107,50,82),(11,123,67,92,33,108,51,83),(12,124,68,93,34,109,35,84),(13,125,52,94,18,110,36,85),(14,126,53,95,19,111,37,69),(15,127,54,96,20,112,38,70),(16,128,55,97,21,113,39,71),(17,129,56,98,22,114,40,72)], [(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)], [(1,17),(2,16),(3,15),(4,14),(5,13),(6,12),(7,11),(8,10),(18,27),(19,26),(20,25),(21,24),(22,23),(28,34),(29,33),(30,32),(35,46),(36,45),(37,44),(38,43),(39,42),(40,41),(47,51),(48,50),(52,61),(53,60),(54,59),(55,58),(56,57),(62,68),(63,67),(64,66),(69,102),(70,101),(71,100),(72,99),(73,98),(74,97),(75,96),(76,95),(77,94),(78,93),(79,92),(80,91),(81,90),(82,89),(83,88),(84,87),(85,86),(103,124),(104,123),(105,122),(106,121),(107,120),(108,136),(109,135),(110,134),(111,133),(112,132),(113,131),(114,130),(115,129),(116,128),(117,127),(118,126),(119,125)]])
74 conjugacy classes
| class | 1 | 2A | 2B | 4A | 4B | 4C | 8A | 8B | 8C | 8D | 17A | ··· | 17H | 34A | ··· | 34H | 68A | ··· | 68P | 136A | ··· | 136AF |
| order | 1 | 2 | 2 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 17 | ··· | 17 | 34 | ··· | 34 | 68 | ··· | 68 | 136 | ··· | 136 |
| size | 1 | 1 | 34 | 1 | 1 | 34 | 2 | 2 | 34 | 34 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
74 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | |||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | M4(2) | D17 | D34 | C4×D17 | C8⋊D17 |
| kernel | C8⋊D17 | C17⋊3C8 | C136 | C4×D17 | Dic17 | D34 | C17 | C8 | C4 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 8 | 8 | 16 | 32 |
Matrix representation of C8⋊D17 ►in GL2(𝔽137) generated by
| 57 | 86 |
| 51 | 80 |
| 30 | 1 |
| 136 | 0 |
| 0 | 1 |
| 1 | 0 |
G:=sub<GL(2,GF(137))| [57,51,86,80],[30,136,1,0],[0,1,1,0] >;
C8⋊D17 in GAP, Magma, Sage, TeX
C_8\rtimes D_{17} % in TeX
G:=Group("C8:D17"); // GroupNames label
G:=SmallGroup(272,5);
// by ID
G=gap.SmallGroup(272,5);
# by ID
G:=PCGroup([5,-2,-2,-2,-2,-17,101,26,42,6404]);
// Polycyclic
G:=Group<a,b,c|a^8=b^17=c^2=1,a*b=b*a,c*a*c=a^5,c*b*c=b^-1>;
// generators/relations
Export