direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C8×D17, C136⋊3C2, D34.5C4, C4.12D34, C68.12C22, Dic17.5C4, C17⋊4(C2×C8), C17⋊3C8⋊6C2, C34.8(C2×C4), C2.1(C4×D17), (C4×D17).7C2, SmallGroup(272,4)
Series: Derived ►Chief ►Lower central ►Upper central
| C17 — C8×D17 |
Generators and relations for C8×D17
G = < a,b,c | a8=b17=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of C8×D17
►On 136 points
Generators in S136
(1 132 62 100 31 117 40 83)(2 133 63 101 32 118 41 84)(3 134 64 102 33 119 42 85)(4 135 65 86 34 103 43 69)(5 136 66 87 18 104 44 70)(6 120 67 88 19 105 45 71)(7 121 68 89 20 106 46 72)(8 122 52 90 21 107 47 73)(9 123 53 91 22 108 48 74)(10 124 54 92 23 109 49 75)(11 125 55 93 24 110 50 76)(12 126 56 94 25 111 51 77)(13 127 57 95 26 112 35 78)(14 128 58 96 27 113 36 79)(15 129 59 97 28 114 37 80)(16 130 60 98 29 115 38 81)(17 131 61 99 30 116 39 82)
(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 30)(2 29)(3 28)(4 27)(5 26)(6 25)(7 24)(8 23)(9 22)(10 21)(11 20)(12 19)(13 18)(14 34)(15 33)(16 32)(17 31)(35 66)(36 65)(37 64)(38 63)(39 62)(40 61)(41 60)(42 59)(43 58)(44 57)(45 56)(46 55)(47 54)(48 53)(49 52)(50 68)(51 67)(69 96)(70 95)(71 94)(72 93)(73 92)(74 91)(75 90)(76 89)(77 88)(78 87)(79 86)(80 102)(81 101)(82 100)(83 99)(84 98)(85 97)(103 128)(104 127)(105 126)(106 125)(107 124)(108 123)(109 122)(110 121)(111 120)(112 136)(113 135)(114 134)(115 133)(116 132)(117 131)(118 130)(119 129)
G:=sub<Sym(136)| (1,132,62,100,31,117,40,83)(2,133,63,101,32,118,41,84)(3,134,64,102,33,119,42,85)(4,135,65,86,34,103,43,69)(5,136,66,87,18,104,44,70)(6,120,67,88,19,105,45,71)(7,121,68,89,20,106,46,72)(8,122,52,90,21,107,47,73)(9,123,53,91,22,108,48,74)(10,124,54,92,23,109,49,75)(11,125,55,93,24,110,50,76)(12,126,56,94,25,111,51,77)(13,127,57,95,26,112,35,78)(14,128,58,96,27,113,36,79)(15,129,59,97,28,114,37,80)(16,130,60,98,29,115,38,81)(17,131,61,99,30,116,39,82), (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), (1,30)(2,29)(3,28)(4,27)(5,26)(6,25)(7,24)(8,23)(9,22)(10,21)(11,20)(12,19)(13,18)(14,34)(15,33)(16,32)(17,31)(35,66)(36,65)(37,64)(38,63)(39,62)(40,61)(41,60)(42,59)(43,58)(44,57)(45,56)(46,55)(47,54)(48,53)(49,52)(50,68)(51,67)(69,96)(70,95)(71,94)(72,93)(73,92)(74,91)(75,90)(76,89)(77,88)(78,87)(79,86)(80,102)(81,101)(82,100)(83,99)(84,98)(85,97)(103,128)(104,127)(105,126)(106,125)(107,124)(108,123)(109,122)(110,121)(111,120)(112,136)(113,135)(114,134)(115,133)(116,132)(117,131)(118,130)(119,129)>;
G:=Group( (1,132,62,100,31,117,40,83)(2,133,63,101,32,118,41,84)(3,134,64,102,33,119,42,85)(4,135,65,86,34,103,43,69)(5,136,66,87,18,104,44,70)(6,120,67,88,19,105,45,71)(7,121,68,89,20,106,46,72)(8,122,52,90,21,107,47,73)(9,123,53,91,22,108,48,74)(10,124,54,92,23,109,49,75)(11,125,55,93,24,110,50,76)(12,126,56,94,25,111,51,77)(13,127,57,95,26,112,35,78)(14,128,58,96,27,113,36,79)(15,129,59,97,28,114,37,80)(16,130,60,98,29,115,38,81)(17,131,61,99,30,116,39,82), (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), (1,30)(2,29)(3,28)(4,27)(5,26)(6,25)(7,24)(8,23)(9,22)(10,21)(11,20)(12,19)(13,18)(14,34)(15,33)(16,32)(17,31)(35,66)(36,65)(37,64)(38,63)(39,62)(40,61)(41,60)(42,59)(43,58)(44,57)(45,56)(46,55)(47,54)(48,53)(49,52)(50,68)(51,67)(69,96)(70,95)(71,94)(72,93)(73,92)(74,91)(75,90)(76,89)(77,88)(78,87)(79,86)(80,102)(81,101)(82,100)(83,99)(84,98)(85,97)(103,128)(104,127)(105,126)(106,125)(107,124)(108,123)(109,122)(110,121)(111,120)(112,136)(113,135)(114,134)(115,133)(116,132)(117,131)(118,130)(119,129) );
G=PermutationGroup([[(1,132,62,100,31,117,40,83),(2,133,63,101,32,118,41,84),(3,134,64,102,33,119,42,85),(4,135,65,86,34,103,43,69),(5,136,66,87,18,104,44,70),(6,120,67,88,19,105,45,71),(7,121,68,89,20,106,46,72),(8,122,52,90,21,107,47,73),(9,123,53,91,22,108,48,74),(10,124,54,92,23,109,49,75),(11,125,55,93,24,110,50,76),(12,126,56,94,25,111,51,77),(13,127,57,95,26,112,35,78),(14,128,58,96,27,113,36,79),(15,129,59,97,28,114,37,80),(16,130,60,98,29,115,38,81),(17,131,61,99,30,116,39,82)], [(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)], [(1,30),(2,29),(3,28),(4,27),(5,26),(6,25),(7,24),(8,23),(9,22),(10,21),(11,20),(12,19),(13,18),(14,34),(15,33),(16,32),(17,31),(35,66),(36,65),(37,64),(38,63),(39,62),(40,61),(41,60),(42,59),(43,58),(44,57),(45,56),(46,55),(47,54),(48,53),(49,52),(50,68),(51,67),(69,96),(70,95),(71,94),(72,93),(73,92),(74,91),(75,90),(76,89),(77,88),(78,87),(79,86),(80,102),(81,101),(82,100),(83,99),(84,98),(85,97),(103,128),(104,127),(105,126),(106,125),(107,124),(108,123),(109,122),(110,121),(111,120),(112,136),(113,135),(114,134),(115,133),(116,132),(117,131),(118,130),(119,129)]])
80 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 17A | ··· | 17H | 34A | ··· | 34H | 68A | ··· | 68P | 136A | ··· | 136AF |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 17 | ··· | 17 | 34 | ··· | 34 | 68 | ··· | 68 | 136 | ··· | 136 |
| size | 1 | 1 | 17 | 17 | 1 | 1 | 17 | 17 | 1 | 1 | 1 | 1 | 17 | 17 | 17 | 17 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
80 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | |||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C8 | D17 | D34 | C4×D17 | C8×D17 |
| kernel | C8×D17 | C17⋊3C8 | C136 | C4×D17 | Dic17 | D34 | D17 | C8 | C4 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 8 | 8 | 8 | 16 | 32 |
Matrix representation of C8×D17 ►in GL3(𝔽137) generated by
| 96 | 0 | 0 |
| 0 | 100 | 0 |
| 0 | 0 | 100 |
| 1 | 0 | 0 |
| 0 | 22 | 11 |
| 0 | 136 | 12 |
| 1 | 0 | 0 |
| 0 | 66 | 77 |
| 0 | 84 | 71 |
G:=sub<GL(3,GF(137))| [96,0,0,0,100,0,0,0,100],[1,0,0,0,22,136,0,11,12],[1,0,0,0,66,84,0,77,71] >;
C8×D17 in GAP, Magma, Sage, TeX
C_8\times D_{17} % in TeX
G:=Group("C8xD17"); // GroupNames label
G:=SmallGroup(272,4);
// by ID
G=gap.SmallGroup(272,4);
# by ID
G:=PCGroup([5,-2,-2,-2,-2,-17,26,42,6404]);
// Polycyclic
G:=Group<a,b,c|a^8=b^17=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export