direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C2×Dic17, C34⋊2C4, C2.2D34, C22.D17, C34.4C22, C17⋊3(C2×C4), (C2×C34).C2, SmallGroup(136,7)
Series: Derived ►Chief ►Lower central ►Upper central
| C17 — C2×Dic17 |
Generators and relations for C2×Dic17
G = < a,b,c | a2=b34=1, c2=b17, ab=ba, ac=ca, cbc-1=b-1 >
Smallest permutation representation of C2×Dic17
►Regular action on 136 points
Generators in S136
(1 63)(2 64)(3 65)(4 66)(5 67)(6 68)(7 35)(8 36)(9 37)(10 38)(11 39)(12 40)(13 41)(14 42)(15 43)(16 44)(17 45)(18 46)(19 47)(20 48)(21 49)(22 50)(23 51)(24 52)(25 53)(26 54)(27 55)(28 56)(29 57)(30 58)(31 59)(32 60)(33 61)(34 62)(69 120)(70 121)(71 122)(72 123)(73 124)(74 125)(75 126)(76 127)(77 128)(78 129)(79 130)(80 131)(81 132)(82 133)(83 134)(84 135)(85 136)(86 103)(87 104)(88 105)(89 106)(90 107)(91 108)(92 109)(93 110)(94 111)(95 112)(96 113)(97 114)(98 115)(99 116)(100 117)(101 118)(102 119)
(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 86 18 69)(2 85 19 102)(3 84 20 101)(4 83 21 100)(5 82 22 99)(6 81 23 98)(7 80 24 97)(8 79 25 96)(9 78 26 95)(10 77 27 94)(11 76 28 93)(12 75 29 92)(13 74 30 91)(14 73 31 90)(15 72 32 89)(16 71 33 88)(17 70 34 87)(35 131 52 114)(36 130 53 113)(37 129 54 112)(38 128 55 111)(39 127 56 110)(40 126 57 109)(41 125 58 108)(42 124 59 107)(43 123 60 106)(44 122 61 105)(45 121 62 104)(46 120 63 103)(47 119 64 136)(48 118 65 135)(49 117 66 134)(50 116 67 133)(51 115 68 132)
G:=sub<Sym(136)| (1,63)(2,64)(3,65)(4,66)(5,67)(6,68)(7,35)(8,36)(9,37)(10,38)(11,39)(12,40)(13,41)(14,42)(15,43)(16,44)(17,45)(18,46)(19,47)(20,48)(21,49)(22,50)(23,51)(24,52)(25,53)(26,54)(27,55)(28,56)(29,57)(30,58)(31,59)(32,60)(33,61)(34,62)(69,120)(70,121)(71,122)(72,123)(73,124)(74,125)(75,126)(76,127)(77,128)(78,129)(79,130)(80,131)(81,132)(82,133)(83,134)(84,135)(85,136)(86,103)(87,104)(88,105)(89,106)(90,107)(91,108)(92,109)(93,110)(94,111)(95,112)(96,113)(97,114)(98,115)(99,116)(100,117)(101,118)(102,119), (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), (1,86,18,69)(2,85,19,102)(3,84,20,101)(4,83,21,100)(5,82,22,99)(6,81,23,98)(7,80,24,97)(8,79,25,96)(9,78,26,95)(10,77,27,94)(11,76,28,93)(12,75,29,92)(13,74,30,91)(14,73,31,90)(15,72,32,89)(16,71,33,88)(17,70,34,87)(35,131,52,114)(36,130,53,113)(37,129,54,112)(38,128,55,111)(39,127,56,110)(40,126,57,109)(41,125,58,108)(42,124,59,107)(43,123,60,106)(44,122,61,105)(45,121,62,104)(46,120,63,103)(47,119,64,136)(48,118,65,135)(49,117,66,134)(50,116,67,133)(51,115,68,132)>;
G:=Group( (1,63)(2,64)(3,65)(4,66)(5,67)(6,68)(7,35)(8,36)(9,37)(10,38)(11,39)(12,40)(13,41)(14,42)(15,43)(16,44)(17,45)(18,46)(19,47)(20,48)(21,49)(22,50)(23,51)(24,52)(25,53)(26,54)(27,55)(28,56)(29,57)(30,58)(31,59)(32,60)(33,61)(34,62)(69,120)(70,121)(71,122)(72,123)(73,124)(74,125)(75,126)(76,127)(77,128)(78,129)(79,130)(80,131)(81,132)(82,133)(83,134)(84,135)(85,136)(86,103)(87,104)(88,105)(89,106)(90,107)(91,108)(92,109)(93,110)(94,111)(95,112)(96,113)(97,114)(98,115)(99,116)(100,117)(101,118)(102,119), (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), (1,86,18,69)(2,85,19,102)(3,84,20,101)(4,83,21,100)(5,82,22,99)(6,81,23,98)(7,80,24,97)(8,79,25,96)(9,78,26,95)(10,77,27,94)(11,76,28,93)(12,75,29,92)(13,74,30,91)(14,73,31,90)(15,72,32,89)(16,71,33,88)(17,70,34,87)(35,131,52,114)(36,130,53,113)(37,129,54,112)(38,128,55,111)(39,127,56,110)(40,126,57,109)(41,125,58,108)(42,124,59,107)(43,123,60,106)(44,122,61,105)(45,121,62,104)(46,120,63,103)(47,119,64,136)(48,118,65,135)(49,117,66,134)(50,116,67,133)(51,115,68,132) );
G=PermutationGroup([[(1,63),(2,64),(3,65),(4,66),(5,67),(6,68),(7,35),(8,36),(9,37),(10,38),(11,39),(12,40),(13,41),(14,42),(15,43),(16,44),(17,45),(18,46),(19,47),(20,48),(21,49),(22,50),(23,51),(24,52),(25,53),(26,54),(27,55),(28,56),(29,57),(30,58),(31,59),(32,60),(33,61),(34,62),(69,120),(70,121),(71,122),(72,123),(73,124),(74,125),(75,126),(76,127),(77,128),(78,129),(79,130),(80,131),(81,132),(82,133),(83,134),(84,135),(85,136),(86,103),(87,104),(88,105),(89,106),(90,107),(91,108),(92,109),(93,110),(94,111),(95,112),(96,113),(97,114),(98,115),(99,116),(100,117),(101,118),(102,119)], [(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)], [(1,86,18,69),(2,85,19,102),(3,84,20,101),(4,83,21,100),(5,82,22,99),(6,81,23,98),(7,80,24,97),(8,79,25,96),(9,78,26,95),(10,77,27,94),(11,76,28,93),(12,75,29,92),(13,74,30,91),(14,73,31,90),(15,72,32,89),(16,71,33,88),(17,70,34,87),(35,131,52,114),(36,130,53,113),(37,129,54,112),(38,128,55,111),(39,127,56,110),(40,126,57,109),(41,125,58,108),(42,124,59,107),(43,123,60,106),(44,122,61,105),(45,121,62,104),(46,120,63,103),(47,119,64,136),(48,118,65,135),(49,117,66,134),(50,116,67,133),(51,115,68,132)]])
C2×Dic17 is a maximal subgroup of
C34.D4 C68⋊3C4 D34⋊C4 C23.D17 C17⋊M4(2) C2×C4×D17 D4⋊2D17
C2×Dic17 is a maximal quotient of C68.4C4 C68⋊3C4 C23.D17
40 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 17A | ··· | 17H | 34A | ··· | 34X |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 17 | ··· | 17 | 34 | ··· | 34 |
| size | 1 | 1 | 1 | 1 | 17 | 17 | 17 | 17 | 2 | ··· | 2 | 2 | ··· | 2 |
40 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 |
| type | + | + | + | + | - | + | |
| image | C1 | C2 | C2 | C4 | D17 | Dic17 | D34 |
| kernel | C2×Dic17 | Dic17 | C2×C34 | C34 | C22 | C2 | C2 |
| # reps | 1 | 2 | 1 | 4 | 8 | 16 | 8 |
Matrix representation of C2×Dic17 ►in GL3(𝔽137) generated by
| 1 | 0 | 0 |
| 0 | 136 | 0 |
| 0 | 0 | 136 |
| 136 | 0 | 0 |
| 0 | 1 | 136 |
| 0 | 126 | 12 |
| 37 | 0 | 0 |
| 0 | 19 | 67 |
| 0 | 13 | 118 |
G:=sub<GL(3,GF(137))| [1,0,0,0,136,0,0,0,136],[136,0,0,0,1,126,0,136,12],[37,0,0,0,19,13,0,67,118] >;
C2×Dic17 in GAP, Magma, Sage, TeX
C_2\times {\rm Dic}_{17} % in TeX
G:=Group("C2xDic17"); // GroupNames label
G:=SmallGroup(136,7);
// by ID
G=gap.SmallGroup(136,7);
# by ID
G:=PCGroup([4,-2,-2,-2,-17,16,2051]);
// Polycyclic
G:=Group<a,b,c|a^2=b^34=1,c^2=b^17,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
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