direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×C17⋊D4, C34⋊2D4, C23⋊D17, C22⋊2D34, D34⋊3C22, C34.10C23, Dic17⋊2C22, C17⋊3(C2×D4), (C2×C34)⋊3C22, (C22×C34)⋊2C2, (C2×Dic17)⋊4C2, (C22×D17)⋊3C2, C2.10(C22×D17), SmallGroup(272,45)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×C17⋊D4
G = < a,b,c,d | a2=b17=c4=d2=1, ab=ba, ac=ca, ad=da, cbc-1=dbd=b-1, dcd=c-1 >
Subgroups: 374 in 54 conjugacy classes, 27 normal (11 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C2×C4, D4, C23, C23, C2×D4, C17, D17, C34, C34, C34, Dic17, D34, D34, C2×C34, C2×C34, C2×C34, C2×Dic17, C17⋊D4, C22×D17, C22×C34, C2×C17⋊D4
Quotients: C1, C2, C22, D4, C23, C2×D4, D17, D34, C17⋊D4, C22×D17, C2×C17⋊D4
►On 136 points
Generators in S136
(1 86)(2 87)(3 88)(4 89)(5 90)(6 91)(7 92)(8 93)(9 94)(10 95)(11 96)(12 97)(13 98)(14 99)(15 100)(16 101)(17 102)(18 69)(19 70)(20 71)(21 72)(22 73)(23 74)(24 75)(25 76)(26 77)(27 78)(28 79)(29 80)(30 81)(31 82)(32 83)(33 84)(34 85)(35 120)(36 121)(37 122)(38 123)(39 124)(40 125)(41 126)(42 127)(43 128)(44 129)(45 130)(46 131)(47 132)(48 133)(49 134)(50 135)(51 136)(52 103)(53 104)(54 105)(55 106)(56 107)(57 108)(58 109)(59 110)(60 111)(61 112)(62 113)(63 114)(64 115)(65 116)(66 117)(67 118)(68 119)
(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 35 18 52)(2 51 19 68)(3 50 20 67)(4 49 21 66)(5 48 22 65)(6 47 23 64)(7 46 24 63)(8 45 25 62)(9 44 26 61)(10 43 27 60)(11 42 28 59)(12 41 29 58)(13 40 30 57)(14 39 31 56)(15 38 32 55)(16 37 33 54)(17 36 34 53)(69 103 86 120)(70 119 87 136)(71 118 88 135)(72 117 89 134)(73 116 90 133)(74 115 91 132)(75 114 92 131)(76 113 93 130)(77 112 94 129)(78 111 95 128)(79 110 96 127)(80 109 97 126)(81 108 98 125)(82 107 99 124)(83 106 100 123)(84 105 101 122)(85 104 102 121)
(2 17)(3 16)(4 15)(5 14)(6 13)(7 12)(8 11)(9 10)(19 34)(20 33)(21 32)(22 31)(23 30)(24 29)(25 28)(26 27)(35 52)(36 68)(37 67)(38 66)(39 65)(40 64)(41 63)(42 62)(43 61)(44 60)(45 59)(46 58)(47 57)(48 56)(49 55)(50 54)(51 53)(70 85)(71 84)(72 83)(73 82)(74 81)(75 80)(76 79)(77 78)(87 102)(88 101)(89 100)(90 99)(91 98)(92 97)(93 96)(94 95)(103 120)(104 136)(105 135)(106 134)(107 133)(108 132)(109 131)(110 130)(111 129)(112 128)(113 127)(114 126)(115 125)(116 124)(117 123)(118 122)(119 121)
G:=sub<Sym(136)| (1,86)(2,87)(3,88)(4,89)(5,90)(6,91)(7,92)(8,93)(9,94)(10,95)(11,96)(12,97)(13,98)(14,99)(15,100)(16,101)(17,102)(18,69)(19,70)(20,71)(21,72)(22,73)(23,74)(24,75)(25,76)(26,77)(27,78)(28,79)(29,80)(30,81)(31,82)(32,83)(33,84)(34,85)(35,120)(36,121)(37,122)(38,123)(39,124)(40,125)(41,126)(42,127)(43,128)(44,129)(45,130)(46,131)(47,132)(48,133)(49,134)(50,135)(51,136)(52,103)(53,104)(54,105)(55,106)(56,107)(57,108)(58,109)(59,110)(60,111)(61,112)(62,113)(63,114)(64,115)(65,116)(66,117)(67,118)(68,119), (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), (1,35,18,52)(2,51,19,68)(3,50,20,67)(4,49,21,66)(5,48,22,65)(6,47,23,64)(7,46,24,63)(8,45,25,62)(9,44,26,61)(10,43,27,60)(11,42,28,59)(12,41,29,58)(13,40,30,57)(14,39,31,56)(15,38,32,55)(16,37,33,54)(17,36,34,53)(69,103,86,120)(70,119,87,136)(71,118,88,135)(72,117,89,134)(73,116,90,133)(74,115,91,132)(75,114,92,131)(76,113,93,130)(77,112,94,129)(78,111,95,128)(79,110,96,127)(80,109,97,126)(81,108,98,125)(82,107,99,124)(83,106,100,123)(84,105,101,122)(85,104,102,121), (2,17)(3,16)(4,15)(5,14)(6,13)(7,12)(8,11)(9,10)(19,34)(20,33)(21,32)(22,31)(23,30)(24,29)(25,28)(26,27)(35,52)(36,68)(37,67)(38,66)(39,65)(40,64)(41,63)(42,62)(43,61)(44,60)(45,59)(46,58)(47,57)(48,56)(49,55)(50,54)(51,53)(70,85)(71,84)(72,83)(73,82)(74,81)(75,80)(76,79)(77,78)(87,102)(88,101)(89,100)(90,99)(91,98)(92,97)(93,96)(94,95)(103,120)(104,136)(105,135)(106,134)(107,133)(108,132)(109,131)(110,130)(111,129)(112,128)(113,127)(114,126)(115,125)(116,124)(117,123)(118,122)(119,121)>;
G:=Group( (1,86)(2,87)(3,88)(4,89)(5,90)(6,91)(7,92)(8,93)(9,94)(10,95)(11,96)(12,97)(13,98)(14,99)(15,100)(16,101)(17,102)(18,69)(19,70)(20,71)(21,72)(22,73)(23,74)(24,75)(25,76)(26,77)(27,78)(28,79)(29,80)(30,81)(31,82)(32,83)(33,84)(34,85)(35,120)(36,121)(37,122)(38,123)(39,124)(40,125)(41,126)(42,127)(43,128)(44,129)(45,130)(46,131)(47,132)(48,133)(49,134)(50,135)(51,136)(52,103)(53,104)(54,105)(55,106)(56,107)(57,108)(58,109)(59,110)(60,111)(61,112)(62,113)(63,114)(64,115)(65,116)(66,117)(67,118)(68,119), (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), (1,35,18,52)(2,51,19,68)(3,50,20,67)(4,49,21,66)(5,48,22,65)(6,47,23,64)(7,46,24,63)(8,45,25,62)(9,44,26,61)(10,43,27,60)(11,42,28,59)(12,41,29,58)(13,40,30,57)(14,39,31,56)(15,38,32,55)(16,37,33,54)(17,36,34,53)(69,103,86,120)(70,119,87,136)(71,118,88,135)(72,117,89,134)(73,116,90,133)(74,115,91,132)(75,114,92,131)(76,113,93,130)(77,112,94,129)(78,111,95,128)(79,110,96,127)(80,109,97,126)(81,108,98,125)(82,107,99,124)(83,106,100,123)(84,105,101,122)(85,104,102,121), (2,17)(3,16)(4,15)(5,14)(6,13)(7,12)(8,11)(9,10)(19,34)(20,33)(21,32)(22,31)(23,30)(24,29)(25,28)(26,27)(35,52)(36,68)(37,67)(38,66)(39,65)(40,64)(41,63)(42,62)(43,61)(44,60)(45,59)(46,58)(47,57)(48,56)(49,55)(50,54)(51,53)(70,85)(71,84)(72,83)(73,82)(74,81)(75,80)(76,79)(77,78)(87,102)(88,101)(89,100)(90,99)(91,98)(92,97)(93,96)(94,95)(103,120)(104,136)(105,135)(106,134)(107,133)(108,132)(109,131)(110,130)(111,129)(112,128)(113,127)(114,126)(115,125)(116,124)(117,123)(118,122)(119,121) );
G=PermutationGroup([[(1,86),(2,87),(3,88),(4,89),(5,90),(6,91),(7,92),(8,93),(9,94),(10,95),(11,96),(12,97),(13,98),(14,99),(15,100),(16,101),(17,102),(18,69),(19,70),(20,71),(21,72),(22,73),(23,74),(24,75),(25,76),(26,77),(27,78),(28,79),(29,80),(30,81),(31,82),(32,83),(33,84),(34,85),(35,120),(36,121),(37,122),(38,123),(39,124),(40,125),(41,126),(42,127),(43,128),(44,129),(45,130),(46,131),(47,132),(48,133),(49,134),(50,135),(51,136),(52,103),(53,104),(54,105),(55,106),(56,107),(57,108),(58,109),(59,110),(60,111),(61,112),(62,113),(63,114),(64,115),(65,116),(66,117),(67,118),(68,119)], [(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)], [(1,35,18,52),(2,51,19,68),(3,50,20,67),(4,49,21,66),(5,48,22,65),(6,47,23,64),(7,46,24,63),(8,45,25,62),(9,44,26,61),(10,43,27,60),(11,42,28,59),(12,41,29,58),(13,40,30,57),(14,39,31,56),(15,38,32,55),(16,37,33,54),(17,36,34,53),(69,103,86,120),(70,119,87,136),(71,118,88,135),(72,117,89,134),(73,116,90,133),(74,115,91,132),(75,114,92,131),(76,113,93,130),(77,112,94,129),(78,111,95,128),(79,110,96,127),(80,109,97,126),(81,108,98,125),(82,107,99,124),(83,106,100,123),(84,105,101,122),(85,104,102,121)], [(2,17),(3,16),(4,15),(5,14),(6,13),(7,12),(8,11),(9,10),(19,34),(20,33),(21,32),(22,31),(23,30),(24,29),(25,28),(26,27),(35,52),(36,68),(37,67),(38,66),(39,65),(40,64),(41,63),(42,62),(43,61),(44,60),(45,59),(46,58),(47,57),(48,56),(49,55),(50,54),(51,53),(70,85),(71,84),(72,83),(73,82),(74,81),(75,80),(76,79),(77,78),(87,102),(88,101),(89,100),(90,99),(91,98),(92,97),(93,96),(94,95),(103,120),(104,136),(105,135),(106,134),(107,133),(108,132),(109,131),(110,130),(111,129),(112,128),(113,127),(114,126),(115,125),(116,124),(117,123),(118,122),(119,121)]])
74 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 17A | ··· | 17H | 34A | ··· | 34BD |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 17 | ··· | 17 | 34 | ··· | 34 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 34 | 34 | 34 | 34 | 2 | ··· | 2 | 2 | ··· | 2 |
74 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | |
| image | C1 | C2 | C2 | C2 | C2 | D4 | D17 | D34 | C17⋊D4 |
| kernel | C2×C17⋊D4 | C2×Dic17 | C17⋊D4 | C22×D17 | C22×C34 | C34 | C23 | C22 | C2 |
| # reps | 1 | 1 | 4 | 1 | 1 | 2 | 8 | 24 | 32 |
Matrix representation of C2×C17⋊D4 ►in GL3(𝔽137) generated by
| 136 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 1 |
| 1 | 0 | 0 |
| 0 | 131 | 136 |
| 0 | 1 | 0 |
| 136 | 0 | 0 |
| 0 | 11 | 95 |
| 0 | 29 | 126 |
| 136 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 131 | 136 |
G:=sub<GL(3,GF(137))| [136,0,0,0,1,0,0,0,1],[1,0,0,0,131,1,0,136,0],[136,0,0,0,11,29,0,95,126],[136,0,0,0,1,131,0,0,136] >;
C2×C17⋊D4 in GAP, Magma, Sage, TeX
C_2\times C_{17}\rtimes D_4 % in TeX
G:=Group("C2xC17:D4"); // GroupNames label
G:=SmallGroup(272,45);
// by ID
G=gap.SmallGroup(272,45);
# by ID
G:=PCGroup([5,-2,-2,-2,-2,-17,182,6404]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^17=c^4=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations