direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C4○D4×C17, D4⋊2C34, Q8⋊2C34, C34.13C23, C68.21C22, (C2×C4)⋊3C34, (C2×C68)⋊7C2, (D4×C17)⋊5C2, C4.5(C2×C34), (Q8×C17)⋊5C2, C22.(C2×C34), (C2×C34).2C22, C2.3(C22×C34), SmallGroup(272,49)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C4○D4×C17
G = < a,b,c,d | a17=b4=d2=1, c2=b2, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=b2c >
Smallest permutation representation of C4○D4×C17
►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 58 33 135)(2 59 34 136)(3 60 18 120)(4 61 19 121)(5 62 20 122)(6 63 21 123)(7 64 22 124)(8 65 23 125)(9 66 24 126)(10 67 25 127)(11 68 26 128)(12 52 27 129)(13 53 28 130)(14 54 29 131)(15 55 30 132)(16 56 31 133)(17 57 32 134)(35 111 102 77)(36 112 86 78)(37 113 87 79)(38 114 88 80)(39 115 89 81)(40 116 90 82)(41 117 91 83)(42 118 92 84)(43 119 93 85)(44 103 94 69)(45 104 95 70)(46 105 96 71)(47 106 97 72)(48 107 98 73)(49 108 99 74)(50 109 100 75)(51 110 101 76)
(1 135 33 58)(2 136 34 59)(3 120 18 60)(4 121 19 61)(5 122 20 62)(6 123 21 63)(7 124 22 64)(8 125 23 65)(9 126 24 66)(10 127 25 67)(11 128 26 68)(12 129 27 52)(13 130 28 53)(14 131 29 54)(15 132 30 55)(16 133 31 56)(17 134 32 57)(35 111 102 77)(36 112 86 78)(37 113 87 79)(38 114 88 80)(39 115 89 81)(40 116 90 82)(41 117 91 83)(42 118 92 84)(43 119 93 85)(44 103 94 69)(45 104 95 70)(46 105 96 71)(47 106 97 72)(48 107 98 73)(49 108 99 74)(50 109 100 75)(51 110 101 76)
(1 103)(2 104)(3 105)(4 106)(5 107)(6 108)(7 109)(8 110)(9 111)(10 112)(11 113)(12 114)(13 115)(14 116)(15 117)(16 118)(17 119)(18 71)(19 72)(20 73)(21 74)(22 75)(23 76)(24 77)(25 78)(26 79)(27 80)(28 81)(29 82)(30 83)(31 84)(32 85)(33 69)(34 70)(35 126)(36 127)(37 128)(38 129)(39 130)(40 131)(41 132)(42 133)(43 134)(44 135)(45 136)(46 120)(47 121)(48 122)(49 123)(50 124)(51 125)(52 88)(53 89)(54 90)(55 91)(56 92)(57 93)(58 94)(59 95)(60 96)(61 97)(62 98)(63 99)(64 100)(65 101)(66 102)(67 86)(68 87)
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), (1,58,33,135)(2,59,34,136)(3,60,18,120)(4,61,19,121)(5,62,20,122)(6,63,21,123)(7,64,22,124)(8,65,23,125)(9,66,24,126)(10,67,25,127)(11,68,26,128)(12,52,27,129)(13,53,28,130)(14,54,29,131)(15,55,30,132)(16,56,31,133)(17,57,32,134)(35,111,102,77)(36,112,86,78)(37,113,87,79)(38,114,88,80)(39,115,89,81)(40,116,90,82)(41,117,91,83)(42,118,92,84)(43,119,93,85)(44,103,94,69)(45,104,95,70)(46,105,96,71)(47,106,97,72)(48,107,98,73)(49,108,99,74)(50,109,100,75)(51,110,101,76), (1,135,33,58)(2,136,34,59)(3,120,18,60)(4,121,19,61)(5,122,20,62)(6,123,21,63)(7,124,22,64)(8,125,23,65)(9,126,24,66)(10,127,25,67)(11,128,26,68)(12,129,27,52)(13,130,28,53)(14,131,29,54)(15,132,30,55)(16,133,31,56)(17,134,32,57)(35,111,102,77)(36,112,86,78)(37,113,87,79)(38,114,88,80)(39,115,89,81)(40,116,90,82)(41,117,91,83)(42,118,92,84)(43,119,93,85)(44,103,94,69)(45,104,95,70)(46,105,96,71)(47,106,97,72)(48,107,98,73)(49,108,99,74)(50,109,100,75)(51,110,101,76), (1,103)(2,104)(3,105)(4,106)(5,107)(6,108)(7,109)(8,110)(9,111)(10,112)(11,113)(12,114)(13,115)(14,116)(15,117)(16,118)(17,119)(18,71)(19,72)(20,73)(21,74)(22,75)(23,76)(24,77)(25,78)(26,79)(27,80)(28,81)(29,82)(30,83)(31,84)(32,85)(33,69)(34,70)(35,126)(36,127)(37,128)(38,129)(39,130)(40,131)(41,132)(42,133)(43,134)(44,135)(45,136)(46,120)(47,121)(48,122)(49,123)(50,124)(51,125)(52,88)(53,89)(54,90)(55,91)(56,92)(57,93)(58,94)(59,95)(60,96)(61,97)(62,98)(63,99)(64,100)(65,101)(66,102)(67,86)(68,87)>;
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), (1,58,33,135)(2,59,34,136)(3,60,18,120)(4,61,19,121)(5,62,20,122)(6,63,21,123)(7,64,22,124)(8,65,23,125)(9,66,24,126)(10,67,25,127)(11,68,26,128)(12,52,27,129)(13,53,28,130)(14,54,29,131)(15,55,30,132)(16,56,31,133)(17,57,32,134)(35,111,102,77)(36,112,86,78)(37,113,87,79)(38,114,88,80)(39,115,89,81)(40,116,90,82)(41,117,91,83)(42,118,92,84)(43,119,93,85)(44,103,94,69)(45,104,95,70)(46,105,96,71)(47,106,97,72)(48,107,98,73)(49,108,99,74)(50,109,100,75)(51,110,101,76), (1,135,33,58)(2,136,34,59)(3,120,18,60)(4,121,19,61)(5,122,20,62)(6,123,21,63)(7,124,22,64)(8,125,23,65)(9,126,24,66)(10,127,25,67)(11,128,26,68)(12,129,27,52)(13,130,28,53)(14,131,29,54)(15,132,30,55)(16,133,31,56)(17,134,32,57)(35,111,102,77)(36,112,86,78)(37,113,87,79)(38,114,88,80)(39,115,89,81)(40,116,90,82)(41,117,91,83)(42,118,92,84)(43,119,93,85)(44,103,94,69)(45,104,95,70)(46,105,96,71)(47,106,97,72)(48,107,98,73)(49,108,99,74)(50,109,100,75)(51,110,101,76), (1,103)(2,104)(3,105)(4,106)(5,107)(6,108)(7,109)(8,110)(9,111)(10,112)(11,113)(12,114)(13,115)(14,116)(15,117)(16,118)(17,119)(18,71)(19,72)(20,73)(21,74)(22,75)(23,76)(24,77)(25,78)(26,79)(27,80)(28,81)(29,82)(30,83)(31,84)(32,85)(33,69)(34,70)(35,126)(36,127)(37,128)(38,129)(39,130)(40,131)(41,132)(42,133)(43,134)(44,135)(45,136)(46,120)(47,121)(48,122)(49,123)(50,124)(51,125)(52,88)(53,89)(54,90)(55,91)(56,92)(57,93)(58,94)(59,95)(60,96)(61,97)(62,98)(63,99)(64,100)(65,101)(66,102)(67,86)(68,87) );
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)], [(1,58,33,135),(2,59,34,136),(3,60,18,120),(4,61,19,121),(5,62,20,122),(6,63,21,123),(7,64,22,124),(8,65,23,125),(9,66,24,126),(10,67,25,127),(11,68,26,128),(12,52,27,129),(13,53,28,130),(14,54,29,131),(15,55,30,132),(16,56,31,133),(17,57,32,134),(35,111,102,77),(36,112,86,78),(37,113,87,79),(38,114,88,80),(39,115,89,81),(40,116,90,82),(41,117,91,83),(42,118,92,84),(43,119,93,85),(44,103,94,69),(45,104,95,70),(46,105,96,71),(47,106,97,72),(48,107,98,73),(49,108,99,74),(50,109,100,75),(51,110,101,76)], [(1,135,33,58),(2,136,34,59),(3,120,18,60),(4,121,19,61),(5,122,20,62),(6,123,21,63),(7,124,22,64),(8,125,23,65),(9,126,24,66),(10,127,25,67),(11,128,26,68),(12,129,27,52),(13,130,28,53),(14,131,29,54),(15,132,30,55),(16,133,31,56),(17,134,32,57),(35,111,102,77),(36,112,86,78),(37,113,87,79),(38,114,88,80),(39,115,89,81),(40,116,90,82),(41,117,91,83),(42,118,92,84),(43,119,93,85),(44,103,94,69),(45,104,95,70),(46,105,96,71),(47,106,97,72),(48,107,98,73),(49,108,99,74),(50,109,100,75),(51,110,101,76)], [(1,103),(2,104),(3,105),(4,106),(5,107),(6,108),(7,109),(8,110),(9,111),(10,112),(11,113),(12,114),(13,115),(14,116),(15,117),(16,118),(17,119),(18,71),(19,72),(20,73),(21,74),(22,75),(23,76),(24,77),(25,78),(26,79),(27,80),(28,81),(29,82),(30,83),(31,84),(32,85),(33,69),(34,70),(35,126),(36,127),(37,128),(38,129),(39,130),(40,131),(41,132),(42,133),(43,134),(44,135),(45,136),(46,120),(47,121),(48,122),(49,123),(50,124),(51,125),(52,88),(53,89),(54,90),(55,91),(56,92),(57,93),(58,94),(59,95),(60,96),(61,97),(62,98),(63,99),(64,100),(65,101),(66,102),(67,86),(68,87)]])
170 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | 4D | 4E | 17A | ··· | 17P | 34A | ··· | 34P | 34Q | ··· | 34BL | 68A | ··· | 68AF | 68AG | ··· | 68CB |
| order | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 17 | ··· | 17 | 34 | ··· | 34 | 34 | ··· | 34 | 68 | ··· | 68 | 68 | ··· | 68 |
| size | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 2 | 2 | 2 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 |
170 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | + | ||||||
| image | C1 | C2 | C2 | C2 | C17 | C34 | C34 | C34 | C4○D4 | C4○D4×C17 |
| kernel | C4○D4×C17 | C2×C68 | D4×C17 | Q8×C17 | C4○D4 | C2×C4 | D4 | Q8 | C17 | C1 |
| # reps | 1 | 3 | 3 | 1 | 16 | 48 | 48 | 16 | 2 | 32 |
Matrix representation of C4○D4×C17 ►in GL2(𝔽137) generated by
| 123 | 0 |
| 0 | 123 |
| 100 | 0 |
| 0 | 100 |
| 37 | 46 |
| 0 | 100 |
| 46 | 123 |
| 63 | 91 |
G:=sub<GL(2,GF(137))| [123,0,0,123],[100,0,0,100],[37,0,46,100],[46,63,123,91] >;
C4○D4×C17 in GAP, Magma, Sage, TeX
C_4\circ D_4\times C_{17} % in TeX
G:=Group("C4oD4xC17"); // GroupNames label
G:=SmallGroup(272,49);
// by ID
G=gap.SmallGroup(272,49);
# by ID
G:=PCGroup([5,-2,-2,-2,-17,-2,1381,522]);
// Polycyclic
G:=Group<a,b,c,d|a^17=b^4=d^2=1,c^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=b^2*c>;
// generators/relations
Export