direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C22⋊C4×C17, C22⋊C68, C23.C34, C34.12D4, (C2×C34)⋊3C4, (C2×C4)⋊1C34, (C2×C68)⋊2C2, C2.1(C2×C68), C2.1(D4×C17), C34.17(C2×C4), C22.2(C2×C34), (C22×C34).1C2, (C2×C34).13C22, SmallGroup(272,21)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C22⋊C4×C17
G = < a,b,c,d | a17=b2=c2=d4=1, ab=ba, ac=ca, ad=da, dbd-1=bc=cb, cd=dc >
Smallest permutation representation of C22⋊C4×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)
(1 133)(2 134)(3 135)(4 136)(5 120)(6 121)(7 122)(8 123)(9 124)(10 125)(11 126)(12 127)(13 128)(14 129)(15 130)(16 131)(17 132)(18 110)(19 111)(20 112)(21 113)(22 114)(23 115)(24 116)(25 117)(26 118)(27 119)(28 103)(29 104)(30 105)(31 106)(32 107)(33 108)(34 109)(35 68)(36 52)(37 53)(38 54)(39 55)(40 56)(41 57)(42 58)(43 59)(44 60)(45 61)(46 62)(47 63)(48 64)(49 65)(50 66)(51 67)(69 100)(70 101)(71 102)(72 86)(73 87)(74 88)(75 89)(76 90)(77 91)(78 92)(79 93)(80 94)(81 95)(82 96)(83 97)(84 98)(85 99)
(1 73)(2 74)(3 75)(4 76)(5 77)(6 78)(7 79)(8 80)(9 81)(10 82)(11 83)(12 84)(13 85)(14 69)(15 70)(16 71)(17 72)(18 58)(19 59)(20 60)(21 61)(22 62)(23 63)(24 64)(25 65)(26 66)(27 67)(28 68)(29 52)(30 53)(31 54)(32 55)(33 56)(34 57)(35 103)(36 104)(37 105)(38 106)(39 107)(40 108)(41 109)(42 110)(43 111)(44 112)(45 113)(46 114)(47 115)(48 116)(49 117)(50 118)(51 119)(86 132)(87 133)(88 134)(89 135)(90 136)(91 120)(92 121)(93 122)(94 123)(95 124)(96 125)(97 126)(98 127)(99 128)(100 129)(101 130)(102 131)
(1 57 133 109)(2 58 134 110)(3 59 135 111)(4 60 136 112)(5 61 120 113)(6 62 121 114)(7 63 122 115)(8 64 123 116)(9 65 124 117)(10 66 125 118)(11 67 126 119)(12 68 127 103)(13 52 128 104)(14 53 129 105)(15 54 130 106)(16 55 131 107)(17 56 132 108)(18 88 42 74)(19 89 43 75)(20 90 44 76)(21 91 45 77)(22 92 46 78)(23 93 47 79)(24 94 48 80)(25 95 49 81)(26 96 50 82)(27 97 51 83)(28 98 35 84)(29 99 36 85)(30 100 37 69)(31 101 38 70)(32 102 39 71)(33 86 40 72)(34 87 41 73)
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), (1,133)(2,134)(3,135)(4,136)(5,120)(6,121)(7,122)(8,123)(9,124)(10,125)(11,126)(12,127)(13,128)(14,129)(15,130)(16,131)(17,132)(18,110)(19,111)(20,112)(21,113)(22,114)(23,115)(24,116)(25,117)(26,118)(27,119)(28,103)(29,104)(30,105)(31,106)(32,107)(33,108)(34,109)(35,68)(36,52)(37,53)(38,54)(39,55)(40,56)(41,57)(42,58)(43,59)(44,60)(45,61)(46,62)(47,63)(48,64)(49,65)(50,66)(51,67)(69,100)(70,101)(71,102)(72,86)(73,87)(74,88)(75,89)(76,90)(77,91)(78,92)(79,93)(80,94)(81,95)(82,96)(83,97)(84,98)(85,99), (1,73)(2,74)(3,75)(4,76)(5,77)(6,78)(7,79)(8,80)(9,81)(10,82)(11,83)(12,84)(13,85)(14,69)(15,70)(16,71)(17,72)(18,58)(19,59)(20,60)(21,61)(22,62)(23,63)(24,64)(25,65)(26,66)(27,67)(28,68)(29,52)(30,53)(31,54)(32,55)(33,56)(34,57)(35,103)(36,104)(37,105)(38,106)(39,107)(40,108)(41,109)(42,110)(43,111)(44,112)(45,113)(46,114)(47,115)(48,116)(49,117)(50,118)(51,119)(86,132)(87,133)(88,134)(89,135)(90,136)(91,120)(92,121)(93,122)(94,123)(95,124)(96,125)(97,126)(98,127)(99,128)(100,129)(101,130)(102,131), (1,57,133,109)(2,58,134,110)(3,59,135,111)(4,60,136,112)(5,61,120,113)(6,62,121,114)(7,63,122,115)(8,64,123,116)(9,65,124,117)(10,66,125,118)(11,67,126,119)(12,68,127,103)(13,52,128,104)(14,53,129,105)(15,54,130,106)(16,55,131,107)(17,56,132,108)(18,88,42,74)(19,89,43,75)(20,90,44,76)(21,91,45,77)(22,92,46,78)(23,93,47,79)(24,94,48,80)(25,95,49,81)(26,96,50,82)(27,97,51,83)(28,98,35,84)(29,99,36,85)(30,100,37,69)(31,101,38,70)(32,102,39,71)(33,86,40,72)(34,87,41,73)>;
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), (1,133)(2,134)(3,135)(4,136)(5,120)(6,121)(7,122)(8,123)(9,124)(10,125)(11,126)(12,127)(13,128)(14,129)(15,130)(16,131)(17,132)(18,110)(19,111)(20,112)(21,113)(22,114)(23,115)(24,116)(25,117)(26,118)(27,119)(28,103)(29,104)(30,105)(31,106)(32,107)(33,108)(34,109)(35,68)(36,52)(37,53)(38,54)(39,55)(40,56)(41,57)(42,58)(43,59)(44,60)(45,61)(46,62)(47,63)(48,64)(49,65)(50,66)(51,67)(69,100)(70,101)(71,102)(72,86)(73,87)(74,88)(75,89)(76,90)(77,91)(78,92)(79,93)(80,94)(81,95)(82,96)(83,97)(84,98)(85,99), (1,73)(2,74)(3,75)(4,76)(5,77)(6,78)(7,79)(8,80)(9,81)(10,82)(11,83)(12,84)(13,85)(14,69)(15,70)(16,71)(17,72)(18,58)(19,59)(20,60)(21,61)(22,62)(23,63)(24,64)(25,65)(26,66)(27,67)(28,68)(29,52)(30,53)(31,54)(32,55)(33,56)(34,57)(35,103)(36,104)(37,105)(38,106)(39,107)(40,108)(41,109)(42,110)(43,111)(44,112)(45,113)(46,114)(47,115)(48,116)(49,117)(50,118)(51,119)(86,132)(87,133)(88,134)(89,135)(90,136)(91,120)(92,121)(93,122)(94,123)(95,124)(96,125)(97,126)(98,127)(99,128)(100,129)(101,130)(102,131), (1,57,133,109)(2,58,134,110)(3,59,135,111)(4,60,136,112)(5,61,120,113)(6,62,121,114)(7,63,122,115)(8,64,123,116)(9,65,124,117)(10,66,125,118)(11,67,126,119)(12,68,127,103)(13,52,128,104)(14,53,129,105)(15,54,130,106)(16,55,131,107)(17,56,132,108)(18,88,42,74)(19,89,43,75)(20,90,44,76)(21,91,45,77)(22,92,46,78)(23,93,47,79)(24,94,48,80)(25,95,49,81)(26,96,50,82)(27,97,51,83)(28,98,35,84)(29,99,36,85)(30,100,37,69)(31,101,38,70)(32,102,39,71)(33,86,40,72)(34,87,41,73) );
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)], [(1,133),(2,134),(3,135),(4,136),(5,120),(6,121),(7,122),(8,123),(9,124),(10,125),(11,126),(12,127),(13,128),(14,129),(15,130),(16,131),(17,132),(18,110),(19,111),(20,112),(21,113),(22,114),(23,115),(24,116),(25,117),(26,118),(27,119),(28,103),(29,104),(30,105),(31,106),(32,107),(33,108),(34,109),(35,68),(36,52),(37,53),(38,54),(39,55),(40,56),(41,57),(42,58),(43,59),(44,60),(45,61),(46,62),(47,63),(48,64),(49,65),(50,66),(51,67),(69,100),(70,101),(71,102),(72,86),(73,87),(74,88),(75,89),(76,90),(77,91),(78,92),(79,93),(80,94),(81,95),(82,96),(83,97),(84,98),(85,99)], [(1,73),(2,74),(3,75),(4,76),(5,77),(6,78),(7,79),(8,80),(9,81),(10,82),(11,83),(12,84),(13,85),(14,69),(15,70),(16,71),(17,72),(18,58),(19,59),(20,60),(21,61),(22,62),(23,63),(24,64),(25,65),(26,66),(27,67),(28,68),(29,52),(30,53),(31,54),(32,55),(33,56),(34,57),(35,103),(36,104),(37,105),(38,106),(39,107),(40,108),(41,109),(42,110),(43,111),(44,112),(45,113),(46,114),(47,115),(48,116),(49,117),(50,118),(51,119),(86,132),(87,133),(88,134),(89,135),(90,136),(91,120),(92,121),(93,122),(94,123),(95,124),(96,125),(97,126),(98,127),(99,128),(100,129),(101,130),(102,131)], [(1,57,133,109),(2,58,134,110),(3,59,135,111),(4,60,136,112),(5,61,120,113),(6,62,121,114),(7,63,122,115),(8,64,123,116),(9,65,124,117),(10,66,125,118),(11,67,126,119),(12,68,127,103),(13,52,128,104),(14,53,129,105),(15,54,130,106),(16,55,131,107),(17,56,132,108),(18,88,42,74),(19,89,43,75),(20,90,44,76),(21,91,45,77),(22,92,46,78),(23,93,47,79),(24,94,48,80),(25,95,49,81),(26,96,50,82),(27,97,51,83),(28,98,35,84),(29,99,36,85),(30,100,37,69),(31,101,38,70),(32,102,39,71),(33,86,40,72),(34,87,41,73)]])
170 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 17A | ··· | 17P | 34A | ··· | 34AV | 34AW | ··· | 34CB | 68A | ··· | 68BL |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 17 | ··· | 17 | 34 | ··· | 34 | 34 | ··· | 34 | 68 | ··· | 68 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 |
170 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | + | ||||||
| image | C1 | C2 | C2 | C4 | C17 | C34 | C34 | C68 | D4 | D4×C17 |
| kernel | C22⋊C4×C17 | C2×C68 | C22×C34 | C2×C34 | C22⋊C4 | C2×C4 | C23 | C22 | C34 | C2 |
| # reps | 1 | 2 | 1 | 4 | 16 | 32 | 16 | 64 | 2 | 32 |
Matrix representation of C22⋊C4×C17 ►in GL3(𝔽137) generated by
| 1 | 0 | 0 |
| 0 | 74 | 0 |
| 0 | 0 | 74 |
| 1 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 98 | 136 |
| 1 | 0 | 0 |
| 0 | 136 | 0 |
| 0 | 0 | 136 |
| 37 | 0 | 0 |
| 0 | 98 | 135 |
| 0 | 75 | 39 |
G:=sub<GL(3,GF(137))| [1,0,0,0,74,0,0,0,74],[1,0,0,0,1,98,0,0,136],[1,0,0,0,136,0,0,0,136],[37,0,0,0,98,75,0,135,39] >;
C22⋊C4×C17 in GAP, Magma, Sage, TeX
C_2^2\rtimes C_4\times C_{17} % in TeX
G:=Group("C2^2:C4xC17"); // GroupNames label
G:=SmallGroup(272,21);
// by ID
G=gap.SmallGroup(272,21);
# by ID
G:=PCGroup([5,-2,-2,-17,-2,-2,680,701]);
// Polycyclic
G:=Group<a,b,c,d|a^17=b^2=c^2=d^4=1,a*b=b*a,a*c=c*a,a*d=d*a,d*b*d^-1=b*c=c*b,c*d=d*c>;
// generators/relations
Export