direct product, metabelian, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C23×D17, C17⋊C24, C34⋊C23, (C22×C34)⋊3C2, (C2×C34)⋊4C22, SmallGroup(272,53)
Series: Derived ►Chief ►Lower central ►Upper central
| C17 — C23×D17 |
Generators and relations for C23×D17
G = < a,b,c,d,e | a2=b2=c2=d17=e2=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede=d-1 >
Subgroups: 950 in 134 conjugacy classes, 83 normal (5 characteristic)
C1, C2, C2, C22, C22, C23, C23, C24, C17, D17, C34, D34, C2×C34, C22×D17, C22×C34, C23×D17
Quotients: C1, C2, C22, C23, C24, D17, D34, C22×D17, C23×D17
►On 136 points
Generators in S136
(1 83)(2 84)(3 85)(4 69)(5 70)(6 71)(7 72)(8 73)(9 74)(10 75)(11 76)(12 77)(13 78)(14 79)(15 80)(16 81)(17 82)(18 93)(19 94)(20 95)(21 96)(22 97)(23 98)(24 99)(25 100)(26 101)(27 102)(28 86)(29 87)(30 88)(31 89)(32 90)(33 91)(34 92)(35 115)(36 116)(37 117)(38 118)(39 119)(40 103)(41 104)(42 105)(43 106)(44 107)(45 108)(46 109)(47 110)(48 111)(49 112)(50 113)(51 114)(52 128)(53 129)(54 130)(55 131)(56 132)(57 133)(58 134)(59 135)(60 136)(61 120)(62 121)(63 122)(64 123)(65 124)(66 125)(67 126)(68 127)
(1 64)(2 65)(3 66)(4 67)(5 68)(6 52)(7 53)(8 54)(9 55)(10 56)(11 57)(12 58)(13 59)(14 60)(15 61)(16 62)(17 63)(18 42)(19 43)(20 44)(21 45)(22 46)(23 47)(24 48)(25 49)(26 50)(27 51)(28 35)(29 36)(30 37)(31 38)(32 39)(33 40)(34 41)(69 126)(70 127)(71 128)(72 129)(73 130)(74 131)(75 132)(76 133)(77 134)(78 135)(79 136)(80 120)(81 121)(82 122)(83 123)(84 124)(85 125)(86 115)(87 116)(88 117)(89 118)(90 119)(91 103)(92 104)(93 105)(94 106)(95 107)(96 108)(97 109)(98 110)(99 111)(100 112)(101 113)(102 114)
(1 32)(2 33)(3 34)(4 18)(5 19)(6 20)(7 21)(8 22)(9 23)(10 24)(11 25)(12 26)(13 27)(14 28)(15 29)(16 30)(17 31)(35 60)(36 61)(37 62)(38 63)(39 64)(40 65)(41 66)(42 67)(43 68)(44 52)(45 53)(46 54)(47 55)(48 56)(49 57)(50 58)(51 59)(69 93)(70 94)(71 95)(72 96)(73 97)(74 98)(75 99)(76 100)(77 101)(78 102)(79 86)(80 87)(81 88)(82 89)(83 90)(84 91)(85 92)(103 124)(104 125)(105 126)(106 127)(107 128)(108 129)(109 130)(110 131)(111 132)(112 133)(113 134)(114 135)(115 136)(116 120)(117 121)(118 122)(119 123)
(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 82)(2 81)(3 80)(4 79)(5 78)(6 77)(7 76)(8 75)(9 74)(10 73)(11 72)(12 71)(13 70)(14 69)(15 85)(16 84)(17 83)(18 86)(19 102)(20 101)(21 100)(22 99)(23 98)(24 97)(25 96)(26 95)(27 94)(28 93)(29 92)(30 91)(31 90)(32 89)(33 88)(34 87)(35 105)(36 104)(37 103)(38 119)(39 118)(40 117)(41 116)(42 115)(43 114)(44 113)(45 112)(46 111)(47 110)(48 109)(49 108)(50 107)(51 106)(52 134)(53 133)(54 132)(55 131)(56 130)(57 129)(58 128)(59 127)(60 126)(61 125)(62 124)(63 123)(64 122)(65 121)(66 120)(67 136)(68 135)
G:=sub<Sym(136)| (1,83)(2,84)(3,85)(4,69)(5,70)(6,71)(7,72)(8,73)(9,74)(10,75)(11,76)(12,77)(13,78)(14,79)(15,80)(16,81)(17,82)(18,93)(19,94)(20,95)(21,96)(22,97)(23,98)(24,99)(25,100)(26,101)(27,102)(28,86)(29,87)(30,88)(31,89)(32,90)(33,91)(34,92)(35,115)(36,116)(37,117)(38,118)(39,119)(40,103)(41,104)(42,105)(43,106)(44,107)(45,108)(46,109)(47,110)(48,111)(49,112)(50,113)(51,114)(52,128)(53,129)(54,130)(55,131)(56,132)(57,133)(58,134)(59,135)(60,136)(61,120)(62,121)(63,122)(64,123)(65,124)(66,125)(67,126)(68,127), (1,64)(2,65)(3,66)(4,67)(5,68)(6,52)(7,53)(8,54)(9,55)(10,56)(11,57)(12,58)(13,59)(14,60)(15,61)(16,62)(17,63)(18,42)(19,43)(20,44)(21,45)(22,46)(23,47)(24,48)(25,49)(26,50)(27,51)(28,35)(29,36)(30,37)(31,38)(32,39)(33,40)(34,41)(69,126)(70,127)(71,128)(72,129)(73,130)(74,131)(75,132)(76,133)(77,134)(78,135)(79,136)(80,120)(81,121)(82,122)(83,123)(84,124)(85,125)(86,115)(87,116)(88,117)(89,118)(90,119)(91,103)(92,104)(93,105)(94,106)(95,107)(96,108)(97,109)(98,110)(99,111)(100,112)(101,113)(102,114), (1,32)(2,33)(3,34)(4,18)(5,19)(6,20)(7,21)(8,22)(9,23)(10,24)(11,25)(12,26)(13,27)(14,28)(15,29)(16,30)(17,31)(35,60)(36,61)(37,62)(38,63)(39,64)(40,65)(41,66)(42,67)(43,68)(44,52)(45,53)(46,54)(47,55)(48,56)(49,57)(50,58)(51,59)(69,93)(70,94)(71,95)(72,96)(73,97)(74,98)(75,99)(76,100)(77,101)(78,102)(79,86)(80,87)(81,88)(82,89)(83,90)(84,91)(85,92)(103,124)(104,125)(105,126)(106,127)(107,128)(108,129)(109,130)(110,131)(111,132)(112,133)(113,134)(114,135)(115,136)(116,120)(117,121)(118,122)(119,123), (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), (1,82)(2,81)(3,80)(4,79)(5,78)(6,77)(7,76)(8,75)(9,74)(10,73)(11,72)(12,71)(13,70)(14,69)(15,85)(16,84)(17,83)(18,86)(19,102)(20,101)(21,100)(22,99)(23,98)(24,97)(25,96)(26,95)(27,94)(28,93)(29,92)(30,91)(31,90)(32,89)(33,88)(34,87)(35,105)(36,104)(37,103)(38,119)(39,118)(40,117)(41,116)(42,115)(43,114)(44,113)(45,112)(46,111)(47,110)(48,109)(49,108)(50,107)(51,106)(52,134)(53,133)(54,132)(55,131)(56,130)(57,129)(58,128)(59,127)(60,126)(61,125)(62,124)(63,123)(64,122)(65,121)(66,120)(67,136)(68,135)>;
G:=Group( (1,83)(2,84)(3,85)(4,69)(5,70)(6,71)(7,72)(8,73)(9,74)(10,75)(11,76)(12,77)(13,78)(14,79)(15,80)(16,81)(17,82)(18,93)(19,94)(20,95)(21,96)(22,97)(23,98)(24,99)(25,100)(26,101)(27,102)(28,86)(29,87)(30,88)(31,89)(32,90)(33,91)(34,92)(35,115)(36,116)(37,117)(38,118)(39,119)(40,103)(41,104)(42,105)(43,106)(44,107)(45,108)(46,109)(47,110)(48,111)(49,112)(50,113)(51,114)(52,128)(53,129)(54,130)(55,131)(56,132)(57,133)(58,134)(59,135)(60,136)(61,120)(62,121)(63,122)(64,123)(65,124)(66,125)(67,126)(68,127), (1,64)(2,65)(3,66)(4,67)(5,68)(6,52)(7,53)(8,54)(9,55)(10,56)(11,57)(12,58)(13,59)(14,60)(15,61)(16,62)(17,63)(18,42)(19,43)(20,44)(21,45)(22,46)(23,47)(24,48)(25,49)(26,50)(27,51)(28,35)(29,36)(30,37)(31,38)(32,39)(33,40)(34,41)(69,126)(70,127)(71,128)(72,129)(73,130)(74,131)(75,132)(76,133)(77,134)(78,135)(79,136)(80,120)(81,121)(82,122)(83,123)(84,124)(85,125)(86,115)(87,116)(88,117)(89,118)(90,119)(91,103)(92,104)(93,105)(94,106)(95,107)(96,108)(97,109)(98,110)(99,111)(100,112)(101,113)(102,114), (1,32)(2,33)(3,34)(4,18)(5,19)(6,20)(7,21)(8,22)(9,23)(10,24)(11,25)(12,26)(13,27)(14,28)(15,29)(16,30)(17,31)(35,60)(36,61)(37,62)(38,63)(39,64)(40,65)(41,66)(42,67)(43,68)(44,52)(45,53)(46,54)(47,55)(48,56)(49,57)(50,58)(51,59)(69,93)(70,94)(71,95)(72,96)(73,97)(74,98)(75,99)(76,100)(77,101)(78,102)(79,86)(80,87)(81,88)(82,89)(83,90)(84,91)(85,92)(103,124)(104,125)(105,126)(106,127)(107,128)(108,129)(109,130)(110,131)(111,132)(112,133)(113,134)(114,135)(115,136)(116,120)(117,121)(118,122)(119,123), (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), (1,82)(2,81)(3,80)(4,79)(5,78)(6,77)(7,76)(8,75)(9,74)(10,73)(11,72)(12,71)(13,70)(14,69)(15,85)(16,84)(17,83)(18,86)(19,102)(20,101)(21,100)(22,99)(23,98)(24,97)(25,96)(26,95)(27,94)(28,93)(29,92)(30,91)(31,90)(32,89)(33,88)(34,87)(35,105)(36,104)(37,103)(38,119)(39,118)(40,117)(41,116)(42,115)(43,114)(44,113)(45,112)(46,111)(47,110)(48,109)(49,108)(50,107)(51,106)(52,134)(53,133)(54,132)(55,131)(56,130)(57,129)(58,128)(59,127)(60,126)(61,125)(62,124)(63,123)(64,122)(65,121)(66,120)(67,136)(68,135) );
G=PermutationGroup([[(1,83),(2,84),(3,85),(4,69),(5,70),(6,71),(7,72),(8,73),(9,74),(10,75),(11,76),(12,77),(13,78),(14,79),(15,80),(16,81),(17,82),(18,93),(19,94),(20,95),(21,96),(22,97),(23,98),(24,99),(25,100),(26,101),(27,102),(28,86),(29,87),(30,88),(31,89),(32,90),(33,91),(34,92),(35,115),(36,116),(37,117),(38,118),(39,119),(40,103),(41,104),(42,105),(43,106),(44,107),(45,108),(46,109),(47,110),(48,111),(49,112),(50,113),(51,114),(52,128),(53,129),(54,130),(55,131),(56,132),(57,133),(58,134),(59,135),(60,136),(61,120),(62,121),(63,122),(64,123),(65,124),(66,125),(67,126),(68,127)], [(1,64),(2,65),(3,66),(4,67),(5,68),(6,52),(7,53),(8,54),(9,55),(10,56),(11,57),(12,58),(13,59),(14,60),(15,61),(16,62),(17,63),(18,42),(19,43),(20,44),(21,45),(22,46),(23,47),(24,48),(25,49),(26,50),(27,51),(28,35),(29,36),(30,37),(31,38),(32,39),(33,40),(34,41),(69,126),(70,127),(71,128),(72,129),(73,130),(74,131),(75,132),(76,133),(77,134),(78,135),(79,136),(80,120),(81,121),(82,122),(83,123),(84,124),(85,125),(86,115),(87,116),(88,117),(89,118),(90,119),(91,103),(92,104),(93,105),(94,106),(95,107),(96,108),(97,109),(98,110),(99,111),(100,112),(101,113),(102,114)], [(1,32),(2,33),(3,34),(4,18),(5,19),(6,20),(7,21),(8,22),(9,23),(10,24),(11,25),(12,26),(13,27),(14,28),(15,29),(16,30),(17,31),(35,60),(36,61),(37,62),(38,63),(39,64),(40,65),(41,66),(42,67),(43,68),(44,52),(45,53),(46,54),(47,55),(48,56),(49,57),(50,58),(51,59),(69,93),(70,94),(71,95),(72,96),(73,97),(74,98),(75,99),(76,100),(77,101),(78,102),(79,86),(80,87),(81,88),(82,89),(83,90),(84,91),(85,92),(103,124),(104,125),(105,126),(106,127),(107,128),(108,129),(109,130),(110,131),(111,132),(112,133),(113,134),(114,135),(115,136),(116,120),(117,121),(118,122),(119,123)], [(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)], [(1,82),(2,81),(3,80),(4,79),(5,78),(6,77),(7,76),(8,75),(9,74),(10,73),(11,72),(12,71),(13,70),(14,69),(15,85),(16,84),(17,83),(18,86),(19,102),(20,101),(21,100),(22,99),(23,98),(24,97),(25,96),(26,95),(27,94),(28,93),(29,92),(30,91),(31,90),(32,89),(33,88),(34,87),(35,105),(36,104),(37,103),(38,119),(39,118),(40,117),(41,116),(42,115),(43,114),(44,113),(45,112),(46,111),(47,110),(48,109),(49,108),(50,107),(51,106),(52,134),(53,133),(54,132),(55,131),(56,130),(57,129),(58,128),(59,127),(60,126),(61,125),(62,124),(63,123),(64,122),(65,121),(66,120),(67,136),(68,135)]])
80 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | ··· | 2O | 17A | ··· | 17H | 34A | ··· | 34BD |
| order | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 17 | ··· | 17 | 34 | ··· | 34 |
| size | 1 | 1 | ··· | 1 | 17 | ··· | 17 | 2 | ··· | 2 | 2 | ··· | 2 |
80 irreducible representations
| dim | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | + | + |
| image | C1 | C2 | C2 | D17 | D34 |
| kernel | C23×D17 | C22×D17 | C22×C34 | C23 | C22 |
| # reps | 1 | 14 | 1 | 8 | 56 |
Matrix representation of C23×D17 ►in GL4(𝔽103) generated by
| 1 | 0 | 0 | 0 |
| 0 | 102 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 102 | 0 | 0 | 0 |
| 0 | 102 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 102 | 0 |
| 0 | 0 | 0 | 102 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 72 | 1 |
| 0 | 0 | 34 | 62 |
| 102 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 82 | 57 |
| 0 | 0 | 23 | 21 |
G:=sub<GL(4,GF(103))| [1,0,0,0,0,102,0,0,0,0,1,0,0,0,0,1],[102,0,0,0,0,102,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,102,0,0,0,0,102],[1,0,0,0,0,1,0,0,0,0,72,34,0,0,1,62],[102,0,0,0,0,1,0,0,0,0,82,23,0,0,57,21] >;
C23×D17 in GAP, Magma, Sage, TeX
C_2^3\times D_{17} % in TeX
G:=Group("C2^3xD17"); // GroupNames label
G:=SmallGroup(272,53);
// by ID
G=gap.SmallGroup(272,53);
# by ID
G:=PCGroup([5,-2,-2,-2,-2,-17,6404]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^2=d^17=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=d^-1>;
// generators/relations