direct product, metabelian, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C2×C4×D17, C68⋊3C22, C34.2C23, C22.9D34, D34.8C22, Dic17⋊3C22, C34⋊2(C2×C4), (C2×C68)⋊5C2, C17⋊2(C22×C4), (C2×Dic17)⋊5C2, (C2×C34).9C22, C2.1(C22×D17), (C22×D17).4C2, SmallGroup(272,37)
Series: Derived ►Chief ►Lower central ►Upper central
| C17 — C2×C4×D17 |
Generators and relations for C2×C4×D17
G = < a,b,c,d | a2=b4=c17=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >
Subgroups: 358 in 54 conjugacy classes, 35 normal (11 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C2×C4, C2×C4, C23, C22×C4, C17, D17, C34, C34, Dic17, C68, D34, C2×C34, C4×D17, C2×Dic17, C2×C68, C22×D17, C2×C4×D17
Quotients: C1, C2, C4, C22, C2×C4, C23, C22×C4, D17, D34, C4×D17, C22×D17, C2×C4×D17
►On 136 points
Generators in S136
(1 85)(2 69)(3 70)(4 71)(5 72)(6 73)(7 74)(8 75)(9 76)(10 77)(11 78)(12 79)(13 80)(14 81)(15 82)(16 83)(17 84)(18 94)(19 95)(20 96)(21 97)(22 98)(23 99)(24 100)(25 101)(26 102)(27 86)(28 87)(29 88)(30 89)(31 90)(32 91)(33 92)(34 93)(35 109)(36 110)(37 111)(38 112)(39 113)(40 114)(41 115)(42 116)(43 117)(44 118)(45 119)(46 103)(47 104)(48 105)(49 106)(50 107)(51 108)(52 132)(53 133)(54 134)(55 135)(56 136)(57 120)(58 121)(59 122)(60 123)(61 124)(62 125)(63 126)(64 127)(65 128)(66 129)(67 130)(68 131)
(1 58 27 42)(2 59 28 43)(3 60 29 44)(4 61 30 45)(5 62 31 46)(6 63 32 47)(7 64 33 48)(8 65 34 49)(9 66 18 50)(10 67 19 51)(11 68 20 35)(12 52 21 36)(13 53 22 37)(14 54 23 38)(15 55 24 39)(16 56 25 40)(17 57 26 41)(69 122 87 117)(70 123 88 118)(71 124 89 119)(72 125 90 103)(73 126 91 104)(74 127 92 105)(75 128 93 106)(76 129 94 107)(77 130 95 108)(78 131 96 109)(79 132 97 110)(80 133 98 111)(81 134 99 112)(82 135 100 113)(83 136 101 114)(84 120 102 115)(85 121 86 116)
(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 17)(2 16)(3 15)(4 14)(5 13)(6 12)(7 11)(8 10)(19 34)(20 33)(21 32)(22 31)(23 30)(24 29)(25 28)(26 27)(35 48)(36 47)(37 46)(38 45)(39 44)(40 43)(41 42)(49 51)(52 63)(53 62)(54 61)(55 60)(56 59)(57 58)(64 68)(65 67)(69 83)(70 82)(71 81)(72 80)(73 79)(74 78)(75 77)(84 85)(86 102)(87 101)(88 100)(89 99)(90 98)(91 97)(92 96)(93 95)(103 111)(104 110)(105 109)(106 108)(112 119)(113 118)(114 117)(115 116)(120 121)(122 136)(123 135)(124 134)(125 133)(126 132)(127 131)(128 130)
G:=sub<Sym(136)| (1,85)(2,69)(3,70)(4,71)(5,72)(6,73)(7,74)(8,75)(9,76)(10,77)(11,78)(12,79)(13,80)(14,81)(15,82)(16,83)(17,84)(18,94)(19,95)(20,96)(21,97)(22,98)(23,99)(24,100)(25,101)(26,102)(27,86)(28,87)(29,88)(30,89)(31,90)(32,91)(33,92)(34,93)(35,109)(36,110)(37,111)(38,112)(39,113)(40,114)(41,115)(42,116)(43,117)(44,118)(45,119)(46,103)(47,104)(48,105)(49,106)(50,107)(51,108)(52,132)(53,133)(54,134)(55,135)(56,136)(57,120)(58,121)(59,122)(60,123)(61,124)(62,125)(63,126)(64,127)(65,128)(66,129)(67,130)(68,131), (1,58,27,42)(2,59,28,43)(3,60,29,44)(4,61,30,45)(5,62,31,46)(6,63,32,47)(7,64,33,48)(8,65,34,49)(9,66,18,50)(10,67,19,51)(11,68,20,35)(12,52,21,36)(13,53,22,37)(14,54,23,38)(15,55,24,39)(16,56,25,40)(17,57,26,41)(69,122,87,117)(70,123,88,118)(71,124,89,119)(72,125,90,103)(73,126,91,104)(74,127,92,105)(75,128,93,106)(76,129,94,107)(77,130,95,108)(78,131,96,109)(79,132,97,110)(80,133,98,111)(81,134,99,112)(82,135,100,113)(83,136,101,114)(84,120,102,115)(85,121,86,116), (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), (1,17)(2,16)(3,15)(4,14)(5,13)(6,12)(7,11)(8,10)(19,34)(20,33)(21,32)(22,31)(23,30)(24,29)(25,28)(26,27)(35,48)(36,47)(37,46)(38,45)(39,44)(40,43)(41,42)(49,51)(52,63)(53,62)(54,61)(55,60)(56,59)(57,58)(64,68)(65,67)(69,83)(70,82)(71,81)(72,80)(73,79)(74,78)(75,77)(84,85)(86,102)(87,101)(88,100)(89,99)(90,98)(91,97)(92,96)(93,95)(103,111)(104,110)(105,109)(106,108)(112,119)(113,118)(114,117)(115,116)(120,121)(122,136)(123,135)(124,134)(125,133)(126,132)(127,131)(128,130)>;
G:=Group( (1,85)(2,69)(3,70)(4,71)(5,72)(6,73)(7,74)(8,75)(9,76)(10,77)(11,78)(12,79)(13,80)(14,81)(15,82)(16,83)(17,84)(18,94)(19,95)(20,96)(21,97)(22,98)(23,99)(24,100)(25,101)(26,102)(27,86)(28,87)(29,88)(30,89)(31,90)(32,91)(33,92)(34,93)(35,109)(36,110)(37,111)(38,112)(39,113)(40,114)(41,115)(42,116)(43,117)(44,118)(45,119)(46,103)(47,104)(48,105)(49,106)(50,107)(51,108)(52,132)(53,133)(54,134)(55,135)(56,136)(57,120)(58,121)(59,122)(60,123)(61,124)(62,125)(63,126)(64,127)(65,128)(66,129)(67,130)(68,131), (1,58,27,42)(2,59,28,43)(3,60,29,44)(4,61,30,45)(5,62,31,46)(6,63,32,47)(7,64,33,48)(8,65,34,49)(9,66,18,50)(10,67,19,51)(11,68,20,35)(12,52,21,36)(13,53,22,37)(14,54,23,38)(15,55,24,39)(16,56,25,40)(17,57,26,41)(69,122,87,117)(70,123,88,118)(71,124,89,119)(72,125,90,103)(73,126,91,104)(74,127,92,105)(75,128,93,106)(76,129,94,107)(77,130,95,108)(78,131,96,109)(79,132,97,110)(80,133,98,111)(81,134,99,112)(82,135,100,113)(83,136,101,114)(84,120,102,115)(85,121,86,116), (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), (1,17)(2,16)(3,15)(4,14)(5,13)(6,12)(7,11)(8,10)(19,34)(20,33)(21,32)(22,31)(23,30)(24,29)(25,28)(26,27)(35,48)(36,47)(37,46)(38,45)(39,44)(40,43)(41,42)(49,51)(52,63)(53,62)(54,61)(55,60)(56,59)(57,58)(64,68)(65,67)(69,83)(70,82)(71,81)(72,80)(73,79)(74,78)(75,77)(84,85)(86,102)(87,101)(88,100)(89,99)(90,98)(91,97)(92,96)(93,95)(103,111)(104,110)(105,109)(106,108)(112,119)(113,118)(114,117)(115,116)(120,121)(122,136)(123,135)(124,134)(125,133)(126,132)(127,131)(128,130) );
G=PermutationGroup([[(1,85),(2,69),(3,70),(4,71),(5,72),(6,73),(7,74),(8,75),(9,76),(10,77),(11,78),(12,79),(13,80),(14,81),(15,82),(16,83),(17,84),(18,94),(19,95),(20,96),(21,97),(22,98),(23,99),(24,100),(25,101),(26,102),(27,86),(28,87),(29,88),(30,89),(31,90),(32,91),(33,92),(34,93),(35,109),(36,110),(37,111),(38,112),(39,113),(40,114),(41,115),(42,116),(43,117),(44,118),(45,119),(46,103),(47,104),(48,105),(49,106),(50,107),(51,108),(52,132),(53,133),(54,134),(55,135),(56,136),(57,120),(58,121),(59,122),(60,123),(61,124),(62,125),(63,126),(64,127),(65,128),(66,129),(67,130),(68,131)], [(1,58,27,42),(2,59,28,43),(3,60,29,44),(4,61,30,45),(5,62,31,46),(6,63,32,47),(7,64,33,48),(8,65,34,49),(9,66,18,50),(10,67,19,51),(11,68,20,35),(12,52,21,36),(13,53,22,37),(14,54,23,38),(15,55,24,39),(16,56,25,40),(17,57,26,41),(69,122,87,117),(70,123,88,118),(71,124,89,119),(72,125,90,103),(73,126,91,104),(74,127,92,105),(75,128,93,106),(76,129,94,107),(77,130,95,108),(78,131,96,109),(79,132,97,110),(80,133,98,111),(81,134,99,112),(82,135,100,113),(83,136,101,114),(84,120,102,115),(85,121,86,116)], [(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)], [(1,17),(2,16),(3,15),(4,14),(5,13),(6,12),(7,11),(8,10),(19,34),(20,33),(21,32),(22,31),(23,30),(24,29),(25,28),(26,27),(35,48),(36,47),(37,46),(38,45),(39,44),(40,43),(41,42),(49,51),(52,63),(53,62),(54,61),(55,60),(56,59),(57,58),(64,68),(65,67),(69,83),(70,82),(71,81),(72,80),(73,79),(74,78),(75,77),(84,85),(86,102),(87,101),(88,100),(89,99),(90,98),(91,97),(92,96),(93,95),(103,111),(104,110),(105,109),(106,108),(112,119),(113,118),(114,117),(115,116),(120,121),(122,136),(123,135),(124,134),(125,133),(126,132),(127,131),(128,130)]])
80 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 17A | ··· | 17H | 34A | ··· | 34X | 68A | ··· | 68AF |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 17 | ··· | 17 | 34 | ··· | 34 | 68 | ··· | 68 |
| size | 1 | 1 | 1 | 1 | 17 | 17 | 17 | 17 | 1 | 1 | 1 | 1 | 17 | 17 | 17 | 17 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
80 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C4 | D17 | D34 | D34 | C4×D17 |
| kernel | C2×C4×D17 | C4×D17 | C2×Dic17 | C2×C68 | C22×D17 | D34 | C2×C4 | C4 | C22 | C2 |
| # reps | 1 | 4 | 1 | 1 | 1 | 8 | 8 | 16 | 8 | 32 |
Matrix representation of C2×C4×D17 ►in GL3(𝔽137) generated by
| 136 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 1 |
| 1 | 0 | 0 |
| 0 | 37 | 0 |
| 0 | 0 | 37 |
| 1 | 0 | 0 |
| 0 | 116 | 40 |
| 0 | 136 | 41 |
| 136 | 0 | 0 |
| 0 | 23 | 84 |
| 0 | 72 | 114 |
G:=sub<GL(3,GF(137))| [136,0,0,0,1,0,0,0,1],[1,0,0,0,37,0,0,0,37],[1,0,0,0,116,136,0,40,41],[136,0,0,0,23,72,0,84,114] >;
C2×C4×D17 in GAP, Magma, Sage, TeX
C_2\times C_4\times D_{17} % in TeX
G:=Group("C2xC4xD17"); // GroupNames label
G:=SmallGroup(272,37);
// by ID
G=gap.SmallGroup(272,37);
# by ID
G:=PCGroup([5,-2,-2,-2,-2,-17,42,6404]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^4=c^17=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations