metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: (C2×C20)⋊17D4, C20⋊D4⋊33C2, C20.454(C2×D4), Dic5⋊4(C4○D4), (C2×D4).237D10, (C2×Q8).194D10, Dic5⋊D4⋊47C2, Dic5⋊Q8⋊34C2, C20.17D4⋊33C2, C20.23D4⋊34C2, (C2×C10).318C24, (C2×C20).889C23, (C22×C4).288D10, C10.168(C22×D4), (D4×C10).277C22, (C2×D20).290C22, C5⋊7(C22.26C24), (Q8×C10).244C22, C22.327(C23×D5), C23.139(C22×D5), (C22×C20).297C22, (C22×C10).244C23, (C2×Dic5).304C23, (C4×Dic5).292C22, (C22×D5).139C23, C23.D5.137C22, D10⋊C4.160C22, (C2×Dic10).319C22, C10.D4.172C22, (C22×Dic5).260C22, (C2×C4○D4)⋊10D5, (C4×C5⋊D4)⋊61C2, (C2×C4×Dic5)⋊15C2, (C2×C4○D20)⋊32C2, (C10×C4○D4)⋊10C2, (C2×C4)⋊11(C5⋊D4), C2.106(D5×C4○D4), (C2×C10).83(C2×D4), C4.146(C2×C5⋊D4), C22.1(C2×C5⋊D4), C10.218(C2×C4○D4), (C2×C4×D5).272C22, C2.41(C22×C5⋊D4), (C2×C4).832(C22×D5), (C2×C5⋊D4).150C22, SmallGroup(320,1504)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for (C2×C20)⋊17D4
G = < a,b,c,d | a2=b20=c4=d2=1, ab=ba, ac=ca, dad=ab10, cbc-1=dbd=b9, dcd=c-1 >
Subgroups: 1022 in 310 conjugacy classes, 115 normal (31 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C5, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, C23, D5, C10, C10, C10, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×D4, C2×Q8, C2×Q8, C4○D4, Dic5, Dic5, C20, C20, D10, C2×C10, C2×C10, C2×C10, C2×C42, C4×D4, C4⋊D4, C4.4D4, C4⋊1D4, C4⋊Q8, C2×C4○D4, C2×C4○D4, Dic10, C4×D5, D20, C2×Dic5, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C2×C20, C5×D4, C5×Q8, C22×D5, C22×C10, C22×C10, C22.26C24, C4×Dic5, C4×Dic5, C10.D4, D10⋊C4, C23.D5, C2×Dic10, C2×C4×D5, C2×D20, C4○D20, C22×Dic5, C2×C5⋊D4, C22×C20, C22×C20, D4×C10, D4×C10, Q8×C10, C5×C4○D4, C2×C4×Dic5, C4×C5⋊D4, C20.17D4, Dic5⋊D4, C20⋊D4, Dic5⋊Q8, C20.23D4, C2×C4○D20, C10×C4○D4, (C2×C20)⋊17D4
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, C4○D4, C24, D10, C22×D4, C2×C4○D4, C5⋊D4, C22×D5, C22.26C24, C2×C5⋊D4, C23×D5, D5×C4○D4, C22×C5⋊D4, (C2×C20)⋊17D4
►On 160 points
Generators in S160
(1 70)(2 71)(3 72)(4 73)(5 74)(6 75)(7 76)(8 77)(9 78)(10 79)(11 80)(12 61)(13 62)(14 63)(15 64)(16 65)(17 66)(18 67)(19 68)(20 69)(21 43)(22 44)(23 45)(24 46)(25 47)(26 48)(27 49)(28 50)(29 51)(30 52)(31 53)(32 54)(33 55)(34 56)(35 57)(36 58)(37 59)(38 60)(39 41)(40 42)(81 126)(82 127)(83 128)(84 129)(85 130)(86 131)(87 132)(88 133)(89 134)(90 135)(91 136)(92 137)(93 138)(94 139)(95 140)(96 121)(97 122)(98 123)(99 124)(100 125)(101 153)(102 154)(103 155)(104 156)(105 157)(106 158)(107 159)(108 160)(109 141)(110 142)(111 143)(112 144)(113 145)(114 146)(115 147)(116 148)(117 149)(118 150)(119 151)(120 152)
(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)(121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140)(141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160)
(1 43 137 102)(2 52 138 111)(3 41 139 120)(4 50 140 109)(5 59 121 118)(6 48 122 107)(7 57 123 116)(8 46 124 105)(9 55 125 114)(10 44 126 103)(11 53 127 112)(12 42 128 101)(13 51 129 110)(14 60 130 119)(15 49 131 108)(16 58 132 117)(17 47 133 106)(18 56 134 115)(19 45 135 104)(20 54 136 113)(21 92 154 70)(22 81 155 79)(23 90 156 68)(24 99 157 77)(25 88 158 66)(26 97 159 75)(27 86 160 64)(28 95 141 73)(29 84 142 62)(30 93 143 71)(31 82 144 80)(32 91 145 69)(33 100 146 78)(34 89 147 67)(35 98 148 76)(36 87 149 65)(37 96 150 74)(38 85 151 63)(39 94 152 72)(40 83 153 61)
(2 10)(3 19)(4 8)(5 17)(7 15)(9 13)(12 20)(14 18)(21 144)(22 153)(23 142)(24 151)(25 160)(26 149)(27 158)(28 147)(29 156)(30 145)(31 154)(32 143)(33 152)(34 141)(35 150)(36 159)(37 148)(38 157)(39 146)(40 155)(41 104)(42 113)(43 102)(44 111)(45 120)(46 109)(47 118)(48 107)(49 116)(50 105)(51 114)(52 103)(53 112)(54 101)(55 110)(56 119)(57 108)(58 117)(59 106)(60 115)(61 79)(62 68)(63 77)(64 66)(65 75)(67 73)(69 71)(70 80)(72 78)(74 76)(81 83)(82 92)(84 90)(85 99)(86 88)(87 97)(89 95)(91 93)(94 100)(96 98)(121 133)(123 131)(124 140)(125 129)(126 138)(128 136)(130 134)(135 139)
G:=sub<Sym(160)| (1,70)(2,71)(3,72)(4,73)(5,74)(6,75)(7,76)(8,77)(9,78)(10,79)(11,80)(12,61)(13,62)(14,63)(15,64)(16,65)(17,66)(18,67)(19,68)(20,69)(21,43)(22,44)(23,45)(24,46)(25,47)(26,48)(27,49)(28,50)(29,51)(30,52)(31,53)(32,54)(33,55)(34,56)(35,57)(36,58)(37,59)(38,60)(39,41)(40,42)(81,126)(82,127)(83,128)(84,129)(85,130)(86,131)(87,132)(88,133)(89,134)(90,135)(91,136)(92,137)(93,138)(94,139)(95,140)(96,121)(97,122)(98,123)(99,124)(100,125)(101,153)(102,154)(103,155)(104,156)(105,157)(106,158)(107,159)(108,160)(109,141)(110,142)(111,143)(112,144)(113,145)(114,146)(115,147)(116,148)(117,149)(118,150)(119,151)(120,152), (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)(121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140)(141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160), (1,43,137,102)(2,52,138,111)(3,41,139,120)(4,50,140,109)(5,59,121,118)(6,48,122,107)(7,57,123,116)(8,46,124,105)(9,55,125,114)(10,44,126,103)(11,53,127,112)(12,42,128,101)(13,51,129,110)(14,60,130,119)(15,49,131,108)(16,58,132,117)(17,47,133,106)(18,56,134,115)(19,45,135,104)(20,54,136,113)(21,92,154,70)(22,81,155,79)(23,90,156,68)(24,99,157,77)(25,88,158,66)(26,97,159,75)(27,86,160,64)(28,95,141,73)(29,84,142,62)(30,93,143,71)(31,82,144,80)(32,91,145,69)(33,100,146,78)(34,89,147,67)(35,98,148,76)(36,87,149,65)(37,96,150,74)(38,85,151,63)(39,94,152,72)(40,83,153,61), (2,10)(3,19)(4,8)(5,17)(7,15)(9,13)(12,20)(14,18)(21,144)(22,153)(23,142)(24,151)(25,160)(26,149)(27,158)(28,147)(29,156)(30,145)(31,154)(32,143)(33,152)(34,141)(35,150)(36,159)(37,148)(38,157)(39,146)(40,155)(41,104)(42,113)(43,102)(44,111)(45,120)(46,109)(47,118)(48,107)(49,116)(50,105)(51,114)(52,103)(53,112)(54,101)(55,110)(56,119)(57,108)(58,117)(59,106)(60,115)(61,79)(62,68)(63,77)(64,66)(65,75)(67,73)(69,71)(70,80)(72,78)(74,76)(81,83)(82,92)(84,90)(85,99)(86,88)(87,97)(89,95)(91,93)(94,100)(96,98)(121,133)(123,131)(124,140)(125,129)(126,138)(128,136)(130,134)(135,139)>;
G:=Group( (1,70)(2,71)(3,72)(4,73)(5,74)(6,75)(7,76)(8,77)(9,78)(10,79)(11,80)(12,61)(13,62)(14,63)(15,64)(16,65)(17,66)(18,67)(19,68)(20,69)(21,43)(22,44)(23,45)(24,46)(25,47)(26,48)(27,49)(28,50)(29,51)(30,52)(31,53)(32,54)(33,55)(34,56)(35,57)(36,58)(37,59)(38,60)(39,41)(40,42)(81,126)(82,127)(83,128)(84,129)(85,130)(86,131)(87,132)(88,133)(89,134)(90,135)(91,136)(92,137)(93,138)(94,139)(95,140)(96,121)(97,122)(98,123)(99,124)(100,125)(101,153)(102,154)(103,155)(104,156)(105,157)(106,158)(107,159)(108,160)(109,141)(110,142)(111,143)(112,144)(113,145)(114,146)(115,147)(116,148)(117,149)(118,150)(119,151)(120,152), (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)(121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140)(141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160), (1,43,137,102)(2,52,138,111)(3,41,139,120)(4,50,140,109)(5,59,121,118)(6,48,122,107)(7,57,123,116)(8,46,124,105)(9,55,125,114)(10,44,126,103)(11,53,127,112)(12,42,128,101)(13,51,129,110)(14,60,130,119)(15,49,131,108)(16,58,132,117)(17,47,133,106)(18,56,134,115)(19,45,135,104)(20,54,136,113)(21,92,154,70)(22,81,155,79)(23,90,156,68)(24,99,157,77)(25,88,158,66)(26,97,159,75)(27,86,160,64)(28,95,141,73)(29,84,142,62)(30,93,143,71)(31,82,144,80)(32,91,145,69)(33,100,146,78)(34,89,147,67)(35,98,148,76)(36,87,149,65)(37,96,150,74)(38,85,151,63)(39,94,152,72)(40,83,153,61), (2,10)(3,19)(4,8)(5,17)(7,15)(9,13)(12,20)(14,18)(21,144)(22,153)(23,142)(24,151)(25,160)(26,149)(27,158)(28,147)(29,156)(30,145)(31,154)(32,143)(33,152)(34,141)(35,150)(36,159)(37,148)(38,157)(39,146)(40,155)(41,104)(42,113)(43,102)(44,111)(45,120)(46,109)(47,118)(48,107)(49,116)(50,105)(51,114)(52,103)(53,112)(54,101)(55,110)(56,119)(57,108)(58,117)(59,106)(60,115)(61,79)(62,68)(63,77)(64,66)(65,75)(67,73)(69,71)(70,80)(72,78)(74,76)(81,83)(82,92)(84,90)(85,99)(86,88)(87,97)(89,95)(91,93)(94,100)(96,98)(121,133)(123,131)(124,140)(125,129)(126,138)(128,136)(130,134)(135,139) );
G=PermutationGroup([[(1,70),(2,71),(3,72),(4,73),(5,74),(6,75),(7,76),(8,77),(9,78),(10,79),(11,80),(12,61),(13,62),(14,63),(15,64),(16,65),(17,66),(18,67),(19,68),(20,69),(21,43),(22,44),(23,45),(24,46),(25,47),(26,48),(27,49),(28,50),(29,51),(30,52),(31,53),(32,54),(33,55),(34,56),(35,57),(36,58),(37,59),(38,60),(39,41),(40,42),(81,126),(82,127),(83,128),(84,129),(85,130),(86,131),(87,132),(88,133),(89,134),(90,135),(91,136),(92,137),(93,138),(94,139),(95,140),(96,121),(97,122),(98,123),(99,124),(100,125),(101,153),(102,154),(103,155),(104,156),(105,157),(106,158),(107,159),(108,160),(109,141),(110,142),(111,143),(112,144),(113,145),(114,146),(115,147),(116,148),(117,149),(118,150),(119,151),(120,152)], [(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),(121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140),(141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160)], [(1,43,137,102),(2,52,138,111),(3,41,139,120),(4,50,140,109),(5,59,121,118),(6,48,122,107),(7,57,123,116),(8,46,124,105),(9,55,125,114),(10,44,126,103),(11,53,127,112),(12,42,128,101),(13,51,129,110),(14,60,130,119),(15,49,131,108),(16,58,132,117),(17,47,133,106),(18,56,134,115),(19,45,135,104),(20,54,136,113),(21,92,154,70),(22,81,155,79),(23,90,156,68),(24,99,157,77),(25,88,158,66),(26,97,159,75),(27,86,160,64),(28,95,141,73),(29,84,142,62),(30,93,143,71),(31,82,144,80),(32,91,145,69),(33,100,146,78),(34,89,147,67),(35,98,148,76),(36,87,149,65),(37,96,150,74),(38,85,151,63),(39,94,152,72),(40,83,153,61)], [(2,10),(3,19),(4,8),(5,17),(7,15),(9,13),(12,20),(14,18),(21,144),(22,153),(23,142),(24,151),(25,160),(26,149),(27,158),(28,147),(29,156),(30,145),(31,154),(32,143),(33,152),(34,141),(35,150),(36,159),(37,148),(38,157),(39,146),(40,155),(41,104),(42,113),(43,102),(44,111),(45,120),(46,109),(47,118),(48,107),(49,116),(50,105),(51,114),(52,103),(53,112),(54,101),(55,110),(56,119),(57,108),(58,117),(59,106),(60,115),(61,79),(62,68),(63,77),(64,66),(65,75),(67,73),(69,71),(70,80),(72,78),(74,76),(81,83),(82,92),(84,90),(85,99),(86,88),(87,97),(89,95),(91,93),(94,100),(96,98),(121,133),(123,131),(124,140),(125,129),(126,138),(128,136),(130,134),(135,139)]])
68 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | ··· | 4P | 4Q | 4R | 5A | 5B | 10A | ··· | 10F | 10G | ··· | 10R | 20A | ··· | 20H | 20I | ··· | 20T |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 5 | 5 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 20 | 20 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 10 | ··· | 10 | 20 | 20 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
68 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D5 | C4○D4 | D10 | D10 | D10 | C5⋊D4 | D5×C4○D4 |
| kernel | (C2×C20)⋊17D4 | C2×C4×Dic5 | C4×C5⋊D4 | C20.17D4 | Dic5⋊D4 | C20⋊D4 | Dic5⋊Q8 | C20.23D4 | C2×C4○D20 | C10×C4○D4 | C2×C20 | C2×C4○D4 | Dic5 | C22×C4 | C2×D4 | C2×Q8 | C2×C4 | C2 |
| # reps | 1 | 1 | 4 | 1 | 4 | 1 | 1 | 1 | 1 | 1 | 4 | 2 | 8 | 6 | 6 | 2 | 16 | 8 |
Matrix representation of (C2×C20)⋊17D4 ►in GL6(𝔽41)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 9 | 2 | 0 | 0 |
| 0 | 0 | 1 | 32 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 6 | 1 | 0 | 0 | 0 | 0 |
| 5 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 32 | 0 | 0 | 0 |
| 0 | 0 | 0 | 32 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 6 | 1 | 0 | 0 | 0 | 0 |
| 6 | 35 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 6 | 1 | 0 | 0 | 0 | 0 |
| 6 | 35 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 32 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,9,1,0,0,0,0,2,32,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[6,5,0,0,0,0,1,1,0,0,0,0,0,0,32,0,0,0,0,0,0,32,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[6,6,0,0,0,0,1,35,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,40,0],[6,6,0,0,0,0,1,35,0,0,0,0,0,0,1,32,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,40] >;
(C2×C20)⋊17D4 in GAP, Magma, Sage, TeX
(C_2\times C_{20})\rtimes_{17}D_4 % in TeX
G:=Group("(C2xC20):17D4"); // GroupNames label
G:=SmallGroup(320,1504);
// by ID
G=gap.SmallGroup(320,1504);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,232,758,675,297,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^20=c^4=d^2=1,a*b=b*a,a*c=c*a,d*a*d=a*b^10,c*b*c^-1=d*b*d=b^9,d*c*d=c^-1>;
// generators/relations