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×C20).889C23, (C2×C10).318C24, (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×C10).244C23, (C22×C20).297C22, (C4×Dic5).292C22, (C2×Dic5).304C23, (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
Subgroups: 1022 in 310 conjugacy classes, 115 normal (31 characteristic)
C1, C2, C2 [×2], C2 [×6], C4 [×4], C4 [×10], C22, C22 [×2], C22 [×14], C5, C2×C4 [×2], C2×C4 [×6], C2×C4 [×18], D4 [×20], Q8 [×4], C23, C23 [×2], C23 [×2], D5 [×2], C10, C10 [×2], C10 [×4], C42 [×4], C22⋊C4 [×8], C4⋊C4 [×4], C22×C4, C22×C4 [×2], C22×C4 [×4], C2×D4, C2×D4 [×2], C2×D4 [×7], C2×Q8, C2×Q8, C4○D4 [×8], Dic5 [×4], Dic5 [×4], C20 [×4], C20 [×2], D10 [×6], C2×C10, C2×C10 [×2], C2×C10 [×8], C2×C42, C4×D4 [×4], C4⋊D4 [×4], C4.4D4 [×2], C4⋊1D4, C4⋊Q8, C2×C4○D4, C2×C4○D4, Dic10 [×2], C4×D5 [×4], D20 [×2], C2×Dic5 [×6], C2×Dic5 [×4], C5⋊D4 [×12], C2×C20 [×2], C2×C20 [×6], C2×C20 [×4], C5×D4 [×6], C5×Q8 [×2], C22×D5 [×2], C22×C10, C22×C10 [×2], C22.26C24, C4×Dic5 [×2], C4×Dic5 [×2], C10.D4 [×4], D10⋊C4 [×4], C23.D5 [×4], C2×Dic10, C2×C4×D5 [×2], C2×D20, C4○D20 [×4], C22×Dic5 [×2], C2×C5⋊D4 [×6], C22×C20, C22×C20 [×2], D4×C10, D4×C10 [×2], Q8×C10, C5×C4○D4 [×4], C2×C4×Dic5, C4×C5⋊D4 [×4], C20.17D4, Dic5⋊D4 [×4], C20⋊D4, Dic5⋊Q8, C20.23D4, C2×C4○D20, C10×C4○D4, (C2×C20)⋊17D4
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D5, C2×D4 [×6], C4○D4 [×4], C24, D10 [×7], C22×D4, C2×C4○D4 [×2], C5⋊D4 [×4], C22×D5 [×7], C22.26C24, C2×C5⋊D4 [×6], C23×D5, D5×C4○D4 [×2], C22×C5⋊D4, (C2×C20)⋊17D4
Generators and relations
G = < a,b,c,d | a2=b20=c4=d2=1, ab=ba, ac=ca, dad=ab10, cbc-1=dbd=b9, dcd=c-1 >
►On 160 points
Generators in S160
(1 101)(2 102)(3 103)(4 104)(5 105)(6 106)(7 107)(8 108)(9 109)(10 110)(11 111)(12 112)(13 113)(14 114)(15 115)(16 116)(17 117)(18 118)(19 119)(20 120)(21 50)(22 51)(23 52)(24 53)(25 54)(26 55)(27 56)(28 57)(29 58)(30 59)(31 60)(32 41)(33 42)(34 43)(35 44)(36 45)(37 46)(38 47)(39 48)(40 49)(61 139)(62 140)(63 121)(64 122)(65 123)(66 124)(67 125)(68 126)(69 127)(70 128)(71 129)(72 130)(73 131)(74 132)(75 133)(76 134)(77 135)(78 136)(79 137)(80 138)(81 160)(82 141)(83 142)(84 143)(85 144)(86 145)(87 146)(88 147)(89 148)(90 149)(91 150)(92 151)(93 152)(94 153)(95 154)(96 155)(97 156)(98 157)(99 158)(100 159)
(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 24 69 93)(2 33 70 82)(3 22 71 91)(4 31 72 100)(5 40 73 89)(6 29 74 98)(7 38 75 87)(8 27 76 96)(9 36 77 85)(10 25 78 94)(11 34 79 83)(12 23 80 92)(13 32 61 81)(14 21 62 90)(15 30 63 99)(16 39 64 88)(17 28 65 97)(18 37 66 86)(19 26 67 95)(20 35 68 84)(41 139 160 113)(42 128 141 102)(43 137 142 111)(44 126 143 120)(45 135 144 109)(46 124 145 118)(47 133 146 107)(48 122 147 116)(49 131 148 105)(50 140 149 114)(51 129 150 103)(52 138 151 112)(53 127 152 101)(54 136 153 110)(55 125 154 119)(56 134 155 108)(57 123 156 117)(58 132 157 106)(59 121 158 115)(60 130 159 104)
(2 10)(3 19)(4 8)(5 17)(7 15)(9 13)(12 20)(14 18)(21 86)(22 95)(23 84)(24 93)(25 82)(26 91)(27 100)(28 89)(29 98)(30 87)(31 96)(32 85)(33 94)(34 83)(35 92)(36 81)(37 90)(38 99)(39 88)(40 97)(41 154)(42 143)(43 152)(44 141)(45 150)(46 159)(47 148)(48 157)(49 146)(50 155)(51 144)(52 153)(53 142)(54 151)(55 160)(56 149)(57 158)(58 147)(59 156)(60 145)(61 77)(62 66)(63 75)(65 73)(67 71)(68 80)(70 78)(72 76)(101 111)(102 120)(103 109)(104 118)(105 107)(106 116)(108 114)(110 112)(113 119)(115 117)(121 123)(122 132)(124 130)(125 139)(126 128)(127 137)(129 135)(131 133)(134 140)(136 138)
G:=sub<Sym(160)| (1,101)(2,102)(3,103)(4,104)(5,105)(6,106)(7,107)(8,108)(9,109)(10,110)(11,111)(12,112)(13,113)(14,114)(15,115)(16,116)(17,117)(18,118)(19,119)(20,120)(21,50)(22,51)(23,52)(24,53)(25,54)(26,55)(27,56)(28,57)(29,58)(30,59)(31,60)(32,41)(33,42)(34,43)(35,44)(36,45)(37,46)(38,47)(39,48)(40,49)(61,139)(62,140)(63,121)(64,122)(65,123)(66,124)(67,125)(68,126)(69,127)(70,128)(71,129)(72,130)(73,131)(74,132)(75,133)(76,134)(77,135)(78,136)(79,137)(80,138)(81,160)(82,141)(83,142)(84,143)(85,144)(86,145)(87,146)(88,147)(89,148)(90,149)(91,150)(92,151)(93,152)(94,153)(95,154)(96,155)(97,156)(98,157)(99,158)(100,159), (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), (1,24,69,93)(2,33,70,82)(3,22,71,91)(4,31,72,100)(5,40,73,89)(6,29,74,98)(7,38,75,87)(8,27,76,96)(9,36,77,85)(10,25,78,94)(11,34,79,83)(12,23,80,92)(13,32,61,81)(14,21,62,90)(15,30,63,99)(16,39,64,88)(17,28,65,97)(18,37,66,86)(19,26,67,95)(20,35,68,84)(41,139,160,113)(42,128,141,102)(43,137,142,111)(44,126,143,120)(45,135,144,109)(46,124,145,118)(47,133,146,107)(48,122,147,116)(49,131,148,105)(50,140,149,114)(51,129,150,103)(52,138,151,112)(53,127,152,101)(54,136,153,110)(55,125,154,119)(56,134,155,108)(57,123,156,117)(58,132,157,106)(59,121,158,115)(60,130,159,104), (2,10)(3,19)(4,8)(5,17)(7,15)(9,13)(12,20)(14,18)(21,86)(22,95)(23,84)(24,93)(25,82)(26,91)(27,100)(28,89)(29,98)(30,87)(31,96)(32,85)(33,94)(34,83)(35,92)(36,81)(37,90)(38,99)(39,88)(40,97)(41,154)(42,143)(43,152)(44,141)(45,150)(46,159)(47,148)(48,157)(49,146)(50,155)(51,144)(52,153)(53,142)(54,151)(55,160)(56,149)(57,158)(58,147)(59,156)(60,145)(61,77)(62,66)(63,75)(65,73)(67,71)(68,80)(70,78)(72,76)(101,111)(102,120)(103,109)(104,118)(105,107)(106,116)(108,114)(110,112)(113,119)(115,117)(121,123)(122,132)(124,130)(125,139)(126,128)(127,137)(129,135)(131,133)(134,140)(136,138)>;
G:=Group( (1,101)(2,102)(3,103)(4,104)(5,105)(6,106)(7,107)(8,108)(9,109)(10,110)(11,111)(12,112)(13,113)(14,114)(15,115)(16,116)(17,117)(18,118)(19,119)(20,120)(21,50)(22,51)(23,52)(24,53)(25,54)(26,55)(27,56)(28,57)(29,58)(30,59)(31,60)(32,41)(33,42)(34,43)(35,44)(36,45)(37,46)(38,47)(39,48)(40,49)(61,139)(62,140)(63,121)(64,122)(65,123)(66,124)(67,125)(68,126)(69,127)(70,128)(71,129)(72,130)(73,131)(74,132)(75,133)(76,134)(77,135)(78,136)(79,137)(80,138)(81,160)(82,141)(83,142)(84,143)(85,144)(86,145)(87,146)(88,147)(89,148)(90,149)(91,150)(92,151)(93,152)(94,153)(95,154)(96,155)(97,156)(98,157)(99,158)(100,159), (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), (1,24,69,93)(2,33,70,82)(3,22,71,91)(4,31,72,100)(5,40,73,89)(6,29,74,98)(7,38,75,87)(8,27,76,96)(9,36,77,85)(10,25,78,94)(11,34,79,83)(12,23,80,92)(13,32,61,81)(14,21,62,90)(15,30,63,99)(16,39,64,88)(17,28,65,97)(18,37,66,86)(19,26,67,95)(20,35,68,84)(41,139,160,113)(42,128,141,102)(43,137,142,111)(44,126,143,120)(45,135,144,109)(46,124,145,118)(47,133,146,107)(48,122,147,116)(49,131,148,105)(50,140,149,114)(51,129,150,103)(52,138,151,112)(53,127,152,101)(54,136,153,110)(55,125,154,119)(56,134,155,108)(57,123,156,117)(58,132,157,106)(59,121,158,115)(60,130,159,104), (2,10)(3,19)(4,8)(5,17)(7,15)(9,13)(12,20)(14,18)(21,86)(22,95)(23,84)(24,93)(25,82)(26,91)(27,100)(28,89)(29,98)(30,87)(31,96)(32,85)(33,94)(34,83)(35,92)(36,81)(37,90)(38,99)(39,88)(40,97)(41,154)(42,143)(43,152)(44,141)(45,150)(46,159)(47,148)(48,157)(49,146)(50,155)(51,144)(52,153)(53,142)(54,151)(55,160)(56,149)(57,158)(58,147)(59,156)(60,145)(61,77)(62,66)(63,75)(65,73)(67,71)(68,80)(70,78)(72,76)(101,111)(102,120)(103,109)(104,118)(105,107)(106,116)(108,114)(110,112)(113,119)(115,117)(121,123)(122,132)(124,130)(125,139)(126,128)(127,137)(129,135)(131,133)(134,140)(136,138) );
G=PermutationGroup([(1,101),(2,102),(3,103),(4,104),(5,105),(6,106),(7,107),(8,108),(9,109),(10,110),(11,111),(12,112),(13,113),(14,114),(15,115),(16,116),(17,117),(18,118),(19,119),(20,120),(21,50),(22,51),(23,52),(24,53),(25,54),(26,55),(27,56),(28,57),(29,58),(30,59),(31,60),(32,41),(33,42),(34,43),(35,44),(36,45),(37,46),(38,47),(39,48),(40,49),(61,139),(62,140),(63,121),(64,122),(65,123),(66,124),(67,125),(68,126),(69,127),(70,128),(71,129),(72,130),(73,131),(74,132),(75,133),(76,134),(77,135),(78,136),(79,137),(80,138),(81,160),(82,141),(83,142),(84,143),(85,144),(86,145),(87,146),(88,147),(89,148),(90,149),(91,150),(92,151),(93,152),(94,153),(95,154),(96,155),(97,156),(98,157),(99,158),(100,159)], [(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)], [(1,24,69,93),(2,33,70,82),(3,22,71,91),(4,31,72,100),(5,40,73,89),(6,29,74,98),(7,38,75,87),(8,27,76,96),(9,36,77,85),(10,25,78,94),(11,34,79,83),(12,23,80,92),(13,32,61,81),(14,21,62,90),(15,30,63,99),(16,39,64,88),(17,28,65,97),(18,37,66,86),(19,26,67,95),(20,35,68,84),(41,139,160,113),(42,128,141,102),(43,137,142,111),(44,126,143,120),(45,135,144,109),(46,124,145,118),(47,133,146,107),(48,122,147,116),(49,131,148,105),(50,140,149,114),(51,129,150,103),(52,138,151,112),(53,127,152,101),(54,136,153,110),(55,125,154,119),(56,134,155,108),(57,123,156,117),(58,132,157,106),(59,121,158,115),(60,130,159,104)], [(2,10),(3,19),(4,8),(5,17),(7,15),(9,13),(12,20),(14,18),(21,86),(22,95),(23,84),(24,93),(25,82),(26,91),(27,100),(28,89),(29,98),(30,87),(31,96),(32,85),(33,94),(34,83),(35,92),(36,81),(37,90),(38,99),(39,88),(40,97),(41,154),(42,143),(43,152),(44,141),(45,150),(46,159),(47,148),(48,157),(49,146),(50,155),(51,144),(52,153),(53,142),(54,151),(55,160),(56,149),(57,158),(58,147),(59,156),(60,145),(61,77),(62,66),(63,75),(65,73),(67,71),(68,80),(70,78),(72,76),(101,111),(102,120),(103,109),(104,118),(105,107),(106,116),(108,114),(110,112),(113,119),(115,117),(121,123),(122,132),(124,130),(125,139),(126,128),(127,137),(129,135),(131,133),(134,140),(136,138)])
Matrix representation ►G ⊆ 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] >;
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 |
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
×
⋊
ℤ
𝔽
○
≀