direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×C20⋊2D4, C24.39D10, C20⋊7(C2×D4), D10⋊4(C2×D4), (C2×C20)⋊12D4, (C2×D4)⋊36D10, (C22×D4)⋊6D5, C10⋊4(C4⋊D4), (C22×D5)⋊12D4, (D4×C10)⋊43C22, C4⋊Dic5⋊77C22, C22.147(D4×D5), (C2×C20).542C23, (C2×C10).295C24, (C22×C4).379D10, C10.142(C22×D4), C23.D5⋊61C22, (C23×C10).76C22, C23.337(C22×D5), C22.308(C23×D5), C22.79(D4⋊2D5), (C22×C20).275C22, (C22×C10).419C23, (C2×Dic5).152C23, (C22×D5).249C23, (C23×D5).125C22, (C22×Dic5).163C22, (D4×C2×C10)⋊4C2, C5⋊5(C2×C4⋊D4), C4⋊3(C2×C5⋊D4), C2.102(C2×D4×D5), (D5×C22×C4)⋊6C2, (C2×C4×D5)⋊57C22, (C2×C4)⋊13(C5⋊D4), (C2×C4⋊Dic5)⋊45C2, C10.105(C2×C4○D4), C2.69(C2×D4⋊2D5), (C2×C10).580(C2×D4), (C2×C5⋊D4)⋊44C22, (C22×C5⋊D4)⋊13C2, (C2×C23.D5)⋊28C2, C2.15(C22×C5⋊D4), (C2×C4).625(C22×D5), C22.110(C2×C5⋊D4), (C2×C10).177(C4○D4), SmallGroup(320,1472)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×C20⋊2D4
G = < a,b,c,d | a2=b20=c4=d2=1, ab=ba, ac=ca, ad=da, cbc-1=b-1, dbd=b9, dcd=c-1 >
Subgroups: 1486 in 426 conjugacy classes, 135 normal (21 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C5, C2×C4, C2×C4, D4, C23, C23, C23, D5, C10, C10, C10, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×D4, C24, C24, Dic5, C20, D10, D10, C2×C10, C2×C10, C2×C10, C2×C22⋊C4, C2×C4⋊C4, C4⋊D4, C23×C4, C22×D4, C22×D4, C4×D5, C2×Dic5, C2×Dic5, C5⋊D4, C2×C20, C5×D4, C22×D5, C22×D5, C22×C10, C22×C10, C22×C10, C2×C4⋊D4, C4⋊Dic5, C23.D5, C2×C4×D5, C2×C4×D5, C22×Dic5, C22×Dic5, C2×C5⋊D4, C2×C5⋊D4, C22×C20, D4×C10, D4×C10, C23×D5, C23×C10, C2×C4⋊Dic5, C20⋊2D4, C2×C23.D5, D5×C22×C4, C22×C5⋊D4, D4×C2×C10, C2×C20⋊2D4
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, C4○D4, C24, D10, C4⋊D4, C22×D4, C2×C4○D4, C5⋊D4, C22×D5, C2×C4⋊D4, D4×D5, D4⋊2D5, C2×C5⋊D4, C23×D5, C20⋊2D4, C2×D4×D5, C2×D4⋊2D5, C22×C5⋊D4, C2×C20⋊2D4
►On 160 points
Generators in S160
(1 45)(2 46)(3 47)(4 48)(5 49)(6 50)(7 51)(8 52)(9 53)(10 54)(11 55)(12 56)(13 57)(14 58)(15 59)(16 60)(17 41)(18 42)(19 43)(20 44)(21 133)(22 134)(23 135)(24 136)(25 137)(26 138)(27 139)(28 140)(29 121)(30 122)(31 123)(32 124)(33 125)(34 126)(35 127)(36 128)(37 129)(38 130)(39 131)(40 132)(61 87)(62 88)(63 89)(64 90)(65 91)(66 92)(67 93)(68 94)(69 95)(70 96)(71 97)(72 98)(73 99)(74 100)(75 81)(76 82)(77 83)(78 84)(79 85)(80 86)(101 141)(102 142)(103 143)(104 144)(105 145)(106 146)(107 147)(108 148)(109 149)(110 150)(111 151)(112 152)(113 153)(114 154)(115 155)(116 156)(117 157)(118 158)(119 159)(120 160)
(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)
(1 101 83 133)(2 120 84 132)(3 119 85 131)(4 118 86 130)(5 117 87 129)(6 116 88 128)(7 115 89 127)(8 114 90 126)(9 113 91 125)(10 112 92 124)(11 111 93 123)(12 110 94 122)(13 109 95 121)(14 108 96 140)(15 107 97 139)(16 106 98 138)(17 105 99 137)(18 104 100 136)(19 103 81 135)(20 102 82 134)(21 45 141 77)(22 44 142 76)(23 43 143 75)(24 42 144 74)(25 41 145 73)(26 60 146 72)(27 59 147 71)(28 58 148 70)(29 57 149 69)(30 56 150 68)(31 55 151 67)(32 54 152 66)(33 53 153 65)(34 52 154 64)(35 51 155 63)(36 50 156 62)(37 49 157 61)(38 48 158 80)(39 47 159 79)(40 46 160 78)
(1 55)(2 44)(3 53)(4 42)(5 51)(6 60)(7 49)(8 58)(9 47)(10 56)(11 45)(12 54)(13 43)(14 52)(15 41)(16 50)(17 59)(18 48)(19 57)(20 46)(21 111)(22 120)(23 109)(24 118)(25 107)(26 116)(27 105)(28 114)(29 103)(30 112)(31 101)(32 110)(33 119)(34 108)(35 117)(36 106)(37 115)(38 104)(39 113)(40 102)(61 89)(62 98)(63 87)(64 96)(65 85)(66 94)(67 83)(68 92)(69 81)(70 90)(71 99)(72 88)(73 97)(74 86)(75 95)(76 84)(77 93)(78 82)(79 91)(80 100)(121 143)(122 152)(123 141)(124 150)(125 159)(126 148)(127 157)(128 146)(129 155)(130 144)(131 153)(132 142)(133 151)(134 160)(135 149)(136 158)(137 147)(138 156)(139 145)(140 154)
G:=sub<Sym(160)| (1,45)(2,46)(3,47)(4,48)(5,49)(6,50)(7,51)(8,52)(9,53)(10,54)(11,55)(12,56)(13,57)(14,58)(15,59)(16,60)(17,41)(18,42)(19,43)(20,44)(21,133)(22,134)(23,135)(24,136)(25,137)(26,138)(27,139)(28,140)(29,121)(30,122)(31,123)(32,124)(33,125)(34,126)(35,127)(36,128)(37,129)(38,130)(39,131)(40,132)(61,87)(62,88)(63,89)(64,90)(65,91)(66,92)(67,93)(68,94)(69,95)(70,96)(71,97)(72,98)(73,99)(74,100)(75,81)(76,82)(77,83)(78,84)(79,85)(80,86)(101,141)(102,142)(103,143)(104,144)(105,145)(106,146)(107,147)(108,148)(109,149)(110,150)(111,151)(112,152)(113,153)(114,154)(115,155)(116,156)(117,157)(118,158)(119,159)(120,160), (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), (1,101,83,133)(2,120,84,132)(3,119,85,131)(4,118,86,130)(5,117,87,129)(6,116,88,128)(7,115,89,127)(8,114,90,126)(9,113,91,125)(10,112,92,124)(11,111,93,123)(12,110,94,122)(13,109,95,121)(14,108,96,140)(15,107,97,139)(16,106,98,138)(17,105,99,137)(18,104,100,136)(19,103,81,135)(20,102,82,134)(21,45,141,77)(22,44,142,76)(23,43,143,75)(24,42,144,74)(25,41,145,73)(26,60,146,72)(27,59,147,71)(28,58,148,70)(29,57,149,69)(30,56,150,68)(31,55,151,67)(32,54,152,66)(33,53,153,65)(34,52,154,64)(35,51,155,63)(36,50,156,62)(37,49,157,61)(38,48,158,80)(39,47,159,79)(40,46,160,78), (1,55)(2,44)(3,53)(4,42)(5,51)(6,60)(7,49)(8,58)(9,47)(10,56)(11,45)(12,54)(13,43)(14,52)(15,41)(16,50)(17,59)(18,48)(19,57)(20,46)(21,111)(22,120)(23,109)(24,118)(25,107)(26,116)(27,105)(28,114)(29,103)(30,112)(31,101)(32,110)(33,119)(34,108)(35,117)(36,106)(37,115)(38,104)(39,113)(40,102)(61,89)(62,98)(63,87)(64,96)(65,85)(66,94)(67,83)(68,92)(69,81)(70,90)(71,99)(72,88)(73,97)(74,86)(75,95)(76,84)(77,93)(78,82)(79,91)(80,100)(121,143)(122,152)(123,141)(124,150)(125,159)(126,148)(127,157)(128,146)(129,155)(130,144)(131,153)(132,142)(133,151)(134,160)(135,149)(136,158)(137,147)(138,156)(139,145)(140,154)>;
G:=Group( (1,45)(2,46)(3,47)(4,48)(5,49)(6,50)(7,51)(8,52)(9,53)(10,54)(11,55)(12,56)(13,57)(14,58)(15,59)(16,60)(17,41)(18,42)(19,43)(20,44)(21,133)(22,134)(23,135)(24,136)(25,137)(26,138)(27,139)(28,140)(29,121)(30,122)(31,123)(32,124)(33,125)(34,126)(35,127)(36,128)(37,129)(38,130)(39,131)(40,132)(61,87)(62,88)(63,89)(64,90)(65,91)(66,92)(67,93)(68,94)(69,95)(70,96)(71,97)(72,98)(73,99)(74,100)(75,81)(76,82)(77,83)(78,84)(79,85)(80,86)(101,141)(102,142)(103,143)(104,144)(105,145)(106,146)(107,147)(108,148)(109,149)(110,150)(111,151)(112,152)(113,153)(114,154)(115,155)(116,156)(117,157)(118,158)(119,159)(120,160), (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), (1,101,83,133)(2,120,84,132)(3,119,85,131)(4,118,86,130)(5,117,87,129)(6,116,88,128)(7,115,89,127)(8,114,90,126)(9,113,91,125)(10,112,92,124)(11,111,93,123)(12,110,94,122)(13,109,95,121)(14,108,96,140)(15,107,97,139)(16,106,98,138)(17,105,99,137)(18,104,100,136)(19,103,81,135)(20,102,82,134)(21,45,141,77)(22,44,142,76)(23,43,143,75)(24,42,144,74)(25,41,145,73)(26,60,146,72)(27,59,147,71)(28,58,148,70)(29,57,149,69)(30,56,150,68)(31,55,151,67)(32,54,152,66)(33,53,153,65)(34,52,154,64)(35,51,155,63)(36,50,156,62)(37,49,157,61)(38,48,158,80)(39,47,159,79)(40,46,160,78), (1,55)(2,44)(3,53)(4,42)(5,51)(6,60)(7,49)(8,58)(9,47)(10,56)(11,45)(12,54)(13,43)(14,52)(15,41)(16,50)(17,59)(18,48)(19,57)(20,46)(21,111)(22,120)(23,109)(24,118)(25,107)(26,116)(27,105)(28,114)(29,103)(30,112)(31,101)(32,110)(33,119)(34,108)(35,117)(36,106)(37,115)(38,104)(39,113)(40,102)(61,89)(62,98)(63,87)(64,96)(65,85)(66,94)(67,83)(68,92)(69,81)(70,90)(71,99)(72,88)(73,97)(74,86)(75,95)(76,84)(77,93)(78,82)(79,91)(80,100)(121,143)(122,152)(123,141)(124,150)(125,159)(126,148)(127,157)(128,146)(129,155)(130,144)(131,153)(132,142)(133,151)(134,160)(135,149)(136,158)(137,147)(138,156)(139,145)(140,154) );
G=PermutationGroup([[(1,45),(2,46),(3,47),(4,48),(5,49),(6,50),(7,51),(8,52),(9,53),(10,54),(11,55),(12,56),(13,57),(14,58),(15,59),(16,60),(17,41),(18,42),(19,43),(20,44),(21,133),(22,134),(23,135),(24,136),(25,137),(26,138),(27,139),(28,140),(29,121),(30,122),(31,123),(32,124),(33,125),(34,126),(35,127),(36,128),(37,129),(38,130),(39,131),(40,132),(61,87),(62,88),(63,89),(64,90),(65,91),(66,92),(67,93),(68,94),(69,95),(70,96),(71,97),(72,98),(73,99),(74,100),(75,81),(76,82),(77,83),(78,84),(79,85),(80,86),(101,141),(102,142),(103,143),(104,144),(105,145),(106,146),(107,147),(108,148),(109,149),(110,150),(111,151),(112,152),(113,153),(114,154),(115,155),(116,156),(117,157),(118,158),(119,159),(120,160)], [(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)], [(1,101,83,133),(2,120,84,132),(3,119,85,131),(4,118,86,130),(5,117,87,129),(6,116,88,128),(7,115,89,127),(8,114,90,126),(9,113,91,125),(10,112,92,124),(11,111,93,123),(12,110,94,122),(13,109,95,121),(14,108,96,140),(15,107,97,139),(16,106,98,138),(17,105,99,137),(18,104,100,136),(19,103,81,135),(20,102,82,134),(21,45,141,77),(22,44,142,76),(23,43,143,75),(24,42,144,74),(25,41,145,73),(26,60,146,72),(27,59,147,71),(28,58,148,70),(29,57,149,69),(30,56,150,68),(31,55,151,67),(32,54,152,66),(33,53,153,65),(34,52,154,64),(35,51,155,63),(36,50,156,62),(37,49,157,61),(38,48,158,80),(39,47,159,79),(40,46,160,78)], [(1,55),(2,44),(3,53),(4,42),(5,51),(6,60),(7,49),(8,58),(9,47),(10,56),(11,45),(12,54),(13,43),(14,52),(15,41),(16,50),(17,59),(18,48),(19,57),(20,46),(21,111),(22,120),(23,109),(24,118),(25,107),(26,116),(27,105),(28,114),(29,103),(30,112),(31,101),(32,110),(33,119),(34,108),(35,117),(36,106),(37,115),(38,104),(39,113),(40,102),(61,89),(62,98),(63,87),(64,96),(65,85),(66,94),(67,83),(68,92),(69,81),(70,90),(71,99),(72,88),(73,97),(74,86),(75,95),(76,84),(77,93),(78,82),(79,91),(80,100),(121,143),(122,152),(123,141),(124,150),(125,159),(126,148),(127,157),(128,146),(129,155),(130,144),(131,153),(132,142),(133,151),(134,160),(135,149),(136,158),(137,147),(138,156),(139,145),(140,154)]])
68 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 2L | 2M | 2N | 2O | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 5A | 5B | 10A | ··· | 10N | 10O | ··· | 10AD | 20A | ··· | 20H |
| order | 1 | 2 | ··· | 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 |
| size | 1 | 1 | ··· | 1 | 4 | 4 | 4 | 4 | 10 | 10 | 10 | 10 | 2 | 2 | 2 | 2 | 10 | 10 | 10 | 10 | 20 | 20 | 20 | 20 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
68 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D5 | C4○D4 | D10 | D10 | D10 | C5⋊D4 | D4×D5 | D4⋊2D5 |
| kernel | C2×C20⋊2D4 | C2×C4⋊Dic5 | C20⋊2D4 | C2×C23.D5 | D5×C22×C4 | C22×C5⋊D4 | D4×C2×C10 | C2×C20 | C22×D5 | C22×D4 | C2×C10 | C22×C4 | C2×D4 | C24 | C2×C4 | C22 | C22 |
| # reps | 1 | 1 | 8 | 2 | 1 | 2 | 1 | 4 | 4 | 2 | 4 | 2 | 8 | 4 | 16 | 4 | 4 |
Matrix representation of C2×C20⋊2D4 ►in GL5(𝔽41)
| 40 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 40 | 0 | 0 | 0 | 0 |
| 0 | 40 | 2 | 0 | 0 |
| 0 | 40 | 1 | 0 | 0 |
| 0 | 0 | 0 | 7 | 40 |
| 0 | 0 | 0 | 1 | 0 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 40 | 2 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 3 | 3 |
| 0 | 0 | 0 | 24 | 38 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 7 | 34 |
| 0 | 0 | 0 | 1 | 34 |
G:=sub<GL(5,GF(41))| [40,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[40,0,0,0,0,0,40,40,0,0,0,2,1,0,0,0,0,0,7,1,0,0,0,40,0],[1,0,0,0,0,0,40,0,0,0,0,2,1,0,0,0,0,0,3,24,0,0,0,3,38],[1,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,7,1,0,0,0,34,34] >;
C2×C20⋊2D4 in GAP, Magma, Sage, TeX
C_2\times C_{20}\rtimes_2D_4 % in TeX
G:=Group("C2xC20:2D4"); // GroupNames label
G:=SmallGroup(320,1472);
// by ID
G=gap.SmallGroup(320,1472);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,184,675,297,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^20=c^4=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^-1,d*b*d=b^9,d*c*d=c^-1>;
// generators/relations