metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: (C2×C4)⋊2D20, (C2×C20)⋊5D4, (C22×D5)⋊3D4, C10.4C22≀C2, (C22×D20)⋊1C2, C5⋊1(C23⋊2D4), C10.1(C4⋊1D4), C2.3(C4⋊D20), C2.7(C4⋊2D20), C22.155(D4×D5), C22.80(C2×D20), (C22×C4).70D10, C2.7(C22⋊D20), C10.35(C4⋊D4), C2.C42⋊10D5, (C23×D5).3C22, (C22×C20).46C22, C23.359(C22×D5), (C22×C10).296C23, C22.44(Q8⋊2D5), (C22×Dic5).18C22, (C2×C10).96(C2×D4), (C2×D10⋊C4)⋊15C2, (C5×C2.C42)⋊8C2, (C2×C10).184(C4○D4), SmallGroup(320,298)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for (C2×C4)⋊2D20
G = < a,b,c,d | a2=b4=c20=d2=1, cbc-1=ab=ba, ac=ca, ad=da, dbd=ab-1, dcd=c-1 >
Subgroups: 1702 in 322 conjugacy classes, 69 normal (12 characteristic)
C1, C2, C2 [×6], C2 [×6], C4 [×7], C22, C22 [×6], C22 [×30], C5, C2×C4 [×6], C2×C4 [×9], D4 [×24], C23, C23 [×24], D5 [×6], C10, C10 [×6], C22⋊C4 [×6], C22×C4 [×3], C22×C4, C2×D4 [×18], C24 [×3], Dic5, C20 [×6], D10 [×30], C2×C10, C2×C10 [×6], C2.C42, C2×C22⋊C4 [×3], C22×D4 [×3], D20 [×24], C2×Dic5 [×3], C2×C20 [×6], C2×C20 [×6], C22×D5 [×6], C22×D5 [×18], C22×C10, C23⋊2D4, D10⋊C4 [×6], C2×D20 [×18], C22×Dic5, C22×C20 [×3], C23×D5 [×3], C5×C2.C42, C2×D10⋊C4 [×3], C22×D20 [×3], (C2×C4)⋊2D20
Quotients: C1, C2 [×7], C22 [×7], D4 [×12], C23, D5, C2×D4 [×6], C4○D4, D10 [×3], C22≀C2 [×3], C4⋊D4 [×3], C4⋊1D4, D20 [×6], C22×D5, C23⋊2D4, C2×D20 [×3], D4×D5 [×3], Q8⋊2D5, C4⋊D20, C22⋊D20 [×3], C4⋊2D20 [×3], (C2×C4)⋊2D20
►On 160 points
Generators in S160
(1 44)(2 45)(3 46)(4 47)(5 48)(6 49)(7 50)(8 51)(9 52)(10 53)(11 54)(12 55)(13 56)(14 57)(15 58)(16 59)(17 60)(18 41)(19 42)(20 43)(21 145)(22 146)(23 147)(24 148)(25 149)(26 150)(27 151)(28 152)(29 153)(30 154)(31 155)(32 156)(33 157)(34 158)(35 159)(36 160)(37 141)(38 142)(39 143)(40 144)(61 95)(62 96)(63 97)(64 98)(65 99)(66 100)(67 81)(68 82)(69 83)(70 84)(71 85)(72 86)(73 87)(74 88)(75 89)(76 90)(77 91)(78 92)(79 93)(80 94)(101 137)(102 138)(103 139)(104 140)(105 121)(106 122)(107 123)(108 124)(109 125)(110 126)(111 127)(112 128)(113 129)(114 130)(115 131)(116 132)(117 133)(118 134)(119 135)(120 136)
(1 22 83 110)(2 147 84 127)(3 24 85 112)(4 149 86 129)(5 26 87 114)(6 151 88 131)(7 28 89 116)(8 153 90 133)(9 30 91 118)(10 155 92 135)(11 32 93 120)(12 157 94 137)(13 34 95 102)(14 159 96 139)(15 36 97 104)(16 141 98 121)(17 38 99 106)(18 143 100 123)(19 40 81 108)(20 145 82 125)(21 68 109 43)(23 70 111 45)(25 72 113 47)(27 74 115 49)(29 76 117 51)(31 78 119 53)(33 80 101 55)(35 62 103 57)(37 64 105 59)(39 66 107 41)(42 144 67 124)(44 146 69 126)(46 148 71 128)(48 150 73 130)(50 152 75 132)(52 154 77 134)(54 156 79 136)(56 158 61 138)(58 160 63 140)(60 142 65 122)
(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 20)(2 19)(3 18)(4 17)(5 16)(6 15)(7 14)(8 13)(9 12)(10 11)(21 110)(22 109)(23 108)(24 107)(25 106)(26 105)(27 104)(28 103)(29 102)(30 101)(31 120)(32 119)(33 118)(34 117)(35 116)(36 115)(37 114)(38 113)(39 112)(40 111)(41 46)(42 45)(43 44)(47 60)(48 59)(49 58)(50 57)(51 56)(52 55)(53 54)(61 76)(62 75)(63 74)(64 73)(65 72)(66 71)(67 70)(68 69)(77 80)(78 79)(81 84)(82 83)(85 100)(86 99)(87 98)(88 97)(89 96)(90 95)(91 94)(92 93)(121 150)(122 149)(123 148)(124 147)(125 146)(126 145)(127 144)(128 143)(129 142)(130 141)(131 160)(132 159)(133 158)(134 157)(135 156)(136 155)(137 154)(138 153)(139 152)(140 151)
G:=sub<Sym(160)| (1,44)(2,45)(3,46)(4,47)(5,48)(6,49)(7,50)(8,51)(9,52)(10,53)(11,54)(12,55)(13,56)(14,57)(15,58)(16,59)(17,60)(18,41)(19,42)(20,43)(21,145)(22,146)(23,147)(24,148)(25,149)(26,150)(27,151)(28,152)(29,153)(30,154)(31,155)(32,156)(33,157)(34,158)(35,159)(36,160)(37,141)(38,142)(39,143)(40,144)(61,95)(62,96)(63,97)(64,98)(65,99)(66,100)(67,81)(68,82)(69,83)(70,84)(71,85)(72,86)(73,87)(74,88)(75,89)(76,90)(77,91)(78,92)(79,93)(80,94)(101,137)(102,138)(103,139)(104,140)(105,121)(106,122)(107,123)(108,124)(109,125)(110,126)(111,127)(112,128)(113,129)(114,130)(115,131)(116,132)(117,133)(118,134)(119,135)(120,136), (1,22,83,110)(2,147,84,127)(3,24,85,112)(4,149,86,129)(5,26,87,114)(6,151,88,131)(7,28,89,116)(8,153,90,133)(9,30,91,118)(10,155,92,135)(11,32,93,120)(12,157,94,137)(13,34,95,102)(14,159,96,139)(15,36,97,104)(16,141,98,121)(17,38,99,106)(18,143,100,123)(19,40,81,108)(20,145,82,125)(21,68,109,43)(23,70,111,45)(25,72,113,47)(27,74,115,49)(29,76,117,51)(31,78,119,53)(33,80,101,55)(35,62,103,57)(37,64,105,59)(39,66,107,41)(42,144,67,124)(44,146,69,126)(46,148,71,128)(48,150,73,130)(50,152,75,132)(52,154,77,134)(54,156,79,136)(56,158,61,138)(58,160,63,140)(60,142,65,122), (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), (1,20)(2,19)(3,18)(4,17)(5,16)(6,15)(7,14)(8,13)(9,12)(10,11)(21,110)(22,109)(23,108)(24,107)(25,106)(26,105)(27,104)(28,103)(29,102)(30,101)(31,120)(32,119)(33,118)(34,117)(35,116)(36,115)(37,114)(38,113)(39,112)(40,111)(41,46)(42,45)(43,44)(47,60)(48,59)(49,58)(50,57)(51,56)(52,55)(53,54)(61,76)(62,75)(63,74)(64,73)(65,72)(66,71)(67,70)(68,69)(77,80)(78,79)(81,84)(82,83)(85,100)(86,99)(87,98)(88,97)(89,96)(90,95)(91,94)(92,93)(121,150)(122,149)(123,148)(124,147)(125,146)(126,145)(127,144)(128,143)(129,142)(130,141)(131,160)(132,159)(133,158)(134,157)(135,156)(136,155)(137,154)(138,153)(139,152)(140,151)>;
G:=Group( (1,44)(2,45)(3,46)(4,47)(5,48)(6,49)(7,50)(8,51)(9,52)(10,53)(11,54)(12,55)(13,56)(14,57)(15,58)(16,59)(17,60)(18,41)(19,42)(20,43)(21,145)(22,146)(23,147)(24,148)(25,149)(26,150)(27,151)(28,152)(29,153)(30,154)(31,155)(32,156)(33,157)(34,158)(35,159)(36,160)(37,141)(38,142)(39,143)(40,144)(61,95)(62,96)(63,97)(64,98)(65,99)(66,100)(67,81)(68,82)(69,83)(70,84)(71,85)(72,86)(73,87)(74,88)(75,89)(76,90)(77,91)(78,92)(79,93)(80,94)(101,137)(102,138)(103,139)(104,140)(105,121)(106,122)(107,123)(108,124)(109,125)(110,126)(111,127)(112,128)(113,129)(114,130)(115,131)(116,132)(117,133)(118,134)(119,135)(120,136), (1,22,83,110)(2,147,84,127)(3,24,85,112)(4,149,86,129)(5,26,87,114)(6,151,88,131)(7,28,89,116)(8,153,90,133)(9,30,91,118)(10,155,92,135)(11,32,93,120)(12,157,94,137)(13,34,95,102)(14,159,96,139)(15,36,97,104)(16,141,98,121)(17,38,99,106)(18,143,100,123)(19,40,81,108)(20,145,82,125)(21,68,109,43)(23,70,111,45)(25,72,113,47)(27,74,115,49)(29,76,117,51)(31,78,119,53)(33,80,101,55)(35,62,103,57)(37,64,105,59)(39,66,107,41)(42,144,67,124)(44,146,69,126)(46,148,71,128)(48,150,73,130)(50,152,75,132)(52,154,77,134)(54,156,79,136)(56,158,61,138)(58,160,63,140)(60,142,65,122), (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), (1,20)(2,19)(3,18)(4,17)(5,16)(6,15)(7,14)(8,13)(9,12)(10,11)(21,110)(22,109)(23,108)(24,107)(25,106)(26,105)(27,104)(28,103)(29,102)(30,101)(31,120)(32,119)(33,118)(34,117)(35,116)(36,115)(37,114)(38,113)(39,112)(40,111)(41,46)(42,45)(43,44)(47,60)(48,59)(49,58)(50,57)(51,56)(52,55)(53,54)(61,76)(62,75)(63,74)(64,73)(65,72)(66,71)(67,70)(68,69)(77,80)(78,79)(81,84)(82,83)(85,100)(86,99)(87,98)(88,97)(89,96)(90,95)(91,94)(92,93)(121,150)(122,149)(123,148)(124,147)(125,146)(126,145)(127,144)(128,143)(129,142)(130,141)(131,160)(132,159)(133,158)(134,157)(135,156)(136,155)(137,154)(138,153)(139,152)(140,151) );
G=PermutationGroup([(1,44),(2,45),(3,46),(4,47),(5,48),(6,49),(7,50),(8,51),(9,52),(10,53),(11,54),(12,55),(13,56),(14,57),(15,58),(16,59),(17,60),(18,41),(19,42),(20,43),(21,145),(22,146),(23,147),(24,148),(25,149),(26,150),(27,151),(28,152),(29,153),(30,154),(31,155),(32,156),(33,157),(34,158),(35,159),(36,160),(37,141),(38,142),(39,143),(40,144),(61,95),(62,96),(63,97),(64,98),(65,99),(66,100),(67,81),(68,82),(69,83),(70,84),(71,85),(72,86),(73,87),(74,88),(75,89),(76,90),(77,91),(78,92),(79,93),(80,94),(101,137),(102,138),(103,139),(104,140),(105,121),(106,122),(107,123),(108,124),(109,125),(110,126),(111,127),(112,128),(113,129),(114,130),(115,131),(116,132),(117,133),(118,134),(119,135),(120,136)], [(1,22,83,110),(2,147,84,127),(3,24,85,112),(4,149,86,129),(5,26,87,114),(6,151,88,131),(7,28,89,116),(8,153,90,133),(9,30,91,118),(10,155,92,135),(11,32,93,120),(12,157,94,137),(13,34,95,102),(14,159,96,139),(15,36,97,104),(16,141,98,121),(17,38,99,106),(18,143,100,123),(19,40,81,108),(20,145,82,125),(21,68,109,43),(23,70,111,45),(25,72,113,47),(27,74,115,49),(29,76,117,51),(31,78,119,53),(33,80,101,55),(35,62,103,57),(37,64,105,59),(39,66,107,41),(42,144,67,124),(44,146,69,126),(46,148,71,128),(48,150,73,130),(50,152,75,132),(52,154,77,134),(54,156,79,136),(56,158,61,138),(58,160,63,140),(60,142,65,122)], [(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)], [(1,20),(2,19),(3,18),(4,17),(5,16),(6,15),(7,14),(8,13),(9,12),(10,11),(21,110),(22,109),(23,108),(24,107),(25,106),(26,105),(27,104),(28,103),(29,102),(30,101),(31,120),(32,119),(33,118),(34,117),(35,116),(36,115),(37,114),(38,113),(39,112),(40,111),(41,46),(42,45),(43,44),(47,60),(48,59),(49,58),(50,57),(51,56),(52,55),(53,54),(61,76),(62,75),(63,74),(64,73),(65,72),(66,71),(67,70),(68,69),(77,80),(78,79),(81,84),(82,83),(85,100),(86,99),(87,98),(88,97),(89,96),(90,95),(91,94),(92,93),(121,150),(122,149),(123,148),(124,147),(125,146),(126,145),(127,144),(128,143),(129,142),(130,141),(131,160),(132,159),(133,158),(134,157),(135,156),(136,155),(137,154),(138,153),(139,152),(140,151)])
62 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | ··· | 2M | 4A | ··· | 4F | 4G | 4H | 5A | 5B | 10A | ··· | 10N | 20A | ··· | 20X |
| order | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | 4 | 5 | 5 | 10 | ··· | 10 | 20 | ··· | 20 |
| size | 1 | 1 | ··· | 1 | 20 | ··· | 20 | 4 | ··· | 4 | 20 | 20 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 |
62 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | |
| image | C1 | C2 | C2 | C2 | D4 | D4 | D5 | C4○D4 | D10 | D20 | D4×D5 | Q8⋊2D5 |
| kernel | (C2×C4)⋊2D20 | C5×C2.C42 | C2×D10⋊C4 | C22×D20 | C2×C20 | C22×D5 | C2.C42 | C2×C10 | C22×C4 | C2×C4 | C22 | C22 |
| # reps | 1 | 1 | 3 | 3 | 6 | 6 | 2 | 2 | 6 | 24 | 6 | 2 |
Matrix representation of (C2×C4)⋊2D20 ►in GL6(𝔽41)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 11 | 9 | 0 | 0 |
| 0 | 0 | 32 | 30 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 37 | 40 |
| 28 | 39 | 0 | 0 | 0 | 0 |
| 2 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 1 | 7 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 8 |
| 0 | 0 | 0 | 0 | 14 | 25 |
| 28 | 39 | 0 | 0 | 0 | 0 |
| 2 | 13 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 8 |
| 0 | 0 | 0 | 0 | 4 | 25 |
G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,11,32,0,0,0,0,9,30,0,0,0,0,0,0,1,37,0,0,0,0,0,40],[28,2,0,0,0,0,39,16,0,0,0,0,0,0,0,1,0,0,0,0,40,7,0,0,0,0,0,0,16,14,0,0,0,0,8,25],[28,2,0,0,0,0,39,13,0,0,0,0,0,0,0,40,0,0,0,0,40,0,0,0,0,0,0,0,16,4,0,0,0,0,8,25] >;
(C2×C4)⋊2D20 in GAP, Magma, Sage, TeX
(C_2\times C_4)\rtimes_2D_{20} % in TeX
G:=Group("(C2xC4):2D20"); // GroupNames label
G:=SmallGroup(320,298);
// by ID
G=gap.SmallGroup(320,298);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,253,120,254,387,58,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^4=c^20=d^2=1,c*b*c^-1=a*b=b*a,a*c=c*a,a*d=d*a,d*b*d=a*b^-1,d*c*d=c^-1>;
// generators/relations