metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: (C2×C4).21D20, (C2×C20).32D4, C2.9(C4⋊D20), (C22×D5).16D4, C22.83(C2×D20), (C22×C4).19D10, C22.158(D4×D5), C10.36(C4⋊D4), C10.3(C4.4D4), C2.8(C4.D20), C2.C42⋊13D5, C5⋊1(C23.11D4), (C23×D5).6C22, C10.10C42⋊6C2, C22.91(C4○D20), (C22×C20).18C22, C23.362(C22×D5), C10.22(C42⋊2C2), C22.89(D4⋊2D5), (C22×C10).299C23, C22.46(Q8⋊2D5), C2.8(C22.D20), C2.10(D10.12D4), C10.11(C22.D4), (C22×Dic5).21C22, (C2×C4⋊Dic5)⋊3C2, (C2×C10).97(C2×D4), C2.10(C4⋊C4⋊D5), (C2×D10⋊C4).9C2, (C2×C10).185(C4○D4), (C5×C2.C42)⋊11C2, SmallGroup(320,301)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for (C2×C4).21D20
G = < a,b,c,d | a2=b4=c20=1, d2=ab2, cbc-1=ab=ba, ac=ca, ad=da, dbd-1=b-1, dcd-1=ab2c-1 >
Subgroups: 742 in 170 conjugacy classes, 57 normal (25 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C5, C2×C4, C2×C4, C23, C23, D5, C10, C10, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C22×C4, C24, Dic5, C20, D10, C2×C10, C2×C10, C2.C42, C2.C42, C2×C22⋊C4, C2×C4⋊C4, C2×Dic5, C2×C20, C2×C20, C22×D5, C22×D5, C22×C10, C23.11D4, C4⋊Dic5, D10⋊C4, C22×Dic5, C22×Dic5, C22×C20, C22×C20, C23×D5, C10.10C42, C5×C2.C42, C2×C4⋊Dic5, C2×D10⋊C4, C2×D10⋊C4, (C2×C4).21D20
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, C4○D4, D10, C4⋊D4, C22.D4, C4.4D4, C42⋊2C2, D20, C22×D5, C23.11D4, C2×D20, C4○D20, D4×D5, D4⋊2D5, Q8⋊2D5, C4.D20, D10.12D4, C22.D20, C4⋊D20, C4⋊C4⋊D5, (C2×C4).21D20
►On 160 points
Generators in S160
(1 67)(2 68)(3 69)(4 70)(5 71)(6 72)(7 73)(8 74)(9 75)(10 76)(11 77)(12 78)(13 79)(14 80)(15 61)(16 62)(17 63)(18 64)(19 65)(20 66)(21 52)(22 53)(23 54)(24 55)(25 56)(26 57)(27 58)(28 59)(29 60)(30 41)(31 42)(32 43)(33 44)(34 45)(35 46)(36 47)(37 48)(38 49)(39 50)(40 51)(81 150)(82 151)(83 152)(84 153)(85 154)(86 155)(87 156)(88 157)(89 158)(90 159)(91 160)(92 141)(93 142)(94 143)(95 144)(96 145)(97 146)(98 147)(99 148)(100 149)(101 139)(102 140)(103 121)(104 122)(105 123)(106 124)(107 125)(108 126)(109 127)(110 128)(111 129)(112 130)(113 131)(114 132)(115 133)(116 134)(117 135)(118 136)(119 137)(120 138)
(1 125 38 88)(2 108 39 158)(3 127 40 90)(4 110 21 160)(5 129 22 92)(6 112 23 142)(7 131 24 94)(8 114 25 144)(9 133 26 96)(10 116 27 146)(11 135 28 98)(12 118 29 148)(13 137 30 100)(14 120 31 150)(15 139 32 82)(16 102 33 152)(17 121 34 84)(18 104 35 154)(19 123 36 86)(20 106 37 156)(41 149 79 119)(42 81 80 138)(43 151 61 101)(44 83 62 140)(45 153 63 103)(46 85 64 122)(47 155 65 105)(48 87 66 124)(49 157 67 107)(50 89 68 126)(51 159 69 109)(52 91 70 128)(53 141 71 111)(54 93 72 130)(55 143 73 113)(56 95 74 132)(57 145 75 115)(58 97 76 134)(59 147 77 117)(60 99 78 136)
(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 48 49 20)(2 19 50 47)(3 46 51 18)(4 17 52 45)(5 44 53 16)(6 15 54 43)(7 42 55 14)(8 13 56 41)(9 60 57 12)(10 11 58 59)(21 34 70 63)(22 62 71 33)(23 32 72 61)(24 80 73 31)(25 30 74 79)(26 78 75 29)(27 28 76 77)(35 40 64 69)(36 68 65 39)(37 38 66 67)(81 113 120 94)(82 93 101 112)(83 111 102 92)(84 91 103 110)(85 109 104 90)(86 89 105 108)(87 107 106 88)(95 119 114 100)(96 99 115 118)(97 117 116 98)(121 128 153 160)(122 159 154 127)(123 126 155 158)(124 157 156 125)(129 140 141 152)(130 151 142 139)(131 138 143 150)(132 149 144 137)(133 136 145 148)(134 147 146 135)
G:=sub<Sym(160)| (1,67)(2,68)(3,69)(4,70)(5,71)(6,72)(7,73)(8,74)(9,75)(10,76)(11,77)(12,78)(13,79)(14,80)(15,61)(16,62)(17,63)(18,64)(19,65)(20,66)(21,52)(22,53)(23,54)(24,55)(25,56)(26,57)(27,58)(28,59)(29,60)(30,41)(31,42)(32,43)(33,44)(34,45)(35,46)(36,47)(37,48)(38,49)(39,50)(40,51)(81,150)(82,151)(83,152)(84,153)(85,154)(86,155)(87,156)(88,157)(89,158)(90,159)(91,160)(92,141)(93,142)(94,143)(95,144)(96,145)(97,146)(98,147)(99,148)(100,149)(101,139)(102,140)(103,121)(104,122)(105,123)(106,124)(107,125)(108,126)(109,127)(110,128)(111,129)(112,130)(113,131)(114,132)(115,133)(116,134)(117,135)(118,136)(119,137)(120,138), (1,125,38,88)(2,108,39,158)(3,127,40,90)(4,110,21,160)(5,129,22,92)(6,112,23,142)(7,131,24,94)(8,114,25,144)(9,133,26,96)(10,116,27,146)(11,135,28,98)(12,118,29,148)(13,137,30,100)(14,120,31,150)(15,139,32,82)(16,102,33,152)(17,121,34,84)(18,104,35,154)(19,123,36,86)(20,106,37,156)(41,149,79,119)(42,81,80,138)(43,151,61,101)(44,83,62,140)(45,153,63,103)(46,85,64,122)(47,155,65,105)(48,87,66,124)(49,157,67,107)(50,89,68,126)(51,159,69,109)(52,91,70,128)(53,141,71,111)(54,93,72,130)(55,143,73,113)(56,95,74,132)(57,145,75,115)(58,97,76,134)(59,147,77,117)(60,99,78,136), (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), (1,48,49,20)(2,19,50,47)(3,46,51,18)(4,17,52,45)(5,44,53,16)(6,15,54,43)(7,42,55,14)(8,13,56,41)(9,60,57,12)(10,11,58,59)(21,34,70,63)(22,62,71,33)(23,32,72,61)(24,80,73,31)(25,30,74,79)(26,78,75,29)(27,28,76,77)(35,40,64,69)(36,68,65,39)(37,38,66,67)(81,113,120,94)(82,93,101,112)(83,111,102,92)(84,91,103,110)(85,109,104,90)(86,89,105,108)(87,107,106,88)(95,119,114,100)(96,99,115,118)(97,117,116,98)(121,128,153,160)(122,159,154,127)(123,126,155,158)(124,157,156,125)(129,140,141,152)(130,151,142,139)(131,138,143,150)(132,149,144,137)(133,136,145,148)(134,147,146,135)>;
G:=Group( (1,67)(2,68)(3,69)(4,70)(5,71)(6,72)(7,73)(8,74)(9,75)(10,76)(11,77)(12,78)(13,79)(14,80)(15,61)(16,62)(17,63)(18,64)(19,65)(20,66)(21,52)(22,53)(23,54)(24,55)(25,56)(26,57)(27,58)(28,59)(29,60)(30,41)(31,42)(32,43)(33,44)(34,45)(35,46)(36,47)(37,48)(38,49)(39,50)(40,51)(81,150)(82,151)(83,152)(84,153)(85,154)(86,155)(87,156)(88,157)(89,158)(90,159)(91,160)(92,141)(93,142)(94,143)(95,144)(96,145)(97,146)(98,147)(99,148)(100,149)(101,139)(102,140)(103,121)(104,122)(105,123)(106,124)(107,125)(108,126)(109,127)(110,128)(111,129)(112,130)(113,131)(114,132)(115,133)(116,134)(117,135)(118,136)(119,137)(120,138), (1,125,38,88)(2,108,39,158)(3,127,40,90)(4,110,21,160)(5,129,22,92)(6,112,23,142)(7,131,24,94)(8,114,25,144)(9,133,26,96)(10,116,27,146)(11,135,28,98)(12,118,29,148)(13,137,30,100)(14,120,31,150)(15,139,32,82)(16,102,33,152)(17,121,34,84)(18,104,35,154)(19,123,36,86)(20,106,37,156)(41,149,79,119)(42,81,80,138)(43,151,61,101)(44,83,62,140)(45,153,63,103)(46,85,64,122)(47,155,65,105)(48,87,66,124)(49,157,67,107)(50,89,68,126)(51,159,69,109)(52,91,70,128)(53,141,71,111)(54,93,72,130)(55,143,73,113)(56,95,74,132)(57,145,75,115)(58,97,76,134)(59,147,77,117)(60,99,78,136), (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), (1,48,49,20)(2,19,50,47)(3,46,51,18)(4,17,52,45)(5,44,53,16)(6,15,54,43)(7,42,55,14)(8,13,56,41)(9,60,57,12)(10,11,58,59)(21,34,70,63)(22,62,71,33)(23,32,72,61)(24,80,73,31)(25,30,74,79)(26,78,75,29)(27,28,76,77)(35,40,64,69)(36,68,65,39)(37,38,66,67)(81,113,120,94)(82,93,101,112)(83,111,102,92)(84,91,103,110)(85,109,104,90)(86,89,105,108)(87,107,106,88)(95,119,114,100)(96,99,115,118)(97,117,116,98)(121,128,153,160)(122,159,154,127)(123,126,155,158)(124,157,156,125)(129,140,141,152)(130,151,142,139)(131,138,143,150)(132,149,144,137)(133,136,145,148)(134,147,146,135) );
G=PermutationGroup([[(1,67),(2,68),(3,69),(4,70),(5,71),(6,72),(7,73),(8,74),(9,75),(10,76),(11,77),(12,78),(13,79),(14,80),(15,61),(16,62),(17,63),(18,64),(19,65),(20,66),(21,52),(22,53),(23,54),(24,55),(25,56),(26,57),(27,58),(28,59),(29,60),(30,41),(31,42),(32,43),(33,44),(34,45),(35,46),(36,47),(37,48),(38,49),(39,50),(40,51),(81,150),(82,151),(83,152),(84,153),(85,154),(86,155),(87,156),(88,157),(89,158),(90,159),(91,160),(92,141),(93,142),(94,143),(95,144),(96,145),(97,146),(98,147),(99,148),(100,149),(101,139),(102,140),(103,121),(104,122),(105,123),(106,124),(107,125),(108,126),(109,127),(110,128),(111,129),(112,130),(113,131),(114,132),(115,133),(116,134),(117,135),(118,136),(119,137),(120,138)], [(1,125,38,88),(2,108,39,158),(3,127,40,90),(4,110,21,160),(5,129,22,92),(6,112,23,142),(7,131,24,94),(8,114,25,144),(9,133,26,96),(10,116,27,146),(11,135,28,98),(12,118,29,148),(13,137,30,100),(14,120,31,150),(15,139,32,82),(16,102,33,152),(17,121,34,84),(18,104,35,154),(19,123,36,86),(20,106,37,156),(41,149,79,119),(42,81,80,138),(43,151,61,101),(44,83,62,140),(45,153,63,103),(46,85,64,122),(47,155,65,105),(48,87,66,124),(49,157,67,107),(50,89,68,126),(51,159,69,109),(52,91,70,128),(53,141,71,111),(54,93,72,130),(55,143,73,113),(56,95,74,132),(57,145,75,115),(58,97,76,134),(59,147,77,117),(60,99,78,136)], [(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)], [(1,48,49,20),(2,19,50,47),(3,46,51,18),(4,17,52,45),(5,44,53,16),(6,15,54,43),(7,42,55,14),(8,13,56,41),(9,60,57,12),(10,11,58,59),(21,34,70,63),(22,62,71,33),(23,32,72,61),(24,80,73,31),(25,30,74,79),(26,78,75,29),(27,28,76,77),(35,40,64,69),(36,68,65,39),(37,38,66,67),(81,113,120,94),(82,93,101,112),(83,111,102,92),(84,91,103,110),(85,109,104,90),(86,89,105,108),(87,107,106,88),(95,119,114,100),(96,99,115,118),(97,117,116,98),(121,128,153,160),(122,159,154,127),(123,126,155,158),(124,157,156,125),(129,140,141,152),(130,151,142,139),(131,138,143,150),(132,149,144,137),(133,136,145,148),(134,147,146,135)]])
62 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 4A | ··· | 4F | 4G | ··· | 4L | 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 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | - | + | ||
| image | C1 | C2 | C2 | C2 | C2 | D4 | D4 | D5 | C4○D4 | D10 | D20 | C4○D20 | D4×D5 | D4⋊2D5 | Q8⋊2D5 |
| kernel | (C2×C4).21D20 | C10.10C42 | C5×C2.C42 | C2×C4⋊Dic5 | C2×D10⋊C4 | C2×C20 | C22×D5 | C2.C42 | C2×C10 | C22×C4 | C2×C4 | C22 | C22 | C22 | C22 |
| # reps | 1 | 2 | 1 | 1 | 3 | 2 | 2 | 2 | 10 | 6 | 8 | 16 | 2 | 4 | 2 |
Matrix representation of (C2×C4).21D20 ►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 | 30 | 32 | 0 | 0 |
| 0 | 0 | 9 | 11 | 0 | 0 |
| 0 | 0 | 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 0 | 0 | 32 |
| 32 | 30 | 0 | 0 | 0 | 0 |
| 11 | 27 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 32 | 0 | 0 |
| 0 | 0 | 9 | 22 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 32 |
| 0 | 0 | 0 | 0 | 32 | 0 |
| 9 | 11 | 0 | 0 | 0 | 0 |
| 30 | 32 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 32 | 0 | 0 |
| 0 | 0 | 32 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 32 |
| 0 | 0 | 0 | 0 | 9 | 0 |
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],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,30,9,0,0,0,0,32,11,0,0,0,0,0,0,9,0,0,0,0,0,0,32],[32,11,0,0,0,0,30,27,0,0,0,0,0,0,0,9,0,0,0,0,32,22,0,0,0,0,0,0,0,32,0,0,0,0,32,0],[9,30,0,0,0,0,11,32,0,0,0,0,0,0,0,32,0,0,0,0,32,0,0,0,0,0,0,0,0,9,0,0,0,0,32,0] >;
(C2×C4).21D20 in GAP, Magma, Sage, TeX
(C_2\times C_4)._{21}D_{20} % in TeX
G:=Group("(C2xC4).21D20"); // GroupNames label
G:=SmallGroup(320,301);
// by ID
G=gap.SmallGroup(320,301);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,253,64,254,387,100,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^4=c^20=1,d^2=a*b^2,c*b*c^-1=a*b=b*a,a*c=c*a,a*d=d*a,d*b*d^-1=b^-1,d*c*d^-1=a*b^2*c^-1>;
// generators/relations