direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×C20.23D4, (C2×Q8)⋊29D10, (C22×Q8)⋊6D5, (C2×C20).215D4, C20.259(C2×D4), C10⋊4(C4.4D4), (Q8×C10)⋊36C22, (C2×C10).306C24, (C2×C20).646C23, (C4×Dic5)⋊69C22, (C22×D20).20C2, (C22×C4).385D10, C10.154(C22×D4), D10⋊C4⋊74C22, (C2×D20).286C22, (C23×D5).78C22, C23.342(C22×D5), C22.317(C23×D5), (C22×C20).439C22, (C22×C10).424C23, C22.40(Q8⋊2D5), (C2×Dic5).299C23, (C22×D5).133C23, (C22×Dic5).257C22, (Q8×C2×C10)⋊5C2, C5⋊5(C2×C4.4D4), (C2×C4×Dic5)⋊13C2, C4.28(C2×C5⋊D4), C10.128(C2×C4○D4), (C2×C10).589(C2×D4), C2.35(C2×Q8⋊2D5), (C2×D10⋊C4)⋊43C2, C2.27(C22×C5⋊D4), (C2×C4).157(C5⋊D4), (C2×C4).243(C22×D5), C22.117(C2×C5⋊D4), (C2×C10).201(C4○D4), SmallGroup(320,1486)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×C20.23D4
G = < a,b,c,d | a2=b20=c4=d2=1, ab=ba, ac=ca, ad=da, cbc-1=b9, dbd=b-1, dcd=b10c-1 >
Subgroups: 1278 in 330 conjugacy classes, 127 normal (15 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C5, C2×C4, C2×C4, D4, Q8, C23, C23, D5, C10, C10, C42, C22⋊C4, C22×C4, C22×C4, C22×C4, C2×D4, C2×Q8, C2×Q8, C24, Dic5, C20, C20, D10, C2×C10, C2×C10, C2×C42, C2×C22⋊C4, C4.4D4, C22×D4, C22×Q8, D20, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C5×Q8, C22×D5, C22×D5, C22×C10, C2×C4.4D4, C4×Dic5, D10⋊C4, C2×D20, C2×D20, C22×Dic5, C22×C20, C22×C20, Q8×C10, Q8×C10, C23×D5, C2×C4×Dic5, C2×D10⋊C4, C20.23D4, C22×D20, Q8×C2×C10, C2×C20.23D4
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, C4○D4, C24, D10, C4.4D4, C22×D4, C2×C4○D4, C5⋊D4, C22×D5, C2×C4.4D4, Q8⋊2D5, C2×C5⋊D4, C23×D5, C20.23D4, C2×Q8⋊2D5, C22×C5⋊D4, C2×C20.23D4
►On 160 points
Generators in S160
(1 136)(2 137)(3 138)(4 139)(5 140)(6 121)(7 122)(8 123)(9 124)(10 125)(11 126)(12 127)(13 128)(14 129)(15 130)(16 131)(17 132)(18 133)(19 134)(20 135)(21 61)(22 62)(23 63)(24 64)(25 65)(26 66)(27 67)(28 68)(29 69)(30 70)(31 71)(32 72)(33 73)(34 74)(35 75)(36 76)(37 77)(38 78)(39 79)(40 80)(41 95)(42 96)(43 97)(44 98)(45 99)(46 100)(47 81)(48 82)(49 83)(50 84)(51 85)(52 86)(53 87)(54 88)(55 89)(56 90)(57 91)(58 92)(59 93)(60 94)(101 144)(102 145)(103 146)(104 147)(105 148)(106 149)(107 150)(108 151)(109 152)(110 153)(111 154)(112 155)(113 156)(114 157)(115 158)(116 159)(117 160)(118 141)(119 142)(120 143)
(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 49 120 67)(2 58 101 76)(3 47 102 65)(4 56 103 74)(5 45 104 63)(6 54 105 72)(7 43 106 61)(8 52 107 70)(9 41 108 79)(10 50 109 68)(11 59 110 77)(12 48 111 66)(13 57 112 75)(14 46 113 64)(15 55 114 73)(16 44 115 62)(17 53 116 71)(18 42 117 80)(19 51 118 69)(20 60 119 78)(21 122 97 149)(22 131 98 158)(23 140 99 147)(24 129 100 156)(25 138 81 145)(26 127 82 154)(27 136 83 143)(28 125 84 152)(29 134 85 141)(30 123 86 150)(31 132 87 159)(32 121 88 148)(33 130 89 157)(34 139 90 146)(35 128 91 155)(36 137 92 144)(37 126 93 153)(38 135 94 142)(39 124 95 151)(40 133 96 160)
(1 126)(2 125)(3 124)(4 123)(5 122)(6 121)(7 140)(8 139)(9 138)(10 137)(11 136)(12 135)(13 134)(14 133)(15 132)(16 131)(17 130)(18 129)(19 128)(20 127)(21 55)(22 54)(23 53)(24 52)(25 51)(26 50)(27 49)(28 48)(29 47)(30 46)(31 45)(32 44)(33 43)(34 42)(35 41)(36 60)(37 59)(38 58)(39 57)(40 56)(61 89)(62 88)(63 87)(64 86)(65 85)(66 84)(67 83)(68 82)(69 81)(70 100)(71 99)(72 98)(73 97)(74 96)(75 95)(76 94)(77 93)(78 92)(79 91)(80 90)(101 152)(102 151)(103 150)(104 149)(105 148)(106 147)(107 146)(108 145)(109 144)(110 143)(111 142)(112 141)(113 160)(114 159)(115 158)(116 157)(117 156)(118 155)(119 154)(120 153)
G:=sub<Sym(160)| (1,136)(2,137)(3,138)(4,139)(5,140)(6,121)(7,122)(8,123)(9,124)(10,125)(11,126)(12,127)(13,128)(14,129)(15,130)(16,131)(17,132)(18,133)(19,134)(20,135)(21,61)(22,62)(23,63)(24,64)(25,65)(26,66)(27,67)(28,68)(29,69)(30,70)(31,71)(32,72)(33,73)(34,74)(35,75)(36,76)(37,77)(38,78)(39,79)(40,80)(41,95)(42,96)(43,97)(44,98)(45,99)(46,100)(47,81)(48,82)(49,83)(50,84)(51,85)(52,86)(53,87)(54,88)(55,89)(56,90)(57,91)(58,92)(59,93)(60,94)(101,144)(102,145)(103,146)(104,147)(105,148)(106,149)(107,150)(108,151)(109,152)(110,153)(111,154)(112,155)(113,156)(114,157)(115,158)(116,159)(117,160)(118,141)(119,142)(120,143), (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), (1,49,120,67)(2,58,101,76)(3,47,102,65)(4,56,103,74)(5,45,104,63)(6,54,105,72)(7,43,106,61)(8,52,107,70)(9,41,108,79)(10,50,109,68)(11,59,110,77)(12,48,111,66)(13,57,112,75)(14,46,113,64)(15,55,114,73)(16,44,115,62)(17,53,116,71)(18,42,117,80)(19,51,118,69)(20,60,119,78)(21,122,97,149)(22,131,98,158)(23,140,99,147)(24,129,100,156)(25,138,81,145)(26,127,82,154)(27,136,83,143)(28,125,84,152)(29,134,85,141)(30,123,86,150)(31,132,87,159)(32,121,88,148)(33,130,89,157)(34,139,90,146)(35,128,91,155)(36,137,92,144)(37,126,93,153)(38,135,94,142)(39,124,95,151)(40,133,96,160), (1,126)(2,125)(3,124)(4,123)(5,122)(6,121)(7,140)(8,139)(9,138)(10,137)(11,136)(12,135)(13,134)(14,133)(15,132)(16,131)(17,130)(18,129)(19,128)(20,127)(21,55)(22,54)(23,53)(24,52)(25,51)(26,50)(27,49)(28,48)(29,47)(30,46)(31,45)(32,44)(33,43)(34,42)(35,41)(36,60)(37,59)(38,58)(39,57)(40,56)(61,89)(62,88)(63,87)(64,86)(65,85)(66,84)(67,83)(68,82)(69,81)(70,100)(71,99)(72,98)(73,97)(74,96)(75,95)(76,94)(77,93)(78,92)(79,91)(80,90)(101,152)(102,151)(103,150)(104,149)(105,148)(106,147)(107,146)(108,145)(109,144)(110,143)(111,142)(112,141)(113,160)(114,159)(115,158)(116,157)(117,156)(118,155)(119,154)(120,153)>;
G:=Group( (1,136)(2,137)(3,138)(4,139)(5,140)(6,121)(7,122)(8,123)(9,124)(10,125)(11,126)(12,127)(13,128)(14,129)(15,130)(16,131)(17,132)(18,133)(19,134)(20,135)(21,61)(22,62)(23,63)(24,64)(25,65)(26,66)(27,67)(28,68)(29,69)(30,70)(31,71)(32,72)(33,73)(34,74)(35,75)(36,76)(37,77)(38,78)(39,79)(40,80)(41,95)(42,96)(43,97)(44,98)(45,99)(46,100)(47,81)(48,82)(49,83)(50,84)(51,85)(52,86)(53,87)(54,88)(55,89)(56,90)(57,91)(58,92)(59,93)(60,94)(101,144)(102,145)(103,146)(104,147)(105,148)(106,149)(107,150)(108,151)(109,152)(110,153)(111,154)(112,155)(113,156)(114,157)(115,158)(116,159)(117,160)(118,141)(119,142)(120,143), (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), (1,49,120,67)(2,58,101,76)(3,47,102,65)(4,56,103,74)(5,45,104,63)(6,54,105,72)(7,43,106,61)(8,52,107,70)(9,41,108,79)(10,50,109,68)(11,59,110,77)(12,48,111,66)(13,57,112,75)(14,46,113,64)(15,55,114,73)(16,44,115,62)(17,53,116,71)(18,42,117,80)(19,51,118,69)(20,60,119,78)(21,122,97,149)(22,131,98,158)(23,140,99,147)(24,129,100,156)(25,138,81,145)(26,127,82,154)(27,136,83,143)(28,125,84,152)(29,134,85,141)(30,123,86,150)(31,132,87,159)(32,121,88,148)(33,130,89,157)(34,139,90,146)(35,128,91,155)(36,137,92,144)(37,126,93,153)(38,135,94,142)(39,124,95,151)(40,133,96,160), (1,126)(2,125)(3,124)(4,123)(5,122)(6,121)(7,140)(8,139)(9,138)(10,137)(11,136)(12,135)(13,134)(14,133)(15,132)(16,131)(17,130)(18,129)(19,128)(20,127)(21,55)(22,54)(23,53)(24,52)(25,51)(26,50)(27,49)(28,48)(29,47)(30,46)(31,45)(32,44)(33,43)(34,42)(35,41)(36,60)(37,59)(38,58)(39,57)(40,56)(61,89)(62,88)(63,87)(64,86)(65,85)(66,84)(67,83)(68,82)(69,81)(70,100)(71,99)(72,98)(73,97)(74,96)(75,95)(76,94)(77,93)(78,92)(79,91)(80,90)(101,152)(102,151)(103,150)(104,149)(105,148)(106,147)(107,146)(108,145)(109,144)(110,143)(111,142)(112,141)(113,160)(114,159)(115,158)(116,157)(117,156)(118,155)(119,154)(120,153) );
G=PermutationGroup([[(1,136),(2,137),(3,138),(4,139),(5,140),(6,121),(7,122),(8,123),(9,124),(10,125),(11,126),(12,127),(13,128),(14,129),(15,130),(16,131),(17,132),(18,133),(19,134),(20,135),(21,61),(22,62),(23,63),(24,64),(25,65),(26,66),(27,67),(28,68),(29,69),(30,70),(31,71),(32,72),(33,73),(34,74),(35,75),(36,76),(37,77),(38,78),(39,79),(40,80),(41,95),(42,96),(43,97),(44,98),(45,99),(46,100),(47,81),(48,82),(49,83),(50,84),(51,85),(52,86),(53,87),(54,88),(55,89),(56,90),(57,91),(58,92),(59,93),(60,94),(101,144),(102,145),(103,146),(104,147),(105,148),(106,149),(107,150),(108,151),(109,152),(110,153),(111,154),(112,155),(113,156),(114,157),(115,158),(116,159),(117,160),(118,141),(119,142),(120,143)], [(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)], [(1,49,120,67),(2,58,101,76),(3,47,102,65),(4,56,103,74),(5,45,104,63),(6,54,105,72),(7,43,106,61),(8,52,107,70),(9,41,108,79),(10,50,109,68),(11,59,110,77),(12,48,111,66),(13,57,112,75),(14,46,113,64),(15,55,114,73),(16,44,115,62),(17,53,116,71),(18,42,117,80),(19,51,118,69),(20,60,119,78),(21,122,97,149),(22,131,98,158),(23,140,99,147),(24,129,100,156),(25,138,81,145),(26,127,82,154),(27,136,83,143),(28,125,84,152),(29,134,85,141),(30,123,86,150),(31,132,87,159),(32,121,88,148),(33,130,89,157),(34,139,90,146),(35,128,91,155),(36,137,92,144),(37,126,93,153),(38,135,94,142),(39,124,95,151),(40,133,96,160)], [(1,126),(2,125),(3,124),(4,123),(5,122),(6,121),(7,140),(8,139),(9,138),(10,137),(11,136),(12,135),(13,134),(14,133),(15,132),(16,131),(17,130),(18,129),(19,128),(20,127),(21,55),(22,54),(23,53),(24,52),(25,51),(26,50),(27,49),(28,48),(29,47),(30,46),(31,45),(32,44),(33,43),(34,42),(35,41),(36,60),(37,59),(38,58),(39,57),(40,56),(61,89),(62,88),(63,87),(64,86),(65,85),(66,84),(67,83),(68,82),(69,81),(70,100),(71,99),(72,98),(73,97),(74,96),(75,95),(76,94),(77,93),(78,92),(79,91),(80,90),(101,152),(102,151),(103,150),(104,149),(105,148),(106,147),(107,146),(108,145),(109,144),(110,143),(111,142),(112,141),(113,160),(114,159),(115,158),(116,157),(117,156),(118,155),(119,154),(120,153)]])
68 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | ··· | 4P | 5A | 5B | 10A | ··· | 10N | 20A | ··· | 20X |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 5 | 5 | 10 | ··· | 10 | 20 | ··· | 20 |
| size | 1 | 1 | ··· | 1 | 20 | 20 | 20 | 20 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 10 | ··· | 10 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 |
68 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | D5 | C4○D4 | D10 | D10 | C5⋊D4 | Q8⋊2D5 |
| kernel | C2×C20.23D4 | C2×C4×Dic5 | C2×D10⋊C4 | C20.23D4 | C22×D20 | Q8×C2×C10 | C2×C20 | C22×Q8 | C2×C10 | C22×C4 | C2×Q8 | C2×C4 | C22 |
| # reps | 1 | 1 | 4 | 8 | 1 | 1 | 4 | 2 | 8 | 6 | 8 | 16 | 8 |
Matrix representation of C2×C20.23D4 ►in GL6(𝔽41)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 34 | 33 | 0 | 0 | 0 | 0 |
| 1 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 6 | 40 | 0 | 0 |
| 0 | 0 | 36 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 32 |
| 0 | 0 | 0 | 0 | 32 | 0 |
| 38 | 37 | 0 | 0 | 0 | 0 |
| 23 | 3 | 0 | 0 | 0 | 0 |
| 0 | 0 | 20 | 3 | 0 | 0 |
| 0 | 0 | 3 | 21 | 0 | 0 |
| 0 | 0 | 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 0 | 0 | 9 |
| 7 | 7 | 0 | 0 | 0 | 0 |
| 40 | 34 | 0 | 0 | 0 | 0 |
| 0 | 0 | 35 | 34 | 0 | 0 |
| 0 | 0 | 5 | 6 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[34,1,0,0,0,0,33,1,0,0,0,0,0,0,6,36,0,0,0,0,40,1,0,0,0,0,0,0,0,32,0,0,0,0,32,0],[38,23,0,0,0,0,37,3,0,0,0,0,0,0,20,3,0,0,0,0,3,21,0,0,0,0,0,0,9,0,0,0,0,0,0,9],[7,40,0,0,0,0,7,34,0,0,0,0,0,0,35,5,0,0,0,0,34,6,0,0,0,0,0,0,1,0,0,0,0,0,0,40] >;
C2×C20.23D4 in GAP, Magma, Sage, TeX
C_2\times C_{20}._{23}D_4 % in TeX
G:=Group("C2xC20.23D4"); // GroupNames label
G:=SmallGroup(320,1486);
// by ID
G=gap.SmallGroup(320,1486);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,758,184,675,297,136,12550]);
// 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^9,d*b*d=b^-1,d*c*d=b^10*c^-1>;
// generators/relations