metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C22.4D40, C23.39D20, C22⋊C8⋊4D5, C40⋊5C4⋊4C2, (C2×C8).4D10, C2.8(C2×D40), (C2×C10).5D8, C10.6(C2×D8), (C2×C4).34D20, (C2×C20).45D4, D20⋊5C4⋊6C2, C20⋊7D4.3C2, (C2×C40).4C22, (C22×C10).56D4, (C22×C4).86D10, C5⋊1(C22.D8), C20.283(C4○D4), (C2×C20).746C23, (C2×D20).13C22, C22.109(C2×D20), C4.107(D4⋊2D5), C2.13(C8.D10), C10.10(C8.C22), C4⋊Dic5.271C22, (C22×C20).53C22, C2.14(C22.D20), C10.18(C22.D4), (C5×C22⋊C8)⋊6C2, (C2×C4⋊Dic5)⋊6C2, (C2×C10).129(C2×D4), (C2×C4).691(C22×D5), SmallGroup(320,363)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C22.D40
G = < a,b,c,d | a2=b2=c40=1, d2=b, cac-1=ab=ba, ad=da, bc=cb, bd=db, dcd-1=bc-1 >
Subgroups: 542 in 114 conjugacy classes, 43 normal (25 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C5, C8, C2×C4, C2×C4, D4, C23, C23, D5, C10, C10, C22⋊C4, C4⋊C4, C2×C8, C22×C4, C22×C4, C2×D4, Dic5, C20, C20, D10, C2×C10, C2×C10, C2×C10, C22⋊C8, D4⋊C4, C2.D8, C2×C4⋊C4, C4⋊D4, C40, D20, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C22×D5, C22×C10, C22.D8, C4⋊Dic5, C4⋊Dic5, C4⋊Dic5, D10⋊C4, C2×C40, C2×D20, C22×Dic5, C2×C5⋊D4, C22×C20, C40⋊5C4, D20⋊5C4, C5×C22⋊C8, C2×C4⋊Dic5, C20⋊7D4, C22.D40
Quotients: C1, C2, C22, D4, C23, D5, D8, C2×D4, C4○D4, D10, C22.D4, C2×D8, C8.C22, D20, C22×D5, C22.D8, D40, C2×D20, D4⋊2D5, C22.D20, C2×D40, C8.D10, C22.D40
►On 160 points
Generators in S160
(1 100)(2 42)(3 102)(4 44)(5 104)(6 46)(7 106)(8 48)(9 108)(10 50)(11 110)(12 52)(13 112)(14 54)(15 114)(16 56)(17 116)(18 58)(19 118)(20 60)(21 120)(22 62)(23 82)(24 64)(25 84)(26 66)(27 86)(28 68)(29 88)(30 70)(31 90)(32 72)(33 92)(34 74)(35 94)(36 76)(37 96)(38 78)(39 98)(40 80)(41 156)(43 158)(45 160)(47 122)(49 124)(51 126)(53 128)(55 130)(57 132)(59 134)(61 136)(63 138)(65 140)(67 142)(69 144)(71 146)(73 148)(75 150)(77 152)(79 154)(81 137)(83 139)(85 141)(87 143)(89 145)(91 147)(93 149)(95 151)(97 153)(99 155)(101 157)(103 159)(105 121)(107 123)(109 125)(111 127)(113 129)(115 131)(117 133)(119 135)
(1 156)(2 157)(3 158)(4 159)(5 160)(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)(21 136)(22 137)(23 138)(24 139)(25 140)(26 141)(27 142)(28 143)(29 144)(30 145)(31 146)(32 147)(33 148)(34 149)(35 150)(36 151)(37 152)(38 153)(39 154)(40 155)(41 100)(42 101)(43 102)(44 103)(45 104)(46 105)(47 106)(48 107)(49 108)(50 109)(51 110)(52 111)(53 112)(54 113)(55 114)(56 115)(57 116)(58 117)(59 118)(60 119)(61 120)(62 81)(63 82)(64 83)(65 84)(66 85)(67 86)(68 87)(69 88)(70 89)(71 90)(72 91)(73 92)(74 93)(75 94)(76 95)(77 96)(78 97)(79 98)(80 99)
(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)
(1 155 156 40)(2 39 157 154)(3 153 158 38)(4 37 159 152)(5 151 160 36)(6 35 121 150)(7 149 122 34)(8 33 123 148)(9 147 124 32)(10 31 125 146)(11 145 126 30)(12 29 127 144)(13 143 128 28)(14 27 129 142)(15 141 130 26)(16 25 131 140)(17 139 132 24)(18 23 133 138)(19 137 134 22)(20 21 135 136)(41 80 100 99)(42 98 101 79)(43 78 102 97)(44 96 103 77)(45 76 104 95)(46 94 105 75)(47 74 106 93)(48 92 107 73)(49 72 108 91)(50 90 109 71)(51 70 110 89)(52 88 111 69)(53 68 112 87)(54 86 113 67)(55 66 114 85)(56 84 115 65)(57 64 116 83)(58 82 117 63)(59 62 118 81)(60 120 119 61)
G:=sub<Sym(160)| (1,100)(2,42)(3,102)(4,44)(5,104)(6,46)(7,106)(8,48)(9,108)(10,50)(11,110)(12,52)(13,112)(14,54)(15,114)(16,56)(17,116)(18,58)(19,118)(20,60)(21,120)(22,62)(23,82)(24,64)(25,84)(26,66)(27,86)(28,68)(29,88)(30,70)(31,90)(32,72)(33,92)(34,74)(35,94)(36,76)(37,96)(38,78)(39,98)(40,80)(41,156)(43,158)(45,160)(47,122)(49,124)(51,126)(53,128)(55,130)(57,132)(59,134)(61,136)(63,138)(65,140)(67,142)(69,144)(71,146)(73,148)(75,150)(77,152)(79,154)(81,137)(83,139)(85,141)(87,143)(89,145)(91,147)(93,149)(95,151)(97,153)(99,155)(101,157)(103,159)(105,121)(107,123)(109,125)(111,127)(113,129)(115,131)(117,133)(119,135), (1,156)(2,157)(3,158)(4,159)(5,160)(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)(21,136)(22,137)(23,138)(24,139)(25,140)(26,141)(27,142)(28,143)(29,144)(30,145)(31,146)(32,147)(33,148)(34,149)(35,150)(36,151)(37,152)(38,153)(39,154)(40,155)(41,100)(42,101)(43,102)(44,103)(45,104)(46,105)(47,106)(48,107)(49,108)(50,109)(51,110)(52,111)(53,112)(54,113)(55,114)(56,115)(57,116)(58,117)(59,118)(60,119)(61,120)(62,81)(63,82)(64,83)(65,84)(66,85)(67,86)(68,87)(69,88)(70,89)(71,90)(72,91)(73,92)(74,93)(75,94)(76,95)(77,96)(78,97)(79,98)(80,99), (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), (1,155,156,40)(2,39,157,154)(3,153,158,38)(4,37,159,152)(5,151,160,36)(6,35,121,150)(7,149,122,34)(8,33,123,148)(9,147,124,32)(10,31,125,146)(11,145,126,30)(12,29,127,144)(13,143,128,28)(14,27,129,142)(15,141,130,26)(16,25,131,140)(17,139,132,24)(18,23,133,138)(19,137,134,22)(20,21,135,136)(41,80,100,99)(42,98,101,79)(43,78,102,97)(44,96,103,77)(45,76,104,95)(46,94,105,75)(47,74,106,93)(48,92,107,73)(49,72,108,91)(50,90,109,71)(51,70,110,89)(52,88,111,69)(53,68,112,87)(54,86,113,67)(55,66,114,85)(56,84,115,65)(57,64,116,83)(58,82,117,63)(59,62,118,81)(60,120,119,61)>;
G:=Group( (1,100)(2,42)(3,102)(4,44)(5,104)(6,46)(7,106)(8,48)(9,108)(10,50)(11,110)(12,52)(13,112)(14,54)(15,114)(16,56)(17,116)(18,58)(19,118)(20,60)(21,120)(22,62)(23,82)(24,64)(25,84)(26,66)(27,86)(28,68)(29,88)(30,70)(31,90)(32,72)(33,92)(34,74)(35,94)(36,76)(37,96)(38,78)(39,98)(40,80)(41,156)(43,158)(45,160)(47,122)(49,124)(51,126)(53,128)(55,130)(57,132)(59,134)(61,136)(63,138)(65,140)(67,142)(69,144)(71,146)(73,148)(75,150)(77,152)(79,154)(81,137)(83,139)(85,141)(87,143)(89,145)(91,147)(93,149)(95,151)(97,153)(99,155)(101,157)(103,159)(105,121)(107,123)(109,125)(111,127)(113,129)(115,131)(117,133)(119,135), (1,156)(2,157)(3,158)(4,159)(5,160)(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)(21,136)(22,137)(23,138)(24,139)(25,140)(26,141)(27,142)(28,143)(29,144)(30,145)(31,146)(32,147)(33,148)(34,149)(35,150)(36,151)(37,152)(38,153)(39,154)(40,155)(41,100)(42,101)(43,102)(44,103)(45,104)(46,105)(47,106)(48,107)(49,108)(50,109)(51,110)(52,111)(53,112)(54,113)(55,114)(56,115)(57,116)(58,117)(59,118)(60,119)(61,120)(62,81)(63,82)(64,83)(65,84)(66,85)(67,86)(68,87)(69,88)(70,89)(71,90)(72,91)(73,92)(74,93)(75,94)(76,95)(77,96)(78,97)(79,98)(80,99), (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), (1,155,156,40)(2,39,157,154)(3,153,158,38)(4,37,159,152)(5,151,160,36)(6,35,121,150)(7,149,122,34)(8,33,123,148)(9,147,124,32)(10,31,125,146)(11,145,126,30)(12,29,127,144)(13,143,128,28)(14,27,129,142)(15,141,130,26)(16,25,131,140)(17,139,132,24)(18,23,133,138)(19,137,134,22)(20,21,135,136)(41,80,100,99)(42,98,101,79)(43,78,102,97)(44,96,103,77)(45,76,104,95)(46,94,105,75)(47,74,106,93)(48,92,107,73)(49,72,108,91)(50,90,109,71)(51,70,110,89)(52,88,111,69)(53,68,112,87)(54,86,113,67)(55,66,114,85)(56,84,115,65)(57,64,116,83)(58,82,117,63)(59,62,118,81)(60,120,119,61) );
G=PermutationGroup([[(1,100),(2,42),(3,102),(4,44),(5,104),(6,46),(7,106),(8,48),(9,108),(10,50),(11,110),(12,52),(13,112),(14,54),(15,114),(16,56),(17,116),(18,58),(19,118),(20,60),(21,120),(22,62),(23,82),(24,64),(25,84),(26,66),(27,86),(28,68),(29,88),(30,70),(31,90),(32,72),(33,92),(34,74),(35,94),(36,76),(37,96),(38,78),(39,98),(40,80),(41,156),(43,158),(45,160),(47,122),(49,124),(51,126),(53,128),(55,130),(57,132),(59,134),(61,136),(63,138),(65,140),(67,142),(69,144),(71,146),(73,148),(75,150),(77,152),(79,154),(81,137),(83,139),(85,141),(87,143),(89,145),(91,147),(93,149),(95,151),(97,153),(99,155),(101,157),(103,159),(105,121),(107,123),(109,125),(111,127),(113,129),(115,131),(117,133),(119,135)], [(1,156),(2,157),(3,158),(4,159),(5,160),(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),(21,136),(22,137),(23,138),(24,139),(25,140),(26,141),(27,142),(28,143),(29,144),(30,145),(31,146),(32,147),(33,148),(34,149),(35,150),(36,151),(37,152),(38,153),(39,154),(40,155),(41,100),(42,101),(43,102),(44,103),(45,104),(46,105),(47,106),(48,107),(49,108),(50,109),(51,110),(52,111),(53,112),(54,113),(55,114),(56,115),(57,116),(58,117),(59,118),(60,119),(61,120),(62,81),(63,82),(64,83),(65,84),(66,85),(67,86),(68,87),(69,88),(70,89),(71,90),(72,91),(73,92),(74,93),(75,94),(76,95),(77,96),(78,97),(79,98),(80,99)], [(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)], [(1,155,156,40),(2,39,157,154),(3,153,158,38),(4,37,159,152),(5,151,160,36),(6,35,121,150),(7,149,122,34),(8,33,123,148),(9,147,124,32),(10,31,125,146),(11,145,126,30),(12,29,127,144),(13,143,128,28),(14,27,129,142),(15,141,130,26),(16,25,131,140),(17,139,132,24),(18,23,133,138),(19,137,134,22),(20,21,135,136),(41,80,100,99),(42,98,101,79),(43,78,102,97),(44,96,103,77),(45,76,104,95),(46,94,105,75),(47,74,106,93),(48,92,107,73),(49,72,108,91),(50,90,109,71),(51,70,110,89),(52,88,111,69),(53,68,112,87),(54,86,113,67),(55,66,114,85),(56,84,115,65),(57,64,116,83),(58,82,117,63),(59,62,118,81),(60,120,119,61)]])
59 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 5A | 5B | 8A | 8B | 8C | 8D | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 20A | ··· | 20H | 20I | 20J | 20K | 20L | 40A | ··· | 40P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 20 | ··· | 20 | 20 | 20 | 20 | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 40 | 2 | 2 | 4 | 20 | 20 | 20 | 20 | 40 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
59 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | - | - | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D5 | C4○D4 | D8 | D10 | D10 | D20 | D20 | D40 | C8.C22 | D4⋊2D5 | C8.D10 |
| kernel | C22.D40 | C40⋊5C4 | D20⋊5C4 | C5×C22⋊C8 | C2×C4⋊Dic5 | C20⋊7D4 | C2×C20 | C22×C10 | C22⋊C8 | C20 | C2×C10 | C2×C8 | C22×C4 | C2×C4 | C23 | C22 | C10 | C4 | C2 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 2 | 4 | 4 | 4 | 2 | 4 | 4 | 16 | 1 | 4 | 4 |
Matrix representation of C22.D40 ►in GL4(𝔽41) generated by
| 13 | 31 | 0 | 0 |
| 25 | 28 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 40 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 32 | 0 | 0 | 0 |
| 34 | 9 | 0 | 0 |
| 0 | 0 | 2 | 35 |
| 0 | 0 | 6 | 3 |
| 32 | 0 | 0 | 0 |
| 0 | 32 | 0 | 0 |
| 0 | 0 | 39 | 6 |
| 0 | 0 | 20 | 2 |
G:=sub<GL(4,GF(41))| [13,25,0,0,31,28,0,0,0,0,1,0,0,0,0,1],[40,0,0,0,0,40,0,0,0,0,1,0,0,0,0,1],[32,34,0,0,0,9,0,0,0,0,2,6,0,0,35,3],[32,0,0,0,0,32,0,0,0,0,39,20,0,0,6,2] >;
C22.D40 in GAP, Magma, Sage, TeX
C_2^2.D_{40} % in TeX
G:=Group("C2^2.D40"); // GroupNames label
G:=SmallGroup(320,363);
// by ID
G=gap.SmallGroup(320,363);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,253,254,219,310,1123,136,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^2=c^40=1,d^2=b,c*a*c^-1=a*b=b*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=b*c^-1>;
// generators/relations