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 [×3], C2 [×3], C4 [×2], C4 [×4], C22, C22 [×2], C22 [×5], C5, C8 [×2], C2×C4 [×2], C2×C4 [×7], D4 [×4], C23, C23, D5, C10 [×3], C10 [×2], C22⋊C4, C4⋊C4 [×4], C2×C8 [×2], C22×C4, C22×C4, C2×D4 [×2], Dic5 [×3], C20 [×2], C20, D10 [×3], C2×C10, C2×C10 [×2], C2×C10 [×2], C22⋊C8, D4⋊C4 [×2], C2.D8 [×2], C2×C4⋊C4, C4⋊D4, C40 [×2], D20 [×2], C2×Dic5 [×5], C5⋊D4 [×2], C2×C20 [×2], C2×C20 [×2], C22×D5, C22×C10, C22.D8, C4⋊Dic5, C4⋊Dic5 [×2], C4⋊Dic5, D10⋊C4, C2×C40 [×2], C2×D20, C22×Dic5, C2×C5⋊D4, C22×C20, C40⋊5C4 [×2], D20⋊5C4 [×2], C5×C22⋊C8, C2×C4⋊Dic5, C20⋊7D4, C22.D40
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], C23, D5, D8 [×2], C2×D4, C4○D4 [×2], D10 [×3], C22.D4, C2×D8, C8.C22, D20 [×2], C22×D5, C22.D8, D40 [×2], C2×D20, D4⋊2D5 [×2], C22.D20, C2×D40, C8.D10, C22.D40
►On 160 points
Generators in S160
(1 134)(2 79)(3 136)(4 41)(5 138)(6 43)(7 140)(8 45)(9 142)(10 47)(11 144)(12 49)(13 146)(14 51)(15 148)(16 53)(17 150)(18 55)(19 152)(20 57)(21 154)(22 59)(23 156)(24 61)(25 158)(26 63)(27 160)(28 65)(29 122)(30 67)(31 124)(32 69)(33 126)(34 71)(35 128)(36 73)(37 130)(38 75)(39 132)(40 77)(42 82)(44 84)(46 86)(48 88)(50 90)(52 92)(54 94)(56 96)(58 98)(60 100)(62 102)(64 104)(66 106)(68 108)(70 110)(72 112)(74 114)(76 116)(78 118)(80 120)(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)
(1 118)(2 119)(3 120)(4 81)(5 82)(6 83)(7 84)(8 85)(9 86)(10 87)(11 88)(12 89)(13 90)(14 91)(15 92)(16 93)(17 94)(18 95)(19 96)(20 97)(21 98)(22 99)(23 100)(24 101)(25 102)(26 103)(27 104)(28 105)(29 106)(30 107)(31 108)(32 109)(33 110)(34 111)(35 112)(36 113)(37 114)(38 115)(39 116)(40 117)(41 137)(42 138)(43 139)(44 140)(45 141)(46 142)(47 143)(48 144)(49 145)(50 146)(51 147)(52 148)(53 149)(54 150)(55 151)(56 152)(57 153)(58 154)(59 155)(60 156)(61 157)(62 158)(63 159)(64 160)(65 121)(66 122)(67 123)(68 124)(69 125)(70 126)(71 127)(72 128)(73 129)(74 130)(75 131)(76 132)(77 133)(78 134)(79 135)(80 136)
(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)
(1 117 118 40)(2 39 119 116)(3 115 120 38)(4 37 81 114)(5 113 82 36)(6 35 83 112)(7 111 84 34)(8 33 85 110)(9 109 86 32)(10 31 87 108)(11 107 88 30)(12 29 89 106)(13 105 90 28)(14 27 91 104)(15 103 92 26)(16 25 93 102)(17 101 94 24)(18 23 95 100)(19 99 96 22)(20 21 97 98)(41 130 137 74)(42 73 138 129)(43 128 139 72)(44 71 140 127)(45 126 141 70)(46 69 142 125)(47 124 143 68)(48 67 144 123)(49 122 145 66)(50 65 146 121)(51 160 147 64)(52 63 148 159)(53 158 149 62)(54 61 150 157)(55 156 151 60)(56 59 152 155)(57 154 153 58)(75 136 131 80)(76 79 132 135)(77 134 133 78)
G:=sub<Sym(160)| (1,134)(2,79)(3,136)(4,41)(5,138)(6,43)(7,140)(8,45)(9,142)(10,47)(11,144)(12,49)(13,146)(14,51)(15,148)(16,53)(17,150)(18,55)(19,152)(20,57)(21,154)(22,59)(23,156)(24,61)(25,158)(26,63)(27,160)(28,65)(29,122)(30,67)(31,124)(32,69)(33,126)(34,71)(35,128)(36,73)(37,130)(38,75)(39,132)(40,77)(42,82)(44,84)(46,86)(48,88)(50,90)(52,92)(54,94)(56,96)(58,98)(60,100)(62,102)(64,104)(66,106)(68,108)(70,110)(72,112)(74,114)(76,116)(78,118)(80,120)(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), (1,118)(2,119)(3,120)(4,81)(5,82)(6,83)(7,84)(8,85)(9,86)(10,87)(11,88)(12,89)(13,90)(14,91)(15,92)(16,93)(17,94)(18,95)(19,96)(20,97)(21,98)(22,99)(23,100)(24,101)(25,102)(26,103)(27,104)(28,105)(29,106)(30,107)(31,108)(32,109)(33,110)(34,111)(35,112)(36,113)(37,114)(38,115)(39,116)(40,117)(41,137)(42,138)(43,139)(44,140)(45,141)(46,142)(47,143)(48,144)(49,145)(50,146)(51,147)(52,148)(53,149)(54,150)(55,151)(56,152)(57,153)(58,154)(59,155)(60,156)(61,157)(62,158)(63,159)(64,160)(65,121)(66,122)(67,123)(68,124)(69,125)(70,126)(71,127)(72,128)(73,129)(74,130)(75,131)(76,132)(77,133)(78,134)(79,135)(80,136), (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), (1,117,118,40)(2,39,119,116)(3,115,120,38)(4,37,81,114)(5,113,82,36)(6,35,83,112)(7,111,84,34)(8,33,85,110)(9,109,86,32)(10,31,87,108)(11,107,88,30)(12,29,89,106)(13,105,90,28)(14,27,91,104)(15,103,92,26)(16,25,93,102)(17,101,94,24)(18,23,95,100)(19,99,96,22)(20,21,97,98)(41,130,137,74)(42,73,138,129)(43,128,139,72)(44,71,140,127)(45,126,141,70)(46,69,142,125)(47,124,143,68)(48,67,144,123)(49,122,145,66)(50,65,146,121)(51,160,147,64)(52,63,148,159)(53,158,149,62)(54,61,150,157)(55,156,151,60)(56,59,152,155)(57,154,153,58)(75,136,131,80)(76,79,132,135)(77,134,133,78)>;
G:=Group( (1,134)(2,79)(3,136)(4,41)(5,138)(6,43)(7,140)(8,45)(9,142)(10,47)(11,144)(12,49)(13,146)(14,51)(15,148)(16,53)(17,150)(18,55)(19,152)(20,57)(21,154)(22,59)(23,156)(24,61)(25,158)(26,63)(27,160)(28,65)(29,122)(30,67)(31,124)(32,69)(33,126)(34,71)(35,128)(36,73)(37,130)(38,75)(39,132)(40,77)(42,82)(44,84)(46,86)(48,88)(50,90)(52,92)(54,94)(56,96)(58,98)(60,100)(62,102)(64,104)(66,106)(68,108)(70,110)(72,112)(74,114)(76,116)(78,118)(80,120)(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), (1,118)(2,119)(3,120)(4,81)(5,82)(6,83)(7,84)(8,85)(9,86)(10,87)(11,88)(12,89)(13,90)(14,91)(15,92)(16,93)(17,94)(18,95)(19,96)(20,97)(21,98)(22,99)(23,100)(24,101)(25,102)(26,103)(27,104)(28,105)(29,106)(30,107)(31,108)(32,109)(33,110)(34,111)(35,112)(36,113)(37,114)(38,115)(39,116)(40,117)(41,137)(42,138)(43,139)(44,140)(45,141)(46,142)(47,143)(48,144)(49,145)(50,146)(51,147)(52,148)(53,149)(54,150)(55,151)(56,152)(57,153)(58,154)(59,155)(60,156)(61,157)(62,158)(63,159)(64,160)(65,121)(66,122)(67,123)(68,124)(69,125)(70,126)(71,127)(72,128)(73,129)(74,130)(75,131)(76,132)(77,133)(78,134)(79,135)(80,136), (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), (1,117,118,40)(2,39,119,116)(3,115,120,38)(4,37,81,114)(5,113,82,36)(6,35,83,112)(7,111,84,34)(8,33,85,110)(9,109,86,32)(10,31,87,108)(11,107,88,30)(12,29,89,106)(13,105,90,28)(14,27,91,104)(15,103,92,26)(16,25,93,102)(17,101,94,24)(18,23,95,100)(19,99,96,22)(20,21,97,98)(41,130,137,74)(42,73,138,129)(43,128,139,72)(44,71,140,127)(45,126,141,70)(46,69,142,125)(47,124,143,68)(48,67,144,123)(49,122,145,66)(50,65,146,121)(51,160,147,64)(52,63,148,159)(53,158,149,62)(54,61,150,157)(55,156,151,60)(56,59,152,155)(57,154,153,58)(75,136,131,80)(76,79,132,135)(77,134,133,78) );
G=PermutationGroup([(1,134),(2,79),(3,136),(4,41),(5,138),(6,43),(7,140),(8,45),(9,142),(10,47),(11,144),(12,49),(13,146),(14,51),(15,148),(16,53),(17,150),(18,55),(19,152),(20,57),(21,154),(22,59),(23,156),(24,61),(25,158),(26,63),(27,160),(28,65),(29,122),(30,67),(31,124),(32,69),(33,126),(34,71),(35,128),(36,73),(37,130),(38,75),(39,132),(40,77),(42,82),(44,84),(46,86),(48,88),(50,90),(52,92),(54,94),(56,96),(58,98),(60,100),(62,102),(64,104),(66,106),(68,108),(70,110),(72,112),(74,114),(76,116),(78,118),(80,120),(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)], [(1,118),(2,119),(3,120),(4,81),(5,82),(6,83),(7,84),(8,85),(9,86),(10,87),(11,88),(12,89),(13,90),(14,91),(15,92),(16,93),(17,94),(18,95),(19,96),(20,97),(21,98),(22,99),(23,100),(24,101),(25,102),(26,103),(27,104),(28,105),(29,106),(30,107),(31,108),(32,109),(33,110),(34,111),(35,112),(36,113),(37,114),(38,115),(39,116),(40,117),(41,137),(42,138),(43,139),(44,140),(45,141),(46,142),(47,143),(48,144),(49,145),(50,146),(51,147),(52,148),(53,149),(54,150),(55,151),(56,152),(57,153),(58,154),(59,155),(60,156),(61,157),(62,158),(63,159),(64,160),(65,121),(66,122),(67,123),(68,124),(69,125),(70,126),(71,127),(72,128),(73,129),(74,130),(75,131),(76,132),(77,133),(78,134),(79,135),(80,136)], [(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)], [(1,117,118,40),(2,39,119,116),(3,115,120,38),(4,37,81,114),(5,113,82,36),(6,35,83,112),(7,111,84,34),(8,33,85,110),(9,109,86,32),(10,31,87,108),(11,107,88,30),(12,29,89,106),(13,105,90,28),(14,27,91,104),(15,103,92,26),(16,25,93,102),(17,101,94,24),(18,23,95,100),(19,99,96,22),(20,21,97,98),(41,130,137,74),(42,73,138,129),(43,128,139,72),(44,71,140,127),(45,126,141,70),(46,69,142,125),(47,124,143,68),(48,67,144,123),(49,122,145,66),(50,65,146,121),(51,160,147,64),(52,63,148,159),(53,158,149,62),(54,61,150,157),(55,156,151,60),(56,59,152,155),(57,154,153,58),(75,136,131,80),(76,79,132,135),(77,134,133,78)])
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