metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C40⋊11D4, (C2×D8)⋊8D5, C5⋊2C8⋊9D4, (C10×D8)⋊9C2, C8⋊3(C5⋊D4), C5⋊4(C8⋊3D4), C40⋊8C4⋊9C2, C4.21(D4×D5), C20⋊D4⋊5C2, (C2×C8).85D10, (C2×D4).62D10, C20.164(C2×D4), C20.17D4⋊4C2, (C2×Dic5).72D4, C22.254(D4×D5), C2.28(D8⋊D5), C2.18(C20⋊D4), C10.27(C4⋊1D4), C10.49(C8⋊C22), (C2×C20).431C23, (C2×C40).147C22, (D4×C10).81C22, (C2×D20).119C22, (C4×Dic5).51C22, (C2×Dic10).124C22, C4.5(C2×C5⋊D4), (C2×D4⋊D5)⋊18C2, (C2×C40⋊C2)⋊23C2, (C2×D4.D5)⋊17C2, (C2×C10).344(C2×D4), (C2×C4).521(C22×D5), (C2×C5⋊2C8).148C22, SmallGroup(320,781)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C40⋊11D4
G = < a,b,c | a40=b4=c2=1, bab-1=a29, cac=a19, cbc=b-1 >
Subgroups: 686 in 144 conjugacy classes, 43 normal (31 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C5, C8, C8, C2×C4, C2×C4, D4, Q8, C23, D5, C10, C10, C10, C42, C22⋊C4, C2×C8, C2×C8, D8, SD16, C2×D4, C2×D4, C2×Q8, Dic5, C20, D10, C2×C10, C2×C10, C8⋊C4, C4.4D4, C4⋊1D4, C2×D8, C2×D8, C2×SD16, C5⋊2C8, C40, Dic10, D20, C2×Dic5, C2×Dic5, C5⋊D4, C2×C20, C5×D4, C22×D5, C22×C10, C8⋊3D4, C40⋊C2, C2×C5⋊2C8, C4×Dic5, D4⋊D5, D4.D5, C23.D5, C2×C40, C5×D8, C2×Dic10, C2×D20, C2×C5⋊D4, D4×C10, C40⋊8C4, C2×C40⋊C2, C2×D4⋊D5, C2×D4.D5, C20.17D4, C20⋊D4, C10×D8, C40⋊11D4
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, D10, C4⋊1D4, C8⋊C22, C5⋊D4, C22×D5, C8⋊3D4, D4×D5, C2×C5⋊D4, D8⋊D5, C20⋊D4, C40⋊11D4
►On 160 points
Generators in S160
(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 81 71 131)(2 110 72 160)(3 99 73 149)(4 88 74 138)(5 117 75 127)(6 106 76 156)(7 95 77 145)(8 84 78 134)(9 113 79 123)(10 102 80 152)(11 91 41 141)(12 120 42 130)(13 109 43 159)(14 98 44 148)(15 87 45 137)(16 116 46 126)(17 105 47 155)(18 94 48 144)(19 83 49 133)(20 112 50 122)(21 101 51 151)(22 90 52 140)(23 119 53 129)(24 108 54 158)(25 97 55 147)(26 86 56 136)(27 115 57 125)(28 104 58 154)(29 93 59 143)(30 82 60 132)(31 111 61 121)(32 100 62 150)(33 89 63 139)(34 118 64 128)(35 107 65 157)(36 96 66 146)(37 85 67 135)(38 114 68 124)(39 103 69 153)(40 92 70 142)
(2 20)(3 39)(4 18)(5 37)(6 16)(7 35)(8 14)(9 33)(10 12)(11 31)(13 29)(15 27)(17 25)(19 23)(22 40)(24 38)(26 36)(28 34)(30 32)(41 61)(42 80)(43 59)(44 78)(45 57)(46 76)(47 55)(48 74)(49 53)(50 72)(52 70)(54 68)(56 66)(58 64)(60 62)(63 79)(65 77)(67 75)(69 73)(81 131)(82 150)(83 129)(84 148)(85 127)(86 146)(87 125)(88 144)(89 123)(90 142)(91 121)(92 140)(93 159)(94 138)(95 157)(96 136)(97 155)(98 134)(99 153)(100 132)(101 151)(102 130)(103 149)(104 128)(105 147)(106 126)(107 145)(108 124)(109 143)(110 122)(111 141)(112 160)(113 139)(114 158)(115 137)(116 156)(117 135)(118 154)(119 133)(120 152)
G:=sub<Sym(160)| (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), (1,81,71,131)(2,110,72,160)(3,99,73,149)(4,88,74,138)(5,117,75,127)(6,106,76,156)(7,95,77,145)(8,84,78,134)(9,113,79,123)(10,102,80,152)(11,91,41,141)(12,120,42,130)(13,109,43,159)(14,98,44,148)(15,87,45,137)(16,116,46,126)(17,105,47,155)(18,94,48,144)(19,83,49,133)(20,112,50,122)(21,101,51,151)(22,90,52,140)(23,119,53,129)(24,108,54,158)(25,97,55,147)(26,86,56,136)(27,115,57,125)(28,104,58,154)(29,93,59,143)(30,82,60,132)(31,111,61,121)(32,100,62,150)(33,89,63,139)(34,118,64,128)(35,107,65,157)(36,96,66,146)(37,85,67,135)(38,114,68,124)(39,103,69,153)(40,92,70,142), (2,20)(3,39)(4,18)(5,37)(6,16)(7,35)(8,14)(9,33)(10,12)(11,31)(13,29)(15,27)(17,25)(19,23)(22,40)(24,38)(26,36)(28,34)(30,32)(41,61)(42,80)(43,59)(44,78)(45,57)(46,76)(47,55)(48,74)(49,53)(50,72)(52,70)(54,68)(56,66)(58,64)(60,62)(63,79)(65,77)(67,75)(69,73)(81,131)(82,150)(83,129)(84,148)(85,127)(86,146)(87,125)(88,144)(89,123)(90,142)(91,121)(92,140)(93,159)(94,138)(95,157)(96,136)(97,155)(98,134)(99,153)(100,132)(101,151)(102,130)(103,149)(104,128)(105,147)(106,126)(107,145)(108,124)(109,143)(110,122)(111,141)(112,160)(113,139)(114,158)(115,137)(116,156)(117,135)(118,154)(119,133)(120,152)>;
G:=Group( (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), (1,81,71,131)(2,110,72,160)(3,99,73,149)(4,88,74,138)(5,117,75,127)(6,106,76,156)(7,95,77,145)(8,84,78,134)(9,113,79,123)(10,102,80,152)(11,91,41,141)(12,120,42,130)(13,109,43,159)(14,98,44,148)(15,87,45,137)(16,116,46,126)(17,105,47,155)(18,94,48,144)(19,83,49,133)(20,112,50,122)(21,101,51,151)(22,90,52,140)(23,119,53,129)(24,108,54,158)(25,97,55,147)(26,86,56,136)(27,115,57,125)(28,104,58,154)(29,93,59,143)(30,82,60,132)(31,111,61,121)(32,100,62,150)(33,89,63,139)(34,118,64,128)(35,107,65,157)(36,96,66,146)(37,85,67,135)(38,114,68,124)(39,103,69,153)(40,92,70,142), (2,20)(3,39)(4,18)(5,37)(6,16)(7,35)(8,14)(9,33)(10,12)(11,31)(13,29)(15,27)(17,25)(19,23)(22,40)(24,38)(26,36)(28,34)(30,32)(41,61)(42,80)(43,59)(44,78)(45,57)(46,76)(47,55)(48,74)(49,53)(50,72)(52,70)(54,68)(56,66)(58,64)(60,62)(63,79)(65,77)(67,75)(69,73)(81,131)(82,150)(83,129)(84,148)(85,127)(86,146)(87,125)(88,144)(89,123)(90,142)(91,121)(92,140)(93,159)(94,138)(95,157)(96,136)(97,155)(98,134)(99,153)(100,132)(101,151)(102,130)(103,149)(104,128)(105,147)(106,126)(107,145)(108,124)(109,143)(110,122)(111,141)(112,160)(113,139)(114,158)(115,137)(116,156)(117,135)(118,154)(119,133)(120,152) );
G=PermutationGroup([[(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)], [(1,81,71,131),(2,110,72,160),(3,99,73,149),(4,88,74,138),(5,117,75,127),(6,106,76,156),(7,95,77,145),(8,84,78,134),(9,113,79,123),(10,102,80,152),(11,91,41,141),(12,120,42,130),(13,109,43,159),(14,98,44,148),(15,87,45,137),(16,116,46,126),(17,105,47,155),(18,94,48,144),(19,83,49,133),(20,112,50,122),(21,101,51,151),(22,90,52,140),(23,119,53,129),(24,108,54,158),(25,97,55,147),(26,86,56,136),(27,115,57,125),(28,104,58,154),(29,93,59,143),(30,82,60,132),(31,111,61,121),(32,100,62,150),(33,89,63,139),(34,118,64,128),(35,107,65,157),(36,96,66,146),(37,85,67,135),(38,114,68,124),(39,103,69,153),(40,92,70,142)], [(2,20),(3,39),(4,18),(5,37),(6,16),(7,35),(8,14),(9,33),(10,12),(11,31),(13,29),(15,27),(17,25),(19,23),(22,40),(24,38),(26,36),(28,34),(30,32),(41,61),(42,80),(43,59),(44,78),(45,57),(46,76),(47,55),(48,74),(49,53),(50,72),(52,70),(54,68),(56,66),(58,64),(60,62),(63,79),(65,77),(67,75),(69,73),(81,131),(82,150),(83,129),(84,148),(85,127),(86,146),(87,125),(88,144),(89,123),(90,142),(91,121),(92,140),(93,159),(94,138),(95,157),(96,136),(97,155),(98,134),(99,153),(100,132),(101,151),(102,130),(103,149),(104,128),(105,147),(106,126),(107,145),(108,124),(109,143),(110,122),(111,141),(112,160),(113,139),(114,158),(115,137),(116,156),(117,135),(118,154),(119,133),(120,152)]])
44 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | 4B | 4C | 4D | 4E | 5A | 5B | 8A | 8B | 8C | 8D | 10A | ··· | 10F | 10G | ··· | 10N | 20A | 20B | 20C | 20D | 40A | ··· | 40H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | 20 | 20 | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 8 | 8 | 40 | 2 | 2 | 20 | 20 | 40 | 2 | 2 | 4 | 4 | 20 | 20 | 2 | ··· | 2 | 8 | ··· | 8 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | D5 | D10 | D10 | C5⋊D4 | C8⋊C22 | D4×D5 | D4×D5 | D8⋊D5 |
| kernel | C40⋊11D4 | C40⋊8C4 | C2×C40⋊C2 | C2×D4⋊D5 | C2×D4.D5 | C20.17D4 | C20⋊D4 | C10×D8 | C5⋊2C8 | C40 | C2×Dic5 | C2×D8 | C2×C8 | C2×D4 | C8 | C10 | C4 | C22 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 8 | 2 | 2 | 2 | 8 |
Matrix representation of C40⋊11D4 ►in GL10(𝔽41)
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 20 | 38 | 20 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 26 | 35 | 20 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 21 | 37 | 21 | 3 | 0 | 0 | 0 | 0 |
| 0 | 0 | 21 | 0 | 15 | 6 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 35 | 38 | 11 | 11 |
| 0 | 0 | 0 | 0 | 0 | 0 | 31 | 11 | 24 | 7 |
| 0 | 0 | 0 | 0 | 0 | 0 | 3 | 36 | 2 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 6 | 13 | 18 | 34 |
| 17 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 19 | 24 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 21 | 24 | 20 | 24 | 0 | 0 | 0 | 0 |
| 0 | 0 | 21 | 20 | 26 | 21 | 0 | 0 | 0 | 0 |
| 0 | 0 | 20 | 24 | 21 | 24 | 0 | 0 | 0 | 0 |
| 0 | 0 | 26 | 21 | 21 | 20 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 34 | 11 | 9 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 25 | 33 | 2 | 2 |
| 0 | 0 | 0 | 0 | 0 | 0 | 37 | 32 | 0 | 34 |
| 0 | 0 | 0 | 0 | 0 | 0 | 7 | 4 | 39 | 15 |
| 40 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 17 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 6 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 6 | 35 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 35 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 35 | 6 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 23 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 22 | 7 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 26 | 17 | 40 | 40 |
G:=sub<GL(10,GF(41))| [1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,20,26,21,21,0,0,0,0,0,0,38,35,37,0,0,0,0,0,0,0,20,20,21,15,0,0,0,0,0,0,4,0,3,6,0,0,0,0,0,0,0,0,0,0,35,31,3,6,0,0,0,0,0,0,38,11,36,13,0,0,0,0,0,0,11,24,2,18,0,0,0,0,0,0,11,7,0,34],[17,19,0,0,0,0,0,0,0,0,2,24,0,0,0,0,0,0,0,0,0,0,21,21,20,26,0,0,0,0,0,0,24,20,24,21,0,0,0,0,0,0,20,26,21,21,0,0,0,0,0,0,24,21,24,20,0,0,0,0,0,0,0,0,0,0,34,25,37,7,0,0,0,0,0,0,11,33,32,4,0,0,0,0,0,0,9,2,0,39,0,0,0,0,0,0,0,2,34,15],[40,17,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,6,6,0,0,0,0,0,0,0,0,1,35,0,0,0,0,0,0,0,0,0,0,35,35,0,0,0,0,0,0,0,0,40,6,0,0,0,0,0,0,0,0,0,0,1,23,22,26,0,0,0,0,0,0,0,40,7,17,0,0,0,0,0,0,0,0,1,40,0,0,0,0,0,0,0,0,0,40] >;
C40⋊11D4 in GAP, Magma, Sage, TeX
C_{40}\rtimes_{11}D_4 % in TeX
G:=Group("C40:11D4"); // GroupNames label
G:=SmallGroup(320,781);
// by ID
G=gap.SmallGroup(320,781);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,253,1094,135,570,297,136,12550]);
// Polycyclic
G:=Group<a,b,c|a^40=b^4=c^2=1,b*a*b^-1=a^29,c*a*c=a^19,c*b*c=b^-1>;
// generators/relations