metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: (C2×C10).40D8, C10.49(C2×D8), C4⋊C4.227D10, C20⋊7D4.9C2, (C2×C20).283D4, D20⋊6C4⋊25C2, C4.86(C4○D20), C20.55D4⋊4C2, C10.D8⋊25C2, C22.9(D4⋊D5), (C22×C4).94D10, C5⋊4(C22.D8), C20.174(C4○D4), (C2×C20).320C23, (C2×D20).94C22, (C22×C10).185D4, C23.77(C5⋊D4), C2.6(C20.C23), C10.84(C8.C22), C4⋊Dic5.130C22, (C22×C20).135C22, C2.8(C23.23D10), C10.58(C22.D4), (C2×C4⋊C4)⋊3D5, (C10×C4⋊C4)⋊3C2, C2.5(C2×D4⋊D5), (C2×C10).440(C2×D4), (C2×C4).31(C5⋊D4), (C5×C4⋊C4).258C22, (C2×C5⋊2C8).81C22, (C2×C4).420(C22×D5), C22.130(C2×C5⋊D4), SmallGroup(320,594)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for (C2×C10).40D8
G = < a,b,c,d | a10=b2=c8=1, d2=a5, ab=ba, cac-1=a-1, ad=da, cbc-1=a5b, bd=db, dcd-1=c-1 >
Subgroups: 462 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 [×2], C4⋊C4 [×2], C2×C8 [×2], C22×C4, C22×C4, C2×D4 [×2], Dic5, C20 [×2], C20 [×3], D10 [×3], C2×C10, C2×C10 [×2], C2×C10 [×2], C22⋊C8, D4⋊C4 [×2], C2.D8 [×2], C2×C4⋊C4, C4⋊D4, C5⋊2C8 [×2], D20 [×2], C2×Dic5, C5⋊D4 [×2], C2×C20 [×2], C2×C20 [×6], C22×D5, C22×C10, C22.D8, C2×C5⋊2C8 [×2], C4⋊Dic5, D10⋊C4, C5×C4⋊C4 [×2], C5×C4⋊C4, C2×D20, C2×C5⋊D4, C22×C20, C22×C20, C10.D8 [×2], D20⋊6C4 [×2], C20.55D4, C20⋊7D4, C10×C4⋊C4, (C2×C10).40D8
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, C5⋊D4 [×2], C22×D5, C22.D8, D4⋊D5 [×2], C4○D20 [×2], C2×C5⋊D4, C23.23D10, C2×D4⋊D5, C20.C23, (C2×C10).40D8
►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)
(1 110)(2 101)(3 102)(4 103)(5 104)(6 105)(7 106)(8 107)(9 108)(10 109)(11 71)(12 72)(13 73)(14 74)(15 75)(16 76)(17 77)(18 78)(19 79)(20 80)(21 97)(22 98)(23 99)(24 100)(25 91)(26 92)(27 93)(28 94)(29 95)(30 96)(31 125)(32 126)(33 127)(34 128)(35 129)(36 130)(37 121)(38 122)(39 123)(40 124)(41 117)(42 118)(43 119)(44 120)(45 111)(46 112)(47 113)(48 114)(49 115)(50 116)(51 145)(52 146)(53 147)(54 148)(55 149)(56 150)(57 141)(58 142)(59 143)(60 144)(61 137)(62 138)(63 139)(64 140)(65 131)(66 132)(67 133)(68 134)(69 135)(70 136)(81 157)(82 158)(83 159)(84 160)(85 151)(86 152)(87 153)(88 154)(89 155)(90 156)
(1 71 45 70 30 85 31 56)(2 80 46 69 21 84 32 55)(3 79 47 68 22 83 33 54)(4 78 48 67 23 82 34 53)(5 77 49 66 24 81 35 52)(6 76 50 65 25 90 36 51)(7 75 41 64 26 89 37 60)(8 74 42 63 27 88 38 59)(9 73 43 62 28 87 39 58)(10 72 44 61 29 86 40 57)(11 116 136 91 151 130 150 105)(12 115 137 100 152 129 141 104)(13 114 138 99 153 128 142 103)(14 113 139 98 154 127 143 102)(15 112 140 97 155 126 144 101)(16 111 131 96 156 125 145 110)(17 120 132 95 157 124 146 109)(18 119 133 94 158 123 147 108)(19 118 134 93 159 122 148 107)(20 117 135 92 160 121 149 106)
(1 131 6 136)(2 132 7 137)(3 133 8 138)(4 134 9 139)(5 135 10 140)(11 45 16 50)(12 46 17 41)(13 47 18 42)(14 48 19 43)(15 49 20 44)(21 146 26 141)(22 147 27 142)(23 148 28 143)(24 149 29 144)(25 150 30 145)(31 156 36 151)(32 157 37 152)(33 158 38 153)(34 159 39 154)(35 160 40 155)(51 91 56 96)(52 92 57 97)(53 93 58 98)(54 94 59 99)(55 95 60 100)(61 101 66 106)(62 102 67 107)(63 103 68 108)(64 104 69 109)(65 105 70 110)(71 111 76 116)(72 112 77 117)(73 113 78 118)(74 114 79 119)(75 115 80 120)(81 121 86 126)(82 122 87 127)(83 123 88 128)(84 124 89 129)(85 125 90 130)
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), (1,110)(2,101)(3,102)(4,103)(5,104)(6,105)(7,106)(8,107)(9,108)(10,109)(11,71)(12,72)(13,73)(14,74)(15,75)(16,76)(17,77)(18,78)(19,79)(20,80)(21,97)(22,98)(23,99)(24,100)(25,91)(26,92)(27,93)(28,94)(29,95)(30,96)(31,125)(32,126)(33,127)(34,128)(35,129)(36,130)(37,121)(38,122)(39,123)(40,124)(41,117)(42,118)(43,119)(44,120)(45,111)(46,112)(47,113)(48,114)(49,115)(50,116)(51,145)(52,146)(53,147)(54,148)(55,149)(56,150)(57,141)(58,142)(59,143)(60,144)(61,137)(62,138)(63,139)(64,140)(65,131)(66,132)(67,133)(68,134)(69,135)(70,136)(81,157)(82,158)(83,159)(84,160)(85,151)(86,152)(87,153)(88,154)(89,155)(90,156), (1,71,45,70,30,85,31,56)(2,80,46,69,21,84,32,55)(3,79,47,68,22,83,33,54)(4,78,48,67,23,82,34,53)(5,77,49,66,24,81,35,52)(6,76,50,65,25,90,36,51)(7,75,41,64,26,89,37,60)(8,74,42,63,27,88,38,59)(9,73,43,62,28,87,39,58)(10,72,44,61,29,86,40,57)(11,116,136,91,151,130,150,105)(12,115,137,100,152,129,141,104)(13,114,138,99,153,128,142,103)(14,113,139,98,154,127,143,102)(15,112,140,97,155,126,144,101)(16,111,131,96,156,125,145,110)(17,120,132,95,157,124,146,109)(18,119,133,94,158,123,147,108)(19,118,134,93,159,122,148,107)(20,117,135,92,160,121,149,106), (1,131,6,136)(2,132,7,137)(3,133,8,138)(4,134,9,139)(5,135,10,140)(11,45,16,50)(12,46,17,41)(13,47,18,42)(14,48,19,43)(15,49,20,44)(21,146,26,141)(22,147,27,142)(23,148,28,143)(24,149,29,144)(25,150,30,145)(31,156,36,151)(32,157,37,152)(33,158,38,153)(34,159,39,154)(35,160,40,155)(51,91,56,96)(52,92,57,97)(53,93,58,98)(54,94,59,99)(55,95,60,100)(61,101,66,106)(62,102,67,107)(63,103,68,108)(64,104,69,109)(65,105,70,110)(71,111,76,116)(72,112,77,117)(73,113,78,118)(74,114,79,119)(75,115,80,120)(81,121,86,126)(82,122,87,127)(83,123,88,128)(84,124,89,129)(85,125,90,130)>;
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), (1,110)(2,101)(3,102)(4,103)(5,104)(6,105)(7,106)(8,107)(9,108)(10,109)(11,71)(12,72)(13,73)(14,74)(15,75)(16,76)(17,77)(18,78)(19,79)(20,80)(21,97)(22,98)(23,99)(24,100)(25,91)(26,92)(27,93)(28,94)(29,95)(30,96)(31,125)(32,126)(33,127)(34,128)(35,129)(36,130)(37,121)(38,122)(39,123)(40,124)(41,117)(42,118)(43,119)(44,120)(45,111)(46,112)(47,113)(48,114)(49,115)(50,116)(51,145)(52,146)(53,147)(54,148)(55,149)(56,150)(57,141)(58,142)(59,143)(60,144)(61,137)(62,138)(63,139)(64,140)(65,131)(66,132)(67,133)(68,134)(69,135)(70,136)(81,157)(82,158)(83,159)(84,160)(85,151)(86,152)(87,153)(88,154)(89,155)(90,156), (1,71,45,70,30,85,31,56)(2,80,46,69,21,84,32,55)(3,79,47,68,22,83,33,54)(4,78,48,67,23,82,34,53)(5,77,49,66,24,81,35,52)(6,76,50,65,25,90,36,51)(7,75,41,64,26,89,37,60)(8,74,42,63,27,88,38,59)(9,73,43,62,28,87,39,58)(10,72,44,61,29,86,40,57)(11,116,136,91,151,130,150,105)(12,115,137,100,152,129,141,104)(13,114,138,99,153,128,142,103)(14,113,139,98,154,127,143,102)(15,112,140,97,155,126,144,101)(16,111,131,96,156,125,145,110)(17,120,132,95,157,124,146,109)(18,119,133,94,158,123,147,108)(19,118,134,93,159,122,148,107)(20,117,135,92,160,121,149,106), (1,131,6,136)(2,132,7,137)(3,133,8,138)(4,134,9,139)(5,135,10,140)(11,45,16,50)(12,46,17,41)(13,47,18,42)(14,48,19,43)(15,49,20,44)(21,146,26,141)(22,147,27,142)(23,148,28,143)(24,149,29,144)(25,150,30,145)(31,156,36,151)(32,157,37,152)(33,158,38,153)(34,159,39,154)(35,160,40,155)(51,91,56,96)(52,92,57,97)(53,93,58,98)(54,94,59,99)(55,95,60,100)(61,101,66,106)(62,102,67,107)(63,103,68,108)(64,104,69,109)(65,105,70,110)(71,111,76,116)(72,112,77,117)(73,113,78,118)(74,114,79,119)(75,115,80,120)(81,121,86,126)(82,122,87,127)(83,123,88,128)(84,124,89,129)(85,125,90,130) );
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)], [(1,110),(2,101),(3,102),(4,103),(5,104),(6,105),(7,106),(8,107),(9,108),(10,109),(11,71),(12,72),(13,73),(14,74),(15,75),(16,76),(17,77),(18,78),(19,79),(20,80),(21,97),(22,98),(23,99),(24,100),(25,91),(26,92),(27,93),(28,94),(29,95),(30,96),(31,125),(32,126),(33,127),(34,128),(35,129),(36,130),(37,121),(38,122),(39,123),(40,124),(41,117),(42,118),(43,119),(44,120),(45,111),(46,112),(47,113),(48,114),(49,115),(50,116),(51,145),(52,146),(53,147),(54,148),(55,149),(56,150),(57,141),(58,142),(59,143),(60,144),(61,137),(62,138),(63,139),(64,140),(65,131),(66,132),(67,133),(68,134),(69,135),(70,136),(81,157),(82,158),(83,159),(84,160),(85,151),(86,152),(87,153),(88,154),(89,155),(90,156)], [(1,71,45,70,30,85,31,56),(2,80,46,69,21,84,32,55),(3,79,47,68,22,83,33,54),(4,78,48,67,23,82,34,53),(5,77,49,66,24,81,35,52),(6,76,50,65,25,90,36,51),(7,75,41,64,26,89,37,60),(8,74,42,63,27,88,38,59),(9,73,43,62,28,87,39,58),(10,72,44,61,29,86,40,57),(11,116,136,91,151,130,150,105),(12,115,137,100,152,129,141,104),(13,114,138,99,153,128,142,103),(14,113,139,98,154,127,143,102),(15,112,140,97,155,126,144,101),(16,111,131,96,156,125,145,110),(17,120,132,95,157,124,146,109),(18,119,133,94,158,123,147,108),(19,118,134,93,159,122,148,107),(20,117,135,92,160,121,149,106)], [(1,131,6,136),(2,132,7,137),(3,133,8,138),(4,134,9,139),(5,135,10,140),(11,45,16,50),(12,46,17,41),(13,47,18,42),(14,48,19,43),(15,49,20,44),(21,146,26,141),(22,147,27,142),(23,148,28,143),(24,149,29,144),(25,150,30,145),(31,156,36,151),(32,157,37,152),(33,158,38,153),(34,159,39,154),(35,160,40,155),(51,91,56,96),(52,92,57,97),(53,93,58,98),(54,94,59,99),(55,95,60,100),(61,101,66,106),(62,102,67,107),(63,103,68,108),(64,104,69,109),(65,105,70,110),(71,111,76,116),(72,112,77,117),(73,113,78,118),(74,114,79,119),(75,115,80,120),(81,121,86,126),(82,122,87,127),(83,123,88,128),(84,124,89,129),(85,125,90,130)])
59 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | 4B | 4C | ··· | 4G | 4H | 5A | 5B | 8A | 8B | 8C | 8D | 10A | ··· | 10N | 20A | ··· | 20X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | ··· | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 40 | 2 | 2 | 4 | ··· | 4 | 40 | 2 | 2 | 20 | 20 | 20 | 20 | 2 | ··· | 2 | 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 | C5⋊D4 | C5⋊D4 | C4○D20 | C8.C22 | D4⋊D5 | C20.C23 |
| kernel | (C2×C10).40D8 | C10.D8 | D20⋊6C4 | C20.55D4 | C20⋊7D4 | C10×C4⋊C4 | C2×C20 | C22×C10 | C2×C4⋊C4 | C20 | C2×C10 | C4⋊C4 | C22×C4 | C2×C4 | C23 | C4 | C10 | C22 | C2 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 2 | 4 | 4 | 4 | 2 | 4 | 4 | 16 | 1 | 4 | 4 |
Matrix representation of (C2×C10).40D8 ►in GL6(𝔽41)
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 7 | 0 | 0 |
| 0 | 0 | 34 | 7 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 25 | 1 | 0 | 0 | 0 | 0 |
| 32 | 16 | 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 | 1 |
| 20 | 9 | 0 | 0 | 0 | 0 |
| 24 | 21 | 0 | 0 | 0 | 0 |
| 0 | 0 | 7 | 34 | 0 | 0 |
| 0 | 0 | 1 | 34 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 12 |
| 0 | 0 | 0 | 0 | 29 | 12 |
| 9 | 0 | 0 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 29 | 29 |
| 0 | 0 | 0 | 0 | 29 | 12 |
G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,34,0,0,0,0,7,7,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[25,32,0,0,0,0,1,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[20,24,0,0,0,0,9,21,0,0,0,0,0,0,7,1,0,0,0,0,34,34,0,0,0,0,0,0,12,29,0,0,0,0,12,12],[9,0,0,0,0,0,0,9,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,29,29,0,0,0,0,29,12] >;
(C2×C10).40D8 in GAP, Magma, Sage, TeX
(C_2\times C_{10})._{40}D_8 % in TeX
G:=Group("(C2xC10).40D8"); // GroupNames label
G:=SmallGroup(320,594);
// by ID
G=gap.SmallGroup(320,594);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,253,254,100,1123,297,136,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^10=b^2=c^8=1,d^2=a^5,a*b=b*a,c*a*c^-1=a^-1,a*d=d*a,c*b*c^-1=a^5*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations