direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×D10⋊1C8, D10⋊8(C2×C8), (C2×C8)⋊31D10, (C22×C8)⋊1D5, (C22×C40)⋊1C2, C4.85(C2×D20), (C22×D5)⋊4C8, C10⋊3(C22⋊C8), (C2×C40)⋊40C22, C20.434(C2×D4), (C2×C4).171D20, (C2×C20).498D4, (C23×D5).9C4, C22.17(C8×D5), C23.63(C4×D5), C10.43(C22×C8), (C2×C20).858C23, (C22×C4).463D10, C10.46(C2×M4(2)), (C2×C10).31M4(2), C4.54(D10⋊C4), C20.112(C22⋊C4), C22.11(C8⋊D5), (C22×Dic5).19C4, (C22×C20).559C22, C22.48(D10⋊C4), C5⋊5(C2×C22⋊C8), C2.19(D5×C2×C8), (C2×C4×D5).25C4, C2.5(C2×C8⋊D5), C22.60(C2×C4×D5), (C2×C10).47(C2×C8), (C2×C4).184(C4×D5), C4.124(C2×C5⋊D4), (D5×C22×C4).20C2, (C2×C20).427(C2×C4), (C2×C5⋊2C8)⋊44C22, (C22×C5⋊2C8)⋊20C2, C2.2(C2×D10⋊C4), C10.91(C2×C22⋊C4), (C2×C4×D5).352C22, (C2×C4).276(C5⋊D4), (C2×C4).800(C22×D5), (C2×C10).229(C22×C4), (C22×C10).161(C2×C4), (C2×Dic5).154(C2×C4), (C22×D5).106(C2×C4), (C2×C10).125(C22⋊C4), SmallGroup(320,735)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×D10⋊1C8
G = < a,b,c,d | a2=b10=c2=d8=1, ab=ba, ac=ca, ad=da, cbc=b-1, bd=db, dcd-1=b5c >
Subgroups: 718 in 202 conjugacy classes, 87 normal (29 characteristic)
C1, C2 [×3], C2 [×4], C2 [×4], C4 [×2], C4 [×2], C4 [×2], C22, C22 [×6], C22 [×16], C5, C8 [×4], C2×C4 [×2], C2×C4 [×4], C2×C4 [×12], C23, C23 [×10], D5 [×4], C10 [×3], C10 [×4], C2×C8 [×2], C2×C8 [×6], C22×C4, C22×C4 [×9], C24, Dic5 [×2], C20 [×2], C20 [×2], D10 [×4], D10 [×12], C2×C10, C2×C10 [×6], C22⋊C8 [×4], C22×C8, C22×C8, C23×C4, C5⋊2C8 [×2], C40 [×2], C4×D5 [×8], C2×Dic5 [×2], C2×Dic5 [×2], C2×C20 [×2], C2×C20 [×4], C22×D5 [×6], C22×D5 [×4], C22×C10, C2×C22⋊C8, C2×C5⋊2C8 [×2], C2×C5⋊2C8 [×2], C2×C40 [×2], C2×C40 [×2], C2×C4×D5 [×4], C2×C4×D5 [×4], C22×Dic5, C22×C20, C23×D5, D10⋊1C8 [×4], C22×C5⋊2C8, C22×C40, D5×C22×C4, C2×D10⋊1C8
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C8 [×4], C2×C4 [×6], D4 [×4], C23, D5, C22⋊C4 [×4], C2×C8 [×6], M4(2) [×2], C22×C4, C2×D4 [×2], D10 [×3], C22⋊C8 [×4], C2×C22⋊C4, C22×C8, C2×M4(2), C4×D5 [×2], D20 [×2], C5⋊D4 [×2], C22×D5, C2×C22⋊C8, C8×D5 [×2], C8⋊D5 [×2], D10⋊C4 [×4], C2×C4×D5, C2×D20, C2×C5⋊D4, D10⋊1C8 [×4], D5×C2×C8, C2×C8⋊D5, C2×D10⋊C4, C2×D10⋊1C8
►On 160 points
Generators in S160
(1 91)(2 92)(3 93)(4 94)(5 95)(6 96)(7 97)(8 98)(9 99)(10 100)(11 90)(12 81)(13 82)(14 83)(15 84)(16 85)(17 86)(18 87)(19 88)(20 89)(21 106)(22 107)(23 108)(24 109)(25 110)(26 101)(27 102)(28 103)(29 104)(30 105)(31 116)(32 117)(33 118)(34 119)(35 120)(36 111)(37 112)(38 113)(39 114)(40 115)(41 126)(42 127)(43 128)(44 129)(45 130)(46 121)(47 122)(48 123)(49 124)(50 125)(51 136)(52 137)(53 138)(54 139)(55 140)(56 131)(57 132)(58 133)(59 134)(60 135)(61 146)(62 147)(63 148)(64 149)(65 150)(66 141)(67 142)(68 143)(69 144)(70 145)(71 156)(72 157)(73 158)(74 159)(75 160)(76 151)(77 152)(78 153)(79 154)(80 155)
(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)
(1 104)(2 103)(3 102)(4 101)(5 110)(6 109)(7 108)(8 107)(9 106)(10 105)(11 80)(12 79)(13 78)(14 77)(15 76)(16 75)(17 74)(18 73)(19 72)(20 71)(21 99)(22 98)(23 97)(24 96)(25 95)(26 94)(27 93)(28 92)(29 91)(30 100)(31 129)(32 128)(33 127)(34 126)(35 125)(36 124)(37 123)(38 122)(39 121)(40 130)(41 119)(42 118)(43 117)(44 116)(45 115)(46 114)(47 113)(48 112)(49 111)(50 120)(51 144)(52 143)(53 142)(54 141)(55 150)(56 149)(57 148)(58 147)(59 146)(60 145)(61 134)(62 133)(63 132)(64 131)(65 140)(66 139)(67 138)(68 137)(69 136)(70 135)(81 154)(82 153)(83 152)(84 151)(85 160)(86 159)(87 158)(88 157)(89 156)(90 155)
(1 11 45 145 25 156 36 136)(2 12 46 146 26 157 37 137)(3 13 47 147 27 158 38 138)(4 14 48 148 28 159 39 139)(5 15 49 149 29 160 40 140)(6 16 50 150 30 151 31 131)(7 17 41 141 21 152 32 132)(8 18 42 142 22 153 33 133)(9 19 43 143 23 154 34 134)(10 20 44 144 24 155 35 135)(51 91 90 130 70 110 71 111)(52 92 81 121 61 101 72 112)(53 93 82 122 62 102 73 113)(54 94 83 123 63 103 74 114)(55 95 84 124 64 104 75 115)(56 96 85 125 65 105 76 116)(57 97 86 126 66 106 77 117)(58 98 87 127 67 107 78 118)(59 99 88 128 68 108 79 119)(60 100 89 129 69 109 80 120)
G:=sub<Sym(160)| (1,91)(2,92)(3,93)(4,94)(5,95)(6,96)(7,97)(8,98)(9,99)(10,100)(11,90)(12,81)(13,82)(14,83)(15,84)(16,85)(17,86)(18,87)(19,88)(20,89)(21,106)(22,107)(23,108)(24,109)(25,110)(26,101)(27,102)(28,103)(29,104)(30,105)(31,116)(32,117)(33,118)(34,119)(35,120)(36,111)(37,112)(38,113)(39,114)(40,115)(41,126)(42,127)(43,128)(44,129)(45,130)(46,121)(47,122)(48,123)(49,124)(50,125)(51,136)(52,137)(53,138)(54,139)(55,140)(56,131)(57,132)(58,133)(59,134)(60,135)(61,146)(62,147)(63,148)(64,149)(65,150)(66,141)(67,142)(68,143)(69,144)(70,145)(71,156)(72,157)(73,158)(74,159)(75,160)(76,151)(77,152)(78,153)(79,154)(80,155), (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), (1,104)(2,103)(3,102)(4,101)(5,110)(6,109)(7,108)(8,107)(9,106)(10,105)(11,80)(12,79)(13,78)(14,77)(15,76)(16,75)(17,74)(18,73)(19,72)(20,71)(21,99)(22,98)(23,97)(24,96)(25,95)(26,94)(27,93)(28,92)(29,91)(30,100)(31,129)(32,128)(33,127)(34,126)(35,125)(36,124)(37,123)(38,122)(39,121)(40,130)(41,119)(42,118)(43,117)(44,116)(45,115)(46,114)(47,113)(48,112)(49,111)(50,120)(51,144)(52,143)(53,142)(54,141)(55,150)(56,149)(57,148)(58,147)(59,146)(60,145)(61,134)(62,133)(63,132)(64,131)(65,140)(66,139)(67,138)(68,137)(69,136)(70,135)(81,154)(82,153)(83,152)(84,151)(85,160)(86,159)(87,158)(88,157)(89,156)(90,155), (1,11,45,145,25,156,36,136)(2,12,46,146,26,157,37,137)(3,13,47,147,27,158,38,138)(4,14,48,148,28,159,39,139)(5,15,49,149,29,160,40,140)(6,16,50,150,30,151,31,131)(7,17,41,141,21,152,32,132)(8,18,42,142,22,153,33,133)(9,19,43,143,23,154,34,134)(10,20,44,144,24,155,35,135)(51,91,90,130,70,110,71,111)(52,92,81,121,61,101,72,112)(53,93,82,122,62,102,73,113)(54,94,83,123,63,103,74,114)(55,95,84,124,64,104,75,115)(56,96,85,125,65,105,76,116)(57,97,86,126,66,106,77,117)(58,98,87,127,67,107,78,118)(59,99,88,128,68,108,79,119)(60,100,89,129,69,109,80,120)>;
G:=Group( (1,91)(2,92)(3,93)(4,94)(5,95)(6,96)(7,97)(8,98)(9,99)(10,100)(11,90)(12,81)(13,82)(14,83)(15,84)(16,85)(17,86)(18,87)(19,88)(20,89)(21,106)(22,107)(23,108)(24,109)(25,110)(26,101)(27,102)(28,103)(29,104)(30,105)(31,116)(32,117)(33,118)(34,119)(35,120)(36,111)(37,112)(38,113)(39,114)(40,115)(41,126)(42,127)(43,128)(44,129)(45,130)(46,121)(47,122)(48,123)(49,124)(50,125)(51,136)(52,137)(53,138)(54,139)(55,140)(56,131)(57,132)(58,133)(59,134)(60,135)(61,146)(62,147)(63,148)(64,149)(65,150)(66,141)(67,142)(68,143)(69,144)(70,145)(71,156)(72,157)(73,158)(74,159)(75,160)(76,151)(77,152)(78,153)(79,154)(80,155), (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), (1,104)(2,103)(3,102)(4,101)(5,110)(6,109)(7,108)(8,107)(9,106)(10,105)(11,80)(12,79)(13,78)(14,77)(15,76)(16,75)(17,74)(18,73)(19,72)(20,71)(21,99)(22,98)(23,97)(24,96)(25,95)(26,94)(27,93)(28,92)(29,91)(30,100)(31,129)(32,128)(33,127)(34,126)(35,125)(36,124)(37,123)(38,122)(39,121)(40,130)(41,119)(42,118)(43,117)(44,116)(45,115)(46,114)(47,113)(48,112)(49,111)(50,120)(51,144)(52,143)(53,142)(54,141)(55,150)(56,149)(57,148)(58,147)(59,146)(60,145)(61,134)(62,133)(63,132)(64,131)(65,140)(66,139)(67,138)(68,137)(69,136)(70,135)(81,154)(82,153)(83,152)(84,151)(85,160)(86,159)(87,158)(88,157)(89,156)(90,155), (1,11,45,145,25,156,36,136)(2,12,46,146,26,157,37,137)(3,13,47,147,27,158,38,138)(4,14,48,148,28,159,39,139)(5,15,49,149,29,160,40,140)(6,16,50,150,30,151,31,131)(7,17,41,141,21,152,32,132)(8,18,42,142,22,153,33,133)(9,19,43,143,23,154,34,134)(10,20,44,144,24,155,35,135)(51,91,90,130,70,110,71,111)(52,92,81,121,61,101,72,112)(53,93,82,122,62,102,73,113)(54,94,83,123,63,103,74,114)(55,95,84,124,64,104,75,115)(56,96,85,125,65,105,76,116)(57,97,86,126,66,106,77,117)(58,98,87,127,67,107,78,118)(59,99,88,128,68,108,79,119)(60,100,89,129,69,109,80,120) );
G=PermutationGroup([(1,91),(2,92),(3,93),(4,94),(5,95),(6,96),(7,97),(8,98),(9,99),(10,100),(11,90),(12,81),(13,82),(14,83),(15,84),(16,85),(17,86),(18,87),(19,88),(20,89),(21,106),(22,107),(23,108),(24,109),(25,110),(26,101),(27,102),(28,103),(29,104),(30,105),(31,116),(32,117),(33,118),(34,119),(35,120),(36,111),(37,112),(38,113),(39,114),(40,115),(41,126),(42,127),(43,128),(44,129),(45,130),(46,121),(47,122),(48,123),(49,124),(50,125),(51,136),(52,137),(53,138),(54,139),(55,140),(56,131),(57,132),(58,133),(59,134),(60,135),(61,146),(62,147),(63,148),(64,149),(65,150),(66,141),(67,142),(68,143),(69,144),(70,145),(71,156),(72,157),(73,158),(74,159),(75,160),(76,151),(77,152),(78,153),(79,154),(80,155)], [(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)], [(1,104),(2,103),(3,102),(4,101),(5,110),(6,109),(7,108),(8,107),(9,106),(10,105),(11,80),(12,79),(13,78),(14,77),(15,76),(16,75),(17,74),(18,73),(19,72),(20,71),(21,99),(22,98),(23,97),(24,96),(25,95),(26,94),(27,93),(28,92),(29,91),(30,100),(31,129),(32,128),(33,127),(34,126),(35,125),(36,124),(37,123),(38,122),(39,121),(40,130),(41,119),(42,118),(43,117),(44,116),(45,115),(46,114),(47,113),(48,112),(49,111),(50,120),(51,144),(52,143),(53,142),(54,141),(55,150),(56,149),(57,148),(58,147),(59,146),(60,145),(61,134),(62,133),(63,132),(64,131),(65,140),(66,139),(67,138),(68,137),(69,136),(70,135),(81,154),(82,153),(83,152),(84,151),(85,160),(86,159),(87,158),(88,157),(89,156),(90,155)], [(1,11,45,145,25,156,36,136),(2,12,46,146,26,157,37,137),(3,13,47,147,27,158,38,138),(4,14,48,148,28,159,39,139),(5,15,49,149,29,160,40,140),(6,16,50,150,30,151,31,131),(7,17,41,141,21,152,32,132),(8,18,42,142,22,153,33,133),(9,19,43,143,23,154,34,134),(10,20,44,144,24,155,35,135),(51,91,90,130,70,110,71,111),(52,92,81,121,61,101,72,112),(53,93,82,122,62,102,73,113),(54,94,83,123,63,103,74,114),(55,95,84,124,64,104,75,115),(56,96,85,125,65,105,76,116),(57,97,86,126,66,106,77,117),(58,98,87,127,67,107,78,118),(59,99,88,128,68,108,79,119),(60,100,89,129,69,109,80,120)])
104 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 4A | ··· | 4H | 4I | 4J | 4K | 4L | 5A | 5B | 8A | ··· | 8H | 8I | ··· | 8P | 10A | ··· | 10N | 20A | ··· | 20P | 40A | ··· | 40AF |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | ··· | 8 | 8 | ··· | 8 | 10 | ··· | 10 | 20 | ··· | 20 | 40 | ··· | 40 |
| size | 1 | 1 | ··· | 1 | 10 | 10 | 10 | 10 | 1 | ··· | 1 | 10 | 10 | 10 | 10 | 2 | 2 | 2 | ··· | 2 | 10 | ··· | 10 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
104 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | ||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | C8 | D4 | D5 | M4(2) | D10 | D10 | C4×D5 | D20 | C5⋊D4 | C4×D5 | C8×D5 | C8⋊D5 |
| kernel | C2×D10⋊1C8 | D10⋊1C8 | C22×C5⋊2C8 | C22×C40 | D5×C22×C4 | C2×C4×D5 | C22×Dic5 | C23×D5 | C22×D5 | C2×C20 | C22×C8 | C2×C10 | C2×C8 | C22×C4 | C2×C4 | C2×C4 | C2×C4 | C23 | C22 | C22 |
| # reps | 1 | 4 | 1 | 1 | 1 | 4 | 2 | 2 | 16 | 4 | 2 | 4 | 4 | 2 | 4 | 8 | 8 | 4 | 16 | 16 |
Matrix representation of C2×D10⋊1C8 ►in GL5(𝔽41)
| 40 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 40 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 7 | 35 |
| 0 | 0 | 0 | 7 | 0 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 40 | 40 | 0 | 0 |
| 0 | 0 | 0 | 40 | 1 |
| 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 40 | 39 | 0 | 0 |
| 0 | 37 | 1 | 0 | 0 |
| 0 | 0 | 0 | 24 | 35 |
| 0 | 0 | 0 | 7 | 17 |
G:=sub<GL(5,GF(41))| [40,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,40],[1,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,7,7,0,0,0,35,0],[1,0,0,0,0,0,1,40,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,1,1],[1,0,0,0,0,0,40,37,0,0,0,39,1,0,0,0,0,0,24,7,0,0,0,35,17] >;
C2×D10⋊1C8 in GAP, Magma, Sage, TeX
C_2\times D_{10}\rtimes_1C_8 % in TeX
G:=Group("C2xD10:1C8"); // GroupNames label
G:=SmallGroup(320,735);
// by ID
G=gap.SmallGroup(320,735);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,422,58,136,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^10=c^2=d^8=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^-1,b*d=d*b,d*c*d^-1=b^5*c>;
// generators/relations