direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C4×C8⋊D5, C20⋊11M4(2), D10.4C42, C42.283D10, Dic5.4C42, (C4×C8)⋊13D5, C8⋊12(C4×D5), C40⋊35(C2×C4), (C4×C40)⋊20C2, C5⋊4(C4×M4(2)), C40⋊8C4⋊30C2, C2.7(D5×C42), (C2×C8).323D10, C10.26(C2×C42), (C4×Dic5).16C4, (D5×C42).12C2, (C4×C20).339C22, C20.182(C22×C4), (C2×C20).805C23, (C2×C40).405C22, C10.37(C2×M4(2)), (C4×Dic5).295C22, C4.97(C2×C4×D5), (C2×C4×D5).17C4, (C4×C5⋊2C8)⋊20C2, C5⋊2C8⋊23(C2×C4), C2.2(C2×C8⋊D5), C22.37(C2×C4×D5), (C4×D5).55(C2×C4), (C2×C4).174(C4×D5), (C2×C20).420(C2×C4), (C2×C8⋊D5).17C2, (C2×C4×D5).337C22, (C22×D5).95(C2×C4), (C2×C4).747(C22×D5), (C2×C10).161(C22×C4), (C2×C5⋊2C8).299C22, (C2×Dic5).135(C2×C4), SmallGroup(320,314)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C4×C8⋊D5
G = < a,b,c,d | a4=b8=c5=d2=1, ab=ba, ac=ca, ad=da, bc=cb, dbd=b5, dcd=c-1 >
Subgroups: 398 in 142 conjugacy classes, 83 normal (19 characteristic)
C1, C2, C2 [×2], C2 [×2], C4 [×6], C4 [×4], C22, C22 [×4], C5, C8 [×4], C8 [×4], C2×C4, C2×C4 [×2], C2×C4 [×11], C23, D5 [×2], C10, C10 [×2], C42, C42 [×3], C2×C8 [×2], C2×C8 [×2], M4(2) [×8], C22×C4 [×3], Dic5 [×2], Dic5 [×2], C20 [×6], D10 [×2], D10 [×2], C2×C10, C4×C8, C4×C8, C8⋊C4 [×2], C2×C42, C2×M4(2) [×2], C5⋊2C8 [×4], C40 [×4], C4×D5 [×4], C4×D5 [×4], C2×Dic5, C2×Dic5 [×2], C2×C20, C2×C20 [×2], C22×D5, C4×M4(2), C8⋊D5 [×8], C2×C5⋊2C8 [×2], C4×Dic5, C4×Dic5 [×2], C4×C20, C2×C40 [×2], C2×C4×D5, C2×C4×D5 [×2], C4×C5⋊2C8, C40⋊8C4 [×2], C4×C40, D5×C42, C2×C8⋊D5 [×2], C4×C8⋊D5
Quotients: C1, C2 [×7], C4 [×12], C22 [×7], C2×C4 [×18], C23, D5, C42 [×4], M4(2) [×4], C22×C4 [×3], D10 [×3], C2×C42, C2×M4(2) [×2], C4×D5 [×6], C22×D5, C4×M4(2), C8⋊D5 [×4], C2×C4×D5 [×3], D5×C42, C2×C8⋊D5 [×2], C4×C8⋊D5
►On 160 points
Generators in S160
(1 18 121 142)(2 19 122 143)(3 20 123 144)(4 21 124 137)(5 22 125 138)(6 23 126 139)(7 24 127 140)(8 17 128 141)(9 48 86 129)(10 41 87 130)(11 42 88 131)(12 43 81 132)(13 44 82 133)(14 45 83 134)(15 46 84 135)(16 47 85 136)(25 111 102 50)(26 112 103 51)(27 105 104 52)(28 106 97 53)(29 107 98 54)(30 108 99 55)(31 109 100 56)(32 110 101 49)(33 62 145 113)(34 63 146 114)(35 64 147 115)(36 57 148 116)(37 58 149 117)(38 59 150 118)(39 60 151 119)(40 61 152 120)(65 94 153 73)(66 95 154 74)(67 96 155 75)(68 89 156 76)(69 90 157 77)(70 91 158 78)(71 92 159 79)(72 93 160 80)
(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 35 81 107 73)(2 36 82 108 74)(3 37 83 109 75)(4 38 84 110 76)(5 39 85 111 77)(6 40 86 112 78)(7 33 87 105 79)(8 34 88 106 80)(9 51 91 126 152)(10 52 92 127 145)(11 53 93 128 146)(12 54 94 121 147)(13 55 95 122 148)(14 56 96 123 149)(15 49 89 124 150)(16 50 90 125 151)(17 63 131 97 72)(18 64 132 98 65)(19 57 133 99 66)(20 58 134 100 67)(21 59 135 101 68)(22 60 136 102 69)(23 61 129 103 70)(24 62 130 104 71)(25 157 138 119 47)(26 158 139 120 48)(27 159 140 113 41)(28 160 141 114 42)(29 153 142 115 43)(30 154 143 116 44)(31 155 144 117 45)(32 156 137 118 46)
(1 94)(2 91)(3 96)(4 93)(5 90)(6 95)(7 92)(8 89)(9 82)(10 87)(11 84)(12 81)(13 86)(14 83)(15 88)(16 85)(17 156)(18 153)(19 158)(20 155)(21 160)(22 157)(23 154)(24 159)(25 60)(26 57)(27 62)(28 59)(29 64)(30 61)(31 58)(32 63)(33 52)(34 49)(35 54)(36 51)(37 56)(38 53)(39 50)(40 55)(41 130)(42 135)(43 132)(44 129)(45 134)(46 131)(47 136)(48 133)(65 142)(66 139)(67 144)(68 141)(69 138)(70 143)(71 140)(72 137)(73 121)(74 126)(75 123)(76 128)(77 125)(78 122)(79 127)(80 124)(97 118)(98 115)(99 120)(100 117)(101 114)(102 119)(103 116)(104 113)(105 145)(106 150)(107 147)(108 152)(109 149)(110 146)(111 151)(112 148)
G:=sub<Sym(160)| (1,18,121,142)(2,19,122,143)(3,20,123,144)(4,21,124,137)(5,22,125,138)(6,23,126,139)(7,24,127,140)(8,17,128,141)(9,48,86,129)(10,41,87,130)(11,42,88,131)(12,43,81,132)(13,44,82,133)(14,45,83,134)(15,46,84,135)(16,47,85,136)(25,111,102,50)(26,112,103,51)(27,105,104,52)(28,106,97,53)(29,107,98,54)(30,108,99,55)(31,109,100,56)(32,110,101,49)(33,62,145,113)(34,63,146,114)(35,64,147,115)(36,57,148,116)(37,58,149,117)(38,59,150,118)(39,60,151,119)(40,61,152,120)(65,94,153,73)(66,95,154,74)(67,96,155,75)(68,89,156,76)(69,90,157,77)(70,91,158,78)(71,92,159,79)(72,93,160,80), (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), (1,35,81,107,73)(2,36,82,108,74)(3,37,83,109,75)(4,38,84,110,76)(5,39,85,111,77)(6,40,86,112,78)(7,33,87,105,79)(8,34,88,106,80)(9,51,91,126,152)(10,52,92,127,145)(11,53,93,128,146)(12,54,94,121,147)(13,55,95,122,148)(14,56,96,123,149)(15,49,89,124,150)(16,50,90,125,151)(17,63,131,97,72)(18,64,132,98,65)(19,57,133,99,66)(20,58,134,100,67)(21,59,135,101,68)(22,60,136,102,69)(23,61,129,103,70)(24,62,130,104,71)(25,157,138,119,47)(26,158,139,120,48)(27,159,140,113,41)(28,160,141,114,42)(29,153,142,115,43)(30,154,143,116,44)(31,155,144,117,45)(32,156,137,118,46), (1,94)(2,91)(3,96)(4,93)(5,90)(6,95)(7,92)(8,89)(9,82)(10,87)(11,84)(12,81)(13,86)(14,83)(15,88)(16,85)(17,156)(18,153)(19,158)(20,155)(21,160)(22,157)(23,154)(24,159)(25,60)(26,57)(27,62)(28,59)(29,64)(30,61)(31,58)(32,63)(33,52)(34,49)(35,54)(36,51)(37,56)(38,53)(39,50)(40,55)(41,130)(42,135)(43,132)(44,129)(45,134)(46,131)(47,136)(48,133)(65,142)(66,139)(67,144)(68,141)(69,138)(70,143)(71,140)(72,137)(73,121)(74,126)(75,123)(76,128)(77,125)(78,122)(79,127)(80,124)(97,118)(98,115)(99,120)(100,117)(101,114)(102,119)(103,116)(104,113)(105,145)(106,150)(107,147)(108,152)(109,149)(110,146)(111,151)(112,148)>;
G:=Group( (1,18,121,142)(2,19,122,143)(3,20,123,144)(4,21,124,137)(5,22,125,138)(6,23,126,139)(7,24,127,140)(8,17,128,141)(9,48,86,129)(10,41,87,130)(11,42,88,131)(12,43,81,132)(13,44,82,133)(14,45,83,134)(15,46,84,135)(16,47,85,136)(25,111,102,50)(26,112,103,51)(27,105,104,52)(28,106,97,53)(29,107,98,54)(30,108,99,55)(31,109,100,56)(32,110,101,49)(33,62,145,113)(34,63,146,114)(35,64,147,115)(36,57,148,116)(37,58,149,117)(38,59,150,118)(39,60,151,119)(40,61,152,120)(65,94,153,73)(66,95,154,74)(67,96,155,75)(68,89,156,76)(69,90,157,77)(70,91,158,78)(71,92,159,79)(72,93,160,80), (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), (1,35,81,107,73)(2,36,82,108,74)(3,37,83,109,75)(4,38,84,110,76)(5,39,85,111,77)(6,40,86,112,78)(7,33,87,105,79)(8,34,88,106,80)(9,51,91,126,152)(10,52,92,127,145)(11,53,93,128,146)(12,54,94,121,147)(13,55,95,122,148)(14,56,96,123,149)(15,49,89,124,150)(16,50,90,125,151)(17,63,131,97,72)(18,64,132,98,65)(19,57,133,99,66)(20,58,134,100,67)(21,59,135,101,68)(22,60,136,102,69)(23,61,129,103,70)(24,62,130,104,71)(25,157,138,119,47)(26,158,139,120,48)(27,159,140,113,41)(28,160,141,114,42)(29,153,142,115,43)(30,154,143,116,44)(31,155,144,117,45)(32,156,137,118,46), (1,94)(2,91)(3,96)(4,93)(5,90)(6,95)(7,92)(8,89)(9,82)(10,87)(11,84)(12,81)(13,86)(14,83)(15,88)(16,85)(17,156)(18,153)(19,158)(20,155)(21,160)(22,157)(23,154)(24,159)(25,60)(26,57)(27,62)(28,59)(29,64)(30,61)(31,58)(32,63)(33,52)(34,49)(35,54)(36,51)(37,56)(38,53)(39,50)(40,55)(41,130)(42,135)(43,132)(44,129)(45,134)(46,131)(47,136)(48,133)(65,142)(66,139)(67,144)(68,141)(69,138)(70,143)(71,140)(72,137)(73,121)(74,126)(75,123)(76,128)(77,125)(78,122)(79,127)(80,124)(97,118)(98,115)(99,120)(100,117)(101,114)(102,119)(103,116)(104,113)(105,145)(106,150)(107,147)(108,152)(109,149)(110,146)(111,151)(112,148) );
G=PermutationGroup([(1,18,121,142),(2,19,122,143),(3,20,123,144),(4,21,124,137),(5,22,125,138),(6,23,126,139),(7,24,127,140),(8,17,128,141),(9,48,86,129),(10,41,87,130),(11,42,88,131),(12,43,81,132),(13,44,82,133),(14,45,83,134),(15,46,84,135),(16,47,85,136),(25,111,102,50),(26,112,103,51),(27,105,104,52),(28,106,97,53),(29,107,98,54),(30,108,99,55),(31,109,100,56),(32,110,101,49),(33,62,145,113),(34,63,146,114),(35,64,147,115),(36,57,148,116),(37,58,149,117),(38,59,150,118),(39,60,151,119),(40,61,152,120),(65,94,153,73),(66,95,154,74),(67,96,155,75),(68,89,156,76),(69,90,157,77),(70,91,158,78),(71,92,159,79),(72,93,160,80)], [(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)], [(1,35,81,107,73),(2,36,82,108,74),(3,37,83,109,75),(4,38,84,110,76),(5,39,85,111,77),(6,40,86,112,78),(7,33,87,105,79),(8,34,88,106,80),(9,51,91,126,152),(10,52,92,127,145),(11,53,93,128,146),(12,54,94,121,147),(13,55,95,122,148),(14,56,96,123,149),(15,49,89,124,150),(16,50,90,125,151),(17,63,131,97,72),(18,64,132,98,65),(19,57,133,99,66),(20,58,134,100,67),(21,59,135,101,68),(22,60,136,102,69),(23,61,129,103,70),(24,62,130,104,71),(25,157,138,119,47),(26,158,139,120,48),(27,159,140,113,41),(28,160,141,114,42),(29,153,142,115,43),(30,154,143,116,44),(31,155,144,117,45),(32,156,137,118,46)], [(1,94),(2,91),(3,96),(4,93),(5,90),(6,95),(7,92),(8,89),(9,82),(10,87),(11,84),(12,81),(13,86),(14,83),(15,88),(16,85),(17,156),(18,153),(19,158),(20,155),(21,160),(22,157),(23,154),(24,159),(25,60),(26,57),(27,62),(28,59),(29,64),(30,61),(31,58),(32,63),(33,52),(34,49),(35,54),(36,51),(37,56),(38,53),(39,50),(40,55),(41,130),(42,135),(43,132),(44,129),(45,134),(46,131),(47,136),(48,133),(65,142),(66,139),(67,144),(68,141),(69,138),(70,143),(71,140),(72,137),(73,121),(74,126),(75,123),(76,128),(77,125),(78,122),(79,127),(80,124),(97,118),(98,115),(99,120),(100,117),(101,114),(102,119),(103,116),(104,113),(105,145),(106,150),(107,147),(108,152),(109,149),(110,146),(111,151),(112,148)])
104 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | ··· | 4L | 4M | ··· | 4R | 5A | 5B | 8A | ··· | 8H | 8I | ··· | 8P | 10A | ··· | 10F | 20A | ··· | 20X | 40A | ··· | 40AF |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 5 | 5 | 8 | ··· | 8 | 8 | ··· | 8 | 10 | ··· | 10 | 20 | ··· | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 10 | 10 | 1 | ··· | 1 | 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 |
| type | + | + | + | + | + | + | + | + | + | |||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | D5 | M4(2) | D10 | D10 | C4×D5 | C4×D5 | C8⋊D5 |
| kernel | C4×C8⋊D5 | C4×C5⋊2C8 | C40⋊8C4 | C4×C40 | D5×C42 | C2×C8⋊D5 | C8⋊D5 | C4×Dic5 | C2×C4×D5 | C4×C8 | C20 | C42 | C2×C8 | C8 | C2×C4 | C4 |
| # reps | 1 | 1 | 2 | 1 | 1 | 2 | 16 | 4 | 4 | 2 | 8 | 2 | 4 | 16 | 8 | 32 |
Matrix representation of C4×C8⋊D5 ►in GL4(𝔽41) generated by
| 9 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 |
| 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 9 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 10 | 18 |
| 0 | 0 | 20 | 31 |
| 6 | 40 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 7 | 40 |
| 0 | 0 | 8 | 40 |
| 35 | 1 | 0 | 0 |
| 6 | 6 | 0 | 0 |
| 0 | 0 | 0 | 35 |
| 0 | 0 | 34 | 0 |
G:=sub<GL(4,GF(41))| [9,0,0,0,0,9,0,0,0,0,9,0,0,0,0,9],[1,0,0,0,0,1,0,0,0,0,10,20,0,0,18,31],[6,1,0,0,40,0,0,0,0,0,7,8,0,0,40,40],[35,6,0,0,1,6,0,0,0,0,0,34,0,0,35,0] >;
C4×C8⋊D5 in GAP, Magma, Sage, TeX
C_4\times C_8\rtimes D_5
% in TeX
G:=Group("C4xC8:D5"); // GroupNames label
G:=SmallGroup(320,314);
// by ID
G=gap.SmallGroup(320,314);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,120,758,58,136,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^8=c^5=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d=b^5,d*c*d=c^-1>;
// generators/relations