direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D5×C8⋊C4, D10.17C42, C42.181D10, D10.12M4(2), Dic5.17C42, (C8×D5)⋊9C4, C8⋊15(C4×D5), C40⋊26(C2×C4), C40⋊8C4⋊24C2, C2.9(D5×C42), (C2×C8).269D10, C2.1(D5×M4(2)), C10.28(C2×C42), (C4×Dic5).17C4, (D5×C42).13C2, (C2×C40).225C22, (C4×C20).226C22, (C2×C20).810C23, C20.184(C22×C4), C42.D5⋊17C2, C10.49(C2×M4(2)), (C4×Dic5).297C22, C5⋊4(C2×C8⋊C4), C4.99(C2×C4×D5), (D5×C2×C8).26C2, (C2×C4×D5).18C4, (C5×C8⋊C4)⋊6C2, C5⋊2C8⋊31(C2×C4), C22.39(C2×C4×D5), (C4×D5).99(C2×C4), (C2×C4).126(C4×D5), (C2×C20).318(C2×C4), (C2×C4×D5).418C22, (C2×C4).752(C22×D5), (C2×C10).166(C22×C4), (C2×C5⋊2C8).302C22, (C2×Dic5).203(C2×C4), (C22×D5).138(C2×C4), SmallGroup(320,331)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D5×C8⋊C4
G = < a,b,c,d | a5=b2=c8=d4=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c5 >
Subgroups: 398 in 146 conjugacy classes, 83 normal (19 characteristic)
C1, C2, C2 [×2], C2 [×4], C4 [×2], C4 [×6], C22, C22 [×6], C5, C8 [×4], C8 [×4], C2×C4, C2×C4 [×2], C2×C4 [×11], C23, D5 [×4], C10, C10 [×2], C42, C42 [×3], C2×C8 [×2], C2×C8 [×10], C22×C4 [×3], Dic5 [×2], Dic5 [×2], C20 [×2], C20 [×2], D10 [×6], C2×C10, C8⋊C4, C8⋊C4 [×3], C2×C42, C22×C8 [×2], C5⋊2C8 [×4], C40 [×4], C4×D5 [×4], C4×D5 [×4], C2×Dic5, C2×Dic5 [×2], C2×C20, C2×C20 [×2], C22×D5, C2×C8⋊C4, C8×D5 [×8], C2×C5⋊2C8 [×2], C4×Dic5, C4×Dic5 [×2], C4×C20, C2×C40 [×2], C2×C4×D5, C2×C4×D5 [×2], C42.D5, C40⋊8C4 [×2], C5×C8⋊C4, D5×C42, D5×C2×C8 [×2], D5×C8⋊C4
Quotients: C1, C2 [×7], C4 [×12], C22 [×7], C2×C4 [×18], C23, D5, C42 [×4], M4(2) [×4], C22×C4 [×3], D10 [×3], C8⋊C4 [×4], C2×C42, C2×M4(2) [×2], C4×D5 [×6], C22×D5, C2×C8⋊C4, C2×C4×D5 [×3], D5×C42, D5×M4(2) [×2], D5×C8⋊C4
►On 160 points
Generators in S160
(1 106 26 102 70)(2 107 27 103 71)(3 108 28 104 72)(4 109 29 97 65)(5 110 30 98 66)(6 111 31 99 67)(7 112 32 100 68)(8 105 25 101 69)(9 90 158 81 120)(10 91 159 82 113)(11 92 160 83 114)(12 93 153 84 115)(13 94 154 85 116)(14 95 155 86 117)(15 96 156 87 118)(16 89 157 88 119)(17 48 124 55 79)(18 41 125 56 80)(19 42 126 49 73)(20 43 127 50 74)(21 44 128 51 75)(22 45 121 52 76)(23 46 122 53 77)(24 47 123 54 78)(33 129 61 151 144)(34 130 62 152 137)(35 131 63 145 138)(36 132 64 146 139)(37 133 57 147 140)(38 134 58 148 141)(39 135 59 149 142)(40 136 60 150 143)
(1 135)(2 136)(3 129)(4 130)(5 131)(6 132)(7 133)(8 134)(9 24)(10 17)(11 18)(12 19)(13 20)(14 21)(15 22)(16 23)(25 141)(26 142)(27 143)(28 144)(29 137)(30 138)(31 139)(32 140)(33 108)(34 109)(35 110)(36 111)(37 112)(38 105)(39 106)(40 107)(41 114)(42 115)(43 116)(44 117)(45 118)(46 119)(47 120)(48 113)(49 153)(50 154)(51 155)(52 156)(53 157)(54 158)(55 159)(56 160)(57 68)(58 69)(59 70)(60 71)(61 72)(62 65)(63 66)(64 67)(73 93)(74 94)(75 95)(76 96)(77 89)(78 90)(79 91)(80 92)(81 123)(82 124)(83 125)(84 126)(85 127)(86 128)(87 121)(88 122)(97 152)(98 145)(99 146)(100 147)(101 148)(102 149)(103 150)(104 151)
(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 24 63 94)(2 21 64 91)(3 18 57 96)(4 23 58 93)(5 20 59 90)(6 17 60 95)(7 22 61 92)(8 19 62 89)(9 66 74 135)(10 71 75 132)(11 68 76 129)(12 65 77 134)(13 70 78 131)(14 67 79 136)(15 72 80 133)(16 69 73 130)(25 126 137 88)(26 123 138 85)(27 128 139 82)(28 125 140 87)(29 122 141 84)(30 127 142 81)(31 124 143 86)(32 121 144 83)(33 114 100 52)(34 119 101 49)(35 116 102 54)(36 113 103 51)(37 118 104 56)(38 115 97 53)(39 120 98 50)(40 117 99 55)(41 147 156 108)(42 152 157 105)(43 149 158 110)(44 146 159 107)(45 151 160 112)(46 148 153 109)(47 145 154 106)(48 150 155 111)
G:=sub<Sym(160)| (1,106,26,102,70)(2,107,27,103,71)(3,108,28,104,72)(4,109,29,97,65)(5,110,30,98,66)(6,111,31,99,67)(7,112,32,100,68)(8,105,25,101,69)(9,90,158,81,120)(10,91,159,82,113)(11,92,160,83,114)(12,93,153,84,115)(13,94,154,85,116)(14,95,155,86,117)(15,96,156,87,118)(16,89,157,88,119)(17,48,124,55,79)(18,41,125,56,80)(19,42,126,49,73)(20,43,127,50,74)(21,44,128,51,75)(22,45,121,52,76)(23,46,122,53,77)(24,47,123,54,78)(33,129,61,151,144)(34,130,62,152,137)(35,131,63,145,138)(36,132,64,146,139)(37,133,57,147,140)(38,134,58,148,141)(39,135,59,149,142)(40,136,60,150,143), (1,135)(2,136)(3,129)(4,130)(5,131)(6,132)(7,133)(8,134)(9,24)(10,17)(11,18)(12,19)(13,20)(14,21)(15,22)(16,23)(25,141)(26,142)(27,143)(28,144)(29,137)(30,138)(31,139)(32,140)(33,108)(34,109)(35,110)(36,111)(37,112)(38,105)(39,106)(40,107)(41,114)(42,115)(43,116)(44,117)(45,118)(46,119)(47,120)(48,113)(49,153)(50,154)(51,155)(52,156)(53,157)(54,158)(55,159)(56,160)(57,68)(58,69)(59,70)(60,71)(61,72)(62,65)(63,66)(64,67)(73,93)(74,94)(75,95)(76,96)(77,89)(78,90)(79,91)(80,92)(81,123)(82,124)(83,125)(84,126)(85,127)(86,128)(87,121)(88,122)(97,152)(98,145)(99,146)(100,147)(101,148)(102,149)(103,150)(104,151), (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), (1,24,63,94)(2,21,64,91)(3,18,57,96)(4,23,58,93)(5,20,59,90)(6,17,60,95)(7,22,61,92)(8,19,62,89)(9,66,74,135)(10,71,75,132)(11,68,76,129)(12,65,77,134)(13,70,78,131)(14,67,79,136)(15,72,80,133)(16,69,73,130)(25,126,137,88)(26,123,138,85)(27,128,139,82)(28,125,140,87)(29,122,141,84)(30,127,142,81)(31,124,143,86)(32,121,144,83)(33,114,100,52)(34,119,101,49)(35,116,102,54)(36,113,103,51)(37,118,104,56)(38,115,97,53)(39,120,98,50)(40,117,99,55)(41,147,156,108)(42,152,157,105)(43,149,158,110)(44,146,159,107)(45,151,160,112)(46,148,153,109)(47,145,154,106)(48,150,155,111)>;
G:=Group( (1,106,26,102,70)(2,107,27,103,71)(3,108,28,104,72)(4,109,29,97,65)(5,110,30,98,66)(6,111,31,99,67)(7,112,32,100,68)(8,105,25,101,69)(9,90,158,81,120)(10,91,159,82,113)(11,92,160,83,114)(12,93,153,84,115)(13,94,154,85,116)(14,95,155,86,117)(15,96,156,87,118)(16,89,157,88,119)(17,48,124,55,79)(18,41,125,56,80)(19,42,126,49,73)(20,43,127,50,74)(21,44,128,51,75)(22,45,121,52,76)(23,46,122,53,77)(24,47,123,54,78)(33,129,61,151,144)(34,130,62,152,137)(35,131,63,145,138)(36,132,64,146,139)(37,133,57,147,140)(38,134,58,148,141)(39,135,59,149,142)(40,136,60,150,143), (1,135)(2,136)(3,129)(4,130)(5,131)(6,132)(7,133)(8,134)(9,24)(10,17)(11,18)(12,19)(13,20)(14,21)(15,22)(16,23)(25,141)(26,142)(27,143)(28,144)(29,137)(30,138)(31,139)(32,140)(33,108)(34,109)(35,110)(36,111)(37,112)(38,105)(39,106)(40,107)(41,114)(42,115)(43,116)(44,117)(45,118)(46,119)(47,120)(48,113)(49,153)(50,154)(51,155)(52,156)(53,157)(54,158)(55,159)(56,160)(57,68)(58,69)(59,70)(60,71)(61,72)(62,65)(63,66)(64,67)(73,93)(74,94)(75,95)(76,96)(77,89)(78,90)(79,91)(80,92)(81,123)(82,124)(83,125)(84,126)(85,127)(86,128)(87,121)(88,122)(97,152)(98,145)(99,146)(100,147)(101,148)(102,149)(103,150)(104,151), (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), (1,24,63,94)(2,21,64,91)(3,18,57,96)(4,23,58,93)(5,20,59,90)(6,17,60,95)(7,22,61,92)(8,19,62,89)(9,66,74,135)(10,71,75,132)(11,68,76,129)(12,65,77,134)(13,70,78,131)(14,67,79,136)(15,72,80,133)(16,69,73,130)(25,126,137,88)(26,123,138,85)(27,128,139,82)(28,125,140,87)(29,122,141,84)(30,127,142,81)(31,124,143,86)(32,121,144,83)(33,114,100,52)(34,119,101,49)(35,116,102,54)(36,113,103,51)(37,118,104,56)(38,115,97,53)(39,120,98,50)(40,117,99,55)(41,147,156,108)(42,152,157,105)(43,149,158,110)(44,146,159,107)(45,151,160,112)(46,148,153,109)(47,145,154,106)(48,150,155,111) );
G=PermutationGroup([(1,106,26,102,70),(2,107,27,103,71),(3,108,28,104,72),(4,109,29,97,65),(5,110,30,98,66),(6,111,31,99,67),(7,112,32,100,68),(8,105,25,101,69),(9,90,158,81,120),(10,91,159,82,113),(11,92,160,83,114),(12,93,153,84,115),(13,94,154,85,116),(14,95,155,86,117),(15,96,156,87,118),(16,89,157,88,119),(17,48,124,55,79),(18,41,125,56,80),(19,42,126,49,73),(20,43,127,50,74),(21,44,128,51,75),(22,45,121,52,76),(23,46,122,53,77),(24,47,123,54,78),(33,129,61,151,144),(34,130,62,152,137),(35,131,63,145,138),(36,132,64,146,139),(37,133,57,147,140),(38,134,58,148,141),(39,135,59,149,142),(40,136,60,150,143)], [(1,135),(2,136),(3,129),(4,130),(5,131),(6,132),(7,133),(8,134),(9,24),(10,17),(11,18),(12,19),(13,20),(14,21),(15,22),(16,23),(25,141),(26,142),(27,143),(28,144),(29,137),(30,138),(31,139),(32,140),(33,108),(34,109),(35,110),(36,111),(37,112),(38,105),(39,106),(40,107),(41,114),(42,115),(43,116),(44,117),(45,118),(46,119),(47,120),(48,113),(49,153),(50,154),(51,155),(52,156),(53,157),(54,158),(55,159),(56,160),(57,68),(58,69),(59,70),(60,71),(61,72),(62,65),(63,66),(64,67),(73,93),(74,94),(75,95),(76,96),(77,89),(78,90),(79,91),(80,92),(81,123),(82,124),(83,125),(84,126),(85,127),(86,128),(87,121),(88,122),(97,152),(98,145),(99,146),(100,147),(101,148),(102,149),(103,150),(104,151)], [(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)], [(1,24,63,94),(2,21,64,91),(3,18,57,96),(4,23,58,93),(5,20,59,90),(6,17,60,95),(7,22,61,92),(8,19,62,89),(9,66,74,135),(10,71,75,132),(11,68,76,129),(12,65,77,134),(13,70,78,131),(14,67,79,136),(15,72,80,133),(16,69,73,130),(25,126,137,88),(26,123,138,85),(27,128,139,82),(28,125,140,87),(29,122,141,84),(30,127,142,81),(31,124,143,86),(32,121,144,83),(33,114,100,52),(34,119,101,49),(35,116,102,54),(36,113,103,51),(37,118,104,56),(38,115,97,53),(39,120,98,50),(40,117,99,55),(41,147,156,108),(42,152,157,105),(43,149,158,110),(44,146,159,107),(45,151,160,112),(46,148,153,109),(47,145,154,106),(48,150,155,111)])
80 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 4O | 4P | 5A | 5B | 8A | ··· | 8H | 8I | ··· | 8P | 10A | ··· | 10F | 20A | ··· | 20H | 20I | ··· | 20P | 40A | ··· | 40P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | ··· | 8 | 8 | ··· | 8 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 5 | 5 | 5 | 5 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 5 | 5 | 5 | 5 | 10 | 10 | 10 | 10 | 2 | 2 | 2 | ··· | 2 | 10 | ··· | 10 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
80 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | |||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | D5 | M4(2) | D10 | D10 | C4×D5 | C4×D5 | D5×M4(2) |
| kernel | D5×C8⋊C4 | C42.D5 | C40⋊8C4 | C5×C8⋊C4 | D5×C42 | D5×C2×C8 | C8×D5 | C4×Dic5 | C2×C4×D5 | C8⋊C4 | D10 | C42 | C2×C8 | C8 | C2×C4 | C2 |
| # reps | 1 | 1 | 2 | 1 | 1 | 2 | 16 | 4 | 4 | 2 | 8 | 2 | 4 | 16 | 8 | 8 |
Matrix representation of D5×C8⋊C4 ►in GL4(𝔽41) generated by
| 40 | 1 | 0 | 0 |
| 5 | 35 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 36 | 40 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 9 | 7 |
| 0 | 0 | 19 | 32 |
| 9 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 |
| 0 | 0 | 34 | 40 |
| 0 | 0 | 9 | 7 |
G:=sub<GL(4,GF(41))| [40,5,0,0,1,35,0,0,0,0,1,0,0,0,0,1],[1,36,0,0,0,40,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,9,19,0,0,7,32],[9,0,0,0,0,9,0,0,0,0,34,9,0,0,40,7] >;
D5×C8⋊C4 in GAP, Magma, Sage, TeX
D_5\times C_8\rtimes C_4
% in TeX
G:=Group("D5xC8:C4"); // GroupNames label
G:=SmallGroup(320,331);
// by ID
G=gap.SmallGroup(320,331);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,120,387,58,136,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^5=b^2=c^8=d^4=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^5>;
// generators/relations