direct product, metabelian, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: D5×C4×C8, D10.16C42, C42.281D10, Dic5.16C42, (C4×C40)⋊18C2, C40⋊34(C2×C4), C20⋊11(C2×C8), C2.1(D5×C42), (C8×Dic5)⋊33C2, Dic5⋊10(C2×C8), D10.19(C2×C8), (C2×C8).338D10, C10.26(C22×C8), C10.25(C2×C42), (C4×Dic5).50C4, (D5×C42).33C2, (C2×C40).404C22, C20.181(C22×C4), (C4×C20).337C22, (C2×C20).802C23, (C4×Dic5).356C22, C5⋊4(C2×C4×C8), C2.1(D5×C2×C8), C4.96(C2×C4×D5), (D5×C2×C8).35C2, (C2×C4×D5).48C4, (C4×C5⋊2C8)⋊27C2, C5⋊2C8⋊38(C2×C4), C22.34(C2×C4×D5), (C4×D5).98(C2×C4), (C2×C4).172(C4×D5), (C2×C20).418(C2×C4), (C2×C4×D5).417C22, (C2×C4).744(C22×D5), (C2×C10).158(C22×C4), (C2×C5⋊2C8).351C22, (C2×Dic5).202(C2×C4), (C22×D5).137(C2×C4), SmallGroup(320,311)
Series: Derived ►Chief ►Lower central ►Upper central
| C5 — D5×C4×C8 |
Generators and relations for D5×C4×C8
G = < a,b,c,d | a4=b8=c5=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >
Subgroups: 398 in 162 conjugacy classes, 103 normal (19 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C5, C8, C8, C2×C4, C2×C4, C2×C4, C23, D5, C10, C10, C42, C42, C2×C8, C2×C8, C22×C4, Dic5, C20, D10, C2×C10, C4×C8, C4×C8, C2×C42, C22×C8, C5⋊2C8, C40, C4×D5, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C22×D5, C2×C4×C8, C8×D5, C2×C5⋊2C8, C4×Dic5, C4×Dic5, C4×C20, C2×C40, C2×C4×D5, C2×C4×D5, C4×C5⋊2C8, C8×Dic5, C4×C40, D5×C42, D5×C2×C8, D5×C4×C8
Quotients: C1, C2, C4, C22, C8, C2×C4, C23, D5, C42, C2×C8, C22×C4, D10, C4×C8, C2×C42, C22×C8, C4×D5, C22×D5, C2×C4×C8, C8×D5, C2×C4×D5, D5×C42, D5×C2×C8, D5×C4×C8
►On 160 points
Generators in S160
(1 94 48 53)(2 95 41 54)(3 96 42 55)(4 89 43 56)(5 90 44 49)(6 91 45 50)(7 92 46 51)(8 93 47 52)(9 26 152 109)(10 27 145 110)(11 28 146 111)(12 29 147 112)(13 30 148 105)(14 31 149 106)(15 32 150 107)(16 25 151 108)(17 128 130 34)(18 121 131 35)(19 122 132 36)(20 123 133 37)(21 124 134 38)(22 125 135 39)(23 126 136 40)(24 127 129 33)(57 74 83 157)(58 75 84 158)(59 76 85 159)(60 77 86 160)(61 78 87 153)(62 79 88 154)(63 80 81 155)(64 73 82 156)(65 104 141 116)(66 97 142 117)(67 98 143 118)(68 99 144 119)(69 100 137 120)(70 101 138 113)(71 102 139 114)(72 103 140 115)
(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 84 117 35 147)(2 85 118 36 148)(3 86 119 37 149)(4 87 120 38 150)(5 88 113 39 151)(6 81 114 40 152)(7 82 115 33 145)(8 83 116 34 146)(9 45 63 102 126)(10 46 64 103 127)(11 47 57 104 128)(12 48 58 97 121)(13 41 59 98 122)(14 42 60 99 123)(15 43 61 100 124)(16 44 62 101 125)(17 111 93 157 65)(18 112 94 158 66)(19 105 95 159 67)(20 106 96 160 68)(21 107 89 153 69)(22 108 90 154 70)(23 109 91 155 71)(24 110 92 156 72)(25 49 79 138 135)(26 50 80 139 136)(27 51 73 140 129)(28 52 74 141 130)(29 53 75 142 131)(30 54 76 143 132)(31 55 77 144 133)(32 56 78 137 134)
(1 16)(2 9)(3 10)(4 11)(5 12)(6 13)(7 14)(8 15)(17 78)(18 79)(19 80)(20 73)(21 74)(22 75)(23 76)(24 77)(25 94)(26 95)(27 96)(28 89)(29 90)(30 91)(31 92)(32 93)(33 60)(34 61)(35 62)(36 63)(37 64)(38 57)(39 58)(40 59)(41 152)(42 145)(43 146)(44 147)(45 148)(46 149)(47 150)(48 151)(49 112)(50 105)(51 106)(52 107)(53 108)(54 109)(55 110)(56 111)(65 137)(66 138)(67 139)(68 140)(69 141)(70 142)(71 143)(72 144)(81 122)(82 123)(83 124)(84 125)(85 126)(86 127)(87 128)(88 121)(97 113)(98 114)(99 115)(100 116)(101 117)(102 118)(103 119)(104 120)(129 160)(130 153)(131 154)(132 155)(133 156)(134 157)(135 158)(136 159)
G:=sub<Sym(160)| (1,94,48,53)(2,95,41,54)(3,96,42,55)(4,89,43,56)(5,90,44,49)(6,91,45,50)(7,92,46,51)(8,93,47,52)(9,26,152,109)(10,27,145,110)(11,28,146,111)(12,29,147,112)(13,30,148,105)(14,31,149,106)(15,32,150,107)(16,25,151,108)(17,128,130,34)(18,121,131,35)(19,122,132,36)(20,123,133,37)(21,124,134,38)(22,125,135,39)(23,126,136,40)(24,127,129,33)(57,74,83,157)(58,75,84,158)(59,76,85,159)(60,77,86,160)(61,78,87,153)(62,79,88,154)(63,80,81,155)(64,73,82,156)(65,104,141,116)(66,97,142,117)(67,98,143,118)(68,99,144,119)(69,100,137,120)(70,101,138,113)(71,102,139,114)(72,103,140,115), (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), (1,84,117,35,147)(2,85,118,36,148)(3,86,119,37,149)(4,87,120,38,150)(5,88,113,39,151)(6,81,114,40,152)(7,82,115,33,145)(8,83,116,34,146)(9,45,63,102,126)(10,46,64,103,127)(11,47,57,104,128)(12,48,58,97,121)(13,41,59,98,122)(14,42,60,99,123)(15,43,61,100,124)(16,44,62,101,125)(17,111,93,157,65)(18,112,94,158,66)(19,105,95,159,67)(20,106,96,160,68)(21,107,89,153,69)(22,108,90,154,70)(23,109,91,155,71)(24,110,92,156,72)(25,49,79,138,135)(26,50,80,139,136)(27,51,73,140,129)(28,52,74,141,130)(29,53,75,142,131)(30,54,76,143,132)(31,55,77,144,133)(32,56,78,137,134), (1,16)(2,9)(3,10)(4,11)(5,12)(6,13)(7,14)(8,15)(17,78)(18,79)(19,80)(20,73)(21,74)(22,75)(23,76)(24,77)(25,94)(26,95)(27,96)(28,89)(29,90)(30,91)(31,92)(32,93)(33,60)(34,61)(35,62)(36,63)(37,64)(38,57)(39,58)(40,59)(41,152)(42,145)(43,146)(44,147)(45,148)(46,149)(47,150)(48,151)(49,112)(50,105)(51,106)(52,107)(53,108)(54,109)(55,110)(56,111)(65,137)(66,138)(67,139)(68,140)(69,141)(70,142)(71,143)(72,144)(81,122)(82,123)(83,124)(84,125)(85,126)(86,127)(87,128)(88,121)(97,113)(98,114)(99,115)(100,116)(101,117)(102,118)(103,119)(104,120)(129,160)(130,153)(131,154)(132,155)(133,156)(134,157)(135,158)(136,159)>;
G:=Group( (1,94,48,53)(2,95,41,54)(3,96,42,55)(4,89,43,56)(5,90,44,49)(6,91,45,50)(7,92,46,51)(8,93,47,52)(9,26,152,109)(10,27,145,110)(11,28,146,111)(12,29,147,112)(13,30,148,105)(14,31,149,106)(15,32,150,107)(16,25,151,108)(17,128,130,34)(18,121,131,35)(19,122,132,36)(20,123,133,37)(21,124,134,38)(22,125,135,39)(23,126,136,40)(24,127,129,33)(57,74,83,157)(58,75,84,158)(59,76,85,159)(60,77,86,160)(61,78,87,153)(62,79,88,154)(63,80,81,155)(64,73,82,156)(65,104,141,116)(66,97,142,117)(67,98,143,118)(68,99,144,119)(69,100,137,120)(70,101,138,113)(71,102,139,114)(72,103,140,115), (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), (1,84,117,35,147)(2,85,118,36,148)(3,86,119,37,149)(4,87,120,38,150)(5,88,113,39,151)(6,81,114,40,152)(7,82,115,33,145)(8,83,116,34,146)(9,45,63,102,126)(10,46,64,103,127)(11,47,57,104,128)(12,48,58,97,121)(13,41,59,98,122)(14,42,60,99,123)(15,43,61,100,124)(16,44,62,101,125)(17,111,93,157,65)(18,112,94,158,66)(19,105,95,159,67)(20,106,96,160,68)(21,107,89,153,69)(22,108,90,154,70)(23,109,91,155,71)(24,110,92,156,72)(25,49,79,138,135)(26,50,80,139,136)(27,51,73,140,129)(28,52,74,141,130)(29,53,75,142,131)(30,54,76,143,132)(31,55,77,144,133)(32,56,78,137,134), (1,16)(2,9)(3,10)(4,11)(5,12)(6,13)(7,14)(8,15)(17,78)(18,79)(19,80)(20,73)(21,74)(22,75)(23,76)(24,77)(25,94)(26,95)(27,96)(28,89)(29,90)(30,91)(31,92)(32,93)(33,60)(34,61)(35,62)(36,63)(37,64)(38,57)(39,58)(40,59)(41,152)(42,145)(43,146)(44,147)(45,148)(46,149)(47,150)(48,151)(49,112)(50,105)(51,106)(52,107)(53,108)(54,109)(55,110)(56,111)(65,137)(66,138)(67,139)(68,140)(69,141)(70,142)(71,143)(72,144)(81,122)(82,123)(83,124)(84,125)(85,126)(86,127)(87,128)(88,121)(97,113)(98,114)(99,115)(100,116)(101,117)(102,118)(103,119)(104,120)(129,160)(130,153)(131,154)(132,155)(133,156)(134,157)(135,158)(136,159) );
G=PermutationGroup([[(1,94,48,53),(2,95,41,54),(3,96,42,55),(4,89,43,56),(5,90,44,49),(6,91,45,50),(7,92,46,51),(8,93,47,52),(9,26,152,109),(10,27,145,110),(11,28,146,111),(12,29,147,112),(13,30,148,105),(14,31,149,106),(15,32,150,107),(16,25,151,108),(17,128,130,34),(18,121,131,35),(19,122,132,36),(20,123,133,37),(21,124,134,38),(22,125,135,39),(23,126,136,40),(24,127,129,33),(57,74,83,157),(58,75,84,158),(59,76,85,159),(60,77,86,160),(61,78,87,153),(62,79,88,154),(63,80,81,155),(64,73,82,156),(65,104,141,116),(66,97,142,117),(67,98,143,118),(68,99,144,119),(69,100,137,120),(70,101,138,113),(71,102,139,114),(72,103,140,115)], [(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)], [(1,84,117,35,147),(2,85,118,36,148),(3,86,119,37,149),(4,87,120,38,150),(5,88,113,39,151),(6,81,114,40,152),(7,82,115,33,145),(8,83,116,34,146),(9,45,63,102,126),(10,46,64,103,127),(11,47,57,104,128),(12,48,58,97,121),(13,41,59,98,122),(14,42,60,99,123),(15,43,61,100,124),(16,44,62,101,125),(17,111,93,157,65),(18,112,94,158,66),(19,105,95,159,67),(20,106,96,160,68),(21,107,89,153,69),(22,108,90,154,70),(23,109,91,155,71),(24,110,92,156,72),(25,49,79,138,135),(26,50,80,139,136),(27,51,73,140,129),(28,52,74,141,130),(29,53,75,142,131),(30,54,76,143,132),(31,55,77,144,133),(32,56,78,137,134)], [(1,16),(2,9),(3,10),(4,11),(5,12),(6,13),(7,14),(8,15),(17,78),(18,79),(19,80),(20,73),(21,74),(22,75),(23,76),(24,77),(25,94),(26,95),(27,96),(28,89),(29,90),(30,91),(31,92),(32,93),(33,60),(34,61),(35,62),(36,63),(37,64),(38,57),(39,58),(40,59),(41,152),(42,145),(43,146),(44,147),(45,148),(46,149),(47,150),(48,151),(49,112),(50,105),(51,106),(52,107),(53,108),(54,109),(55,110),(56,111),(65,137),(66,138),(67,139),(68,140),(69,141),(70,142),(71,143),(72,144),(81,122),(82,123),(83,124),(84,125),(85,126),(86,127),(87,128),(88,121),(97,113),(98,114),(99,115),(100,116),(101,117),(102,118),(103,119),(104,120),(129,160),(130,153),(131,154),(132,155),(133,156),(134,157),(135,158),(136,159)]])
128 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | ··· | 4L | 4M | ··· | 4X | 5A | 5B | 8A | ··· | 8P | 8Q | ··· | 8AF | 10A | ··· | 10F | 20A | ··· | 20X | 40A | ··· | 40AF |
| order | 1 | 2 | 2 | 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 | 5 | 5 | 5 | 5 | 1 | ··· | 1 | 5 | ··· | 5 | 2 | 2 | 1 | ··· | 1 | 5 | ··· | 5 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
128 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | |||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | C8 | D5 | D10 | D10 | C4×D5 | C4×D5 | C8×D5 |
| kernel | D5×C4×C8 | C4×C5⋊2C8 | C8×Dic5 | C4×C40 | D5×C42 | D5×C2×C8 | C8×D5 | C4×Dic5 | C2×C4×D5 | C4×D5 | C4×C8 | C42 | C2×C8 | C8 | C2×C4 | C4 |
| # reps | 1 | 1 | 2 | 1 | 1 | 2 | 16 | 4 | 4 | 32 | 2 | 2 | 4 | 16 | 8 | 32 |
Matrix representation of D5×C4×C8 ►in GL4(𝔽41) generated by
| 40 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 40 |
| 14 | 0 | 0 | 0 |
| 0 | 14 | 0 | 0 |
| 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 9 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 40 | 34 |
| 1 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
G:=sub<GL(4,GF(41))| [40,0,0,0,0,9,0,0,0,0,40,0,0,0,0,40],[14,0,0,0,0,14,0,0,0,0,9,0,0,0,0,9],[1,0,0,0,0,1,0,0,0,0,0,40,0,0,1,34],[1,0,0,0,0,40,0,0,0,0,0,1,0,0,1,0] >;
D5×C4×C8 in GAP, Magma, Sage, TeX
D_5\times C_4\times C_8
% in TeX
G:=Group("D5xC4xC8"); // GroupNames label
G:=SmallGroup(320,311);
// by ID
G=gap.SmallGroup(320,311);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,120,58,136,12550]);
// 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,b*d=d*b,d*c*d=c^-1>;
// generators/relations