direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C5×SD16⋊C4, SD16⋊1C20, C8⋊2(C2×C20), C40⋊30(C2×C4), (C4×Q8)⋊2C10, Q8⋊2(C2×C20), C8⋊C4⋊1C10, (Q8×C20)⋊22C2, (C5×SD16)⋊9C4, D4.2(C2×C20), (C4×D4).5C10, C2.D8⋊11C10, C2.15(D4×C20), (D4×C20).20C2, C10.147(C4×D4), (C2×C20).456D4, C42.8(C2×C10), Q8⋊C4⋊16C10, D4⋊C4.6C10, C4.12(C22×C20), (C10×SD16).4C2, (C2×SD16).1C10, C22.54(D4×C10), C20.259(C4○D4), C20.216(C22×C4), (C2×C40).329C22, (C2×C20).907C23, (C4×C20).249C22, C10.129(C8⋊C22), (D4×C10).292C22, (Q8×C10).256C22, C10.129(C8.C22), C4.4(C5×C4○D4), (C5×Q8)⋊23(C2×C4), (C5×C2.D8)⋊26C2, (C5×C8⋊C4)⋊10C2, C2.4(C5×C8⋊C22), C4⋊C4.48(C2×C10), (C2×C8).18(C2×C10), (C5×D4).33(C2×C4), (C2×C4).102(C5×D4), C2.4(C5×C8.C22), (C5×Q8⋊C4)⋊39C2, (C2×D4).50(C2×C10), (C2×C10).630(C2×D4), (C2×Q8).41(C2×C10), (C5×D4⋊C4).15C2, (C5×C4⋊C4).369C22, (C2×C4).82(C22×C10), SmallGroup(320,941)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5×SD16⋊C4
G = < a,b,c,d | a5=b8=c2=d4=1, ab=ba, ac=ca, ad=da, cbc=b3, dbd-1=b5, cd=dc >
Subgroups: 202 in 120 conjugacy classes, 70 normal (50 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C10, C10, C42, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, SD16, C22×C4, C2×D4, C2×Q8, C20, C20, C2×C10, C2×C10, C8⋊C4, D4⋊C4, Q8⋊C4, C2.D8, C4×D4, C4×Q8, C2×SD16, C40, C40, C2×C20, C2×C20, C5×D4, C5×D4, C5×Q8, C5×Q8, C22×C10, SD16⋊C4, C4×C20, C4×C20, C5×C22⋊C4, C5×C4⋊C4, C5×C4⋊C4, C2×C40, C5×SD16, C22×C20, D4×C10, Q8×C10, C5×C8⋊C4, C5×D4⋊C4, C5×Q8⋊C4, C5×C2.D8, D4×C20, Q8×C20, C10×SD16, C5×SD16⋊C4
Quotients: C1, C2, C4, C22, C5, C2×C4, D4, C23, C10, C22×C4, C2×D4, C4○D4, C20, C2×C10, C4×D4, C8⋊C22, C8.C22, C2×C20, C5×D4, C22×C10, SD16⋊C4, C22×C20, D4×C10, C5×C4○D4, D4×C20, C5×C8⋊C22, C5×C8.C22, C5×SD16⋊C4
►On 160 points
Generators in S160
(1 122 155 40 147)(2 123 156 33 148)(3 124 157 34 149)(4 125 158 35 150)(5 126 159 36 151)(6 127 160 37 152)(7 128 153 38 145)(8 121 154 39 146)(9 116 140 17 132)(10 117 141 18 133)(11 118 142 19 134)(12 119 143 20 135)(13 120 144 21 136)(14 113 137 22 129)(15 114 138 23 130)(16 115 139 24 131)(25 54 96 46 88)(26 55 89 47 81)(27 56 90 48 82)(28 49 91 41 83)(29 50 92 42 84)(30 51 93 43 85)(31 52 94 44 86)(32 53 95 45 87)(57 78 111 70 103)(58 79 112 71 104)(59 80 105 72 97)(60 73 106 65 98)(61 74 107 66 99)(62 75 108 67 100)(63 76 109 68 101)(64 77 110 69 102)
(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)
(2 4)(3 7)(6 8)(9 11)(10 14)(13 15)(17 19)(18 22)(21 23)(25 27)(26 30)(29 31)(33 35)(34 38)(37 39)(42 44)(43 47)(46 48)(50 52)(51 55)(54 56)(58 60)(59 63)(62 64)(65 71)(67 69)(68 72)(73 79)(75 77)(76 80)(81 85)(82 88)(84 86)(89 93)(90 96)(92 94)(97 101)(98 104)(100 102)(105 109)(106 112)(108 110)(113 117)(114 120)(116 118)(121 127)(123 125)(124 128)(129 133)(130 136)(132 134)(137 141)(138 144)(140 142)(145 149)(146 152)(148 150)(153 157)(154 160)(156 158)
(1 115 87 99)(2 120 88 104)(3 117 81 101)(4 114 82 98)(5 119 83 103)(6 116 84 100)(7 113 85 97)(8 118 86 102)(9 42 67 152)(10 47 68 149)(11 44 69 146)(12 41 70 151)(13 46 71 148)(14 43 72 145)(15 48 65 150)(16 45 66 147)(17 50 75 160)(18 55 76 157)(19 52 77 154)(20 49 78 159)(21 54 79 156)(22 51 80 153)(23 56 73 158)(24 53 74 155)(25 58 123 144)(26 63 124 141)(27 60 125 138)(28 57 126 143)(29 62 127 140)(30 59 128 137)(31 64 121 142)(32 61 122 139)(33 136 96 112)(34 133 89 109)(35 130 90 106)(36 135 91 111)(37 132 92 108)(38 129 93 105)(39 134 94 110)(40 131 95 107)
G:=sub<Sym(160)| (1,122,155,40,147)(2,123,156,33,148)(3,124,157,34,149)(4,125,158,35,150)(5,126,159,36,151)(6,127,160,37,152)(7,128,153,38,145)(8,121,154,39,146)(9,116,140,17,132)(10,117,141,18,133)(11,118,142,19,134)(12,119,143,20,135)(13,120,144,21,136)(14,113,137,22,129)(15,114,138,23,130)(16,115,139,24,131)(25,54,96,46,88)(26,55,89,47,81)(27,56,90,48,82)(28,49,91,41,83)(29,50,92,42,84)(30,51,93,43,85)(31,52,94,44,86)(32,53,95,45,87)(57,78,111,70,103)(58,79,112,71,104)(59,80,105,72,97)(60,73,106,65,98)(61,74,107,66,99)(62,75,108,67,100)(63,76,109,68,101)(64,77,110,69,102), (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), (2,4)(3,7)(6,8)(9,11)(10,14)(13,15)(17,19)(18,22)(21,23)(25,27)(26,30)(29,31)(33,35)(34,38)(37,39)(42,44)(43,47)(46,48)(50,52)(51,55)(54,56)(58,60)(59,63)(62,64)(65,71)(67,69)(68,72)(73,79)(75,77)(76,80)(81,85)(82,88)(84,86)(89,93)(90,96)(92,94)(97,101)(98,104)(100,102)(105,109)(106,112)(108,110)(113,117)(114,120)(116,118)(121,127)(123,125)(124,128)(129,133)(130,136)(132,134)(137,141)(138,144)(140,142)(145,149)(146,152)(148,150)(153,157)(154,160)(156,158), (1,115,87,99)(2,120,88,104)(3,117,81,101)(4,114,82,98)(5,119,83,103)(6,116,84,100)(7,113,85,97)(8,118,86,102)(9,42,67,152)(10,47,68,149)(11,44,69,146)(12,41,70,151)(13,46,71,148)(14,43,72,145)(15,48,65,150)(16,45,66,147)(17,50,75,160)(18,55,76,157)(19,52,77,154)(20,49,78,159)(21,54,79,156)(22,51,80,153)(23,56,73,158)(24,53,74,155)(25,58,123,144)(26,63,124,141)(27,60,125,138)(28,57,126,143)(29,62,127,140)(30,59,128,137)(31,64,121,142)(32,61,122,139)(33,136,96,112)(34,133,89,109)(35,130,90,106)(36,135,91,111)(37,132,92,108)(38,129,93,105)(39,134,94,110)(40,131,95,107)>;
G:=Group( (1,122,155,40,147)(2,123,156,33,148)(3,124,157,34,149)(4,125,158,35,150)(5,126,159,36,151)(6,127,160,37,152)(7,128,153,38,145)(8,121,154,39,146)(9,116,140,17,132)(10,117,141,18,133)(11,118,142,19,134)(12,119,143,20,135)(13,120,144,21,136)(14,113,137,22,129)(15,114,138,23,130)(16,115,139,24,131)(25,54,96,46,88)(26,55,89,47,81)(27,56,90,48,82)(28,49,91,41,83)(29,50,92,42,84)(30,51,93,43,85)(31,52,94,44,86)(32,53,95,45,87)(57,78,111,70,103)(58,79,112,71,104)(59,80,105,72,97)(60,73,106,65,98)(61,74,107,66,99)(62,75,108,67,100)(63,76,109,68,101)(64,77,110,69,102), (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), (2,4)(3,7)(6,8)(9,11)(10,14)(13,15)(17,19)(18,22)(21,23)(25,27)(26,30)(29,31)(33,35)(34,38)(37,39)(42,44)(43,47)(46,48)(50,52)(51,55)(54,56)(58,60)(59,63)(62,64)(65,71)(67,69)(68,72)(73,79)(75,77)(76,80)(81,85)(82,88)(84,86)(89,93)(90,96)(92,94)(97,101)(98,104)(100,102)(105,109)(106,112)(108,110)(113,117)(114,120)(116,118)(121,127)(123,125)(124,128)(129,133)(130,136)(132,134)(137,141)(138,144)(140,142)(145,149)(146,152)(148,150)(153,157)(154,160)(156,158), (1,115,87,99)(2,120,88,104)(3,117,81,101)(4,114,82,98)(5,119,83,103)(6,116,84,100)(7,113,85,97)(8,118,86,102)(9,42,67,152)(10,47,68,149)(11,44,69,146)(12,41,70,151)(13,46,71,148)(14,43,72,145)(15,48,65,150)(16,45,66,147)(17,50,75,160)(18,55,76,157)(19,52,77,154)(20,49,78,159)(21,54,79,156)(22,51,80,153)(23,56,73,158)(24,53,74,155)(25,58,123,144)(26,63,124,141)(27,60,125,138)(28,57,126,143)(29,62,127,140)(30,59,128,137)(31,64,121,142)(32,61,122,139)(33,136,96,112)(34,133,89,109)(35,130,90,106)(36,135,91,111)(37,132,92,108)(38,129,93,105)(39,134,94,110)(40,131,95,107) );
G=PermutationGroup([[(1,122,155,40,147),(2,123,156,33,148),(3,124,157,34,149),(4,125,158,35,150),(5,126,159,36,151),(6,127,160,37,152),(7,128,153,38,145),(8,121,154,39,146),(9,116,140,17,132),(10,117,141,18,133),(11,118,142,19,134),(12,119,143,20,135),(13,120,144,21,136),(14,113,137,22,129),(15,114,138,23,130),(16,115,139,24,131),(25,54,96,46,88),(26,55,89,47,81),(27,56,90,48,82),(28,49,91,41,83),(29,50,92,42,84),(30,51,93,43,85),(31,52,94,44,86),(32,53,95,45,87),(57,78,111,70,103),(58,79,112,71,104),(59,80,105,72,97),(60,73,106,65,98),(61,74,107,66,99),(62,75,108,67,100),(63,76,109,68,101),(64,77,110,69,102)], [(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)], [(2,4),(3,7),(6,8),(9,11),(10,14),(13,15),(17,19),(18,22),(21,23),(25,27),(26,30),(29,31),(33,35),(34,38),(37,39),(42,44),(43,47),(46,48),(50,52),(51,55),(54,56),(58,60),(59,63),(62,64),(65,71),(67,69),(68,72),(73,79),(75,77),(76,80),(81,85),(82,88),(84,86),(89,93),(90,96),(92,94),(97,101),(98,104),(100,102),(105,109),(106,112),(108,110),(113,117),(114,120),(116,118),(121,127),(123,125),(124,128),(129,133),(130,136),(132,134),(137,141),(138,144),(140,142),(145,149),(146,152),(148,150),(153,157),(154,160),(156,158)], [(1,115,87,99),(2,120,88,104),(3,117,81,101),(4,114,82,98),(5,119,83,103),(6,116,84,100),(7,113,85,97),(8,118,86,102),(9,42,67,152),(10,47,68,149),(11,44,69,146),(12,41,70,151),(13,46,71,148),(14,43,72,145),(15,48,65,150),(16,45,66,147),(17,50,75,160),(18,55,76,157),(19,52,77,154),(20,49,78,159),(21,54,79,156),(22,51,80,153),(23,56,73,158),(24,53,74,155),(25,58,123,144),(26,63,124,141),(27,60,125,138),(28,57,126,143),(29,62,127,140),(30,59,128,137),(31,64,121,142),(32,61,122,139),(33,136,96,112),(34,133,89,109),(35,130,90,106),(36,135,91,111),(37,132,92,108),(38,129,93,105),(39,134,94,110),(40,131,95,107)]])
110 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | ··· | 4F | 4G | ··· | 4L | 5A | 5B | 5C | 5D | 8A | 8B | 8C | 8D | 10A | ··· | 10L | 10M | ··· | 10T | 20A | ··· | 20X | 20Y | ··· | 20AV | 40A | ··· | 40P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 5 | 5 | 5 | 5 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 4 | 1 | ··· | 1 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
110 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | - | |||||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | C5 | C10 | C10 | C10 | C10 | C10 | C10 | C10 | C20 | D4 | C4○D4 | C5×D4 | C5×C4○D4 | C8⋊C22 | C8.C22 | C5×C8⋊C22 | C5×C8.C22 |
| kernel | C5×SD16⋊C4 | C5×C8⋊C4 | C5×D4⋊C4 | C5×Q8⋊C4 | C5×C2.D8 | D4×C20 | Q8×C20 | C10×SD16 | C5×SD16 | SD16⋊C4 | C8⋊C4 | D4⋊C4 | Q8⋊C4 | C2.D8 | C4×D4 | C4×Q8 | C2×SD16 | SD16 | C2×C20 | C20 | C2×C4 | C4 | C10 | C10 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 8 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 32 | 2 | 2 | 8 | 8 | 1 | 1 | 4 | 4 |
Matrix representation of C5×SD16⋊C4 ►in GL6(𝔽41)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 37 | 0 | 0 | 0 |
| 0 | 0 | 0 | 37 | 0 | 0 |
| 0 | 0 | 0 | 0 | 37 | 0 |
| 0 | 0 | 0 | 0 | 0 | 37 |
| 2 | 23 | 0 | 0 | 0 | 0 |
| 39 | 39 | 0 | 0 | 0 | 0 |
| 0 | 0 | 5 | 36 | 13 | 28 |
| 0 | 0 | 5 | 5 | 13 | 13 |
| 0 | 0 | 28 | 13 | 36 | 5 |
| 0 | 0 | 28 | 28 | 36 | 36 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 23 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 9 | 0 | 0 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,37,0,0,0,0,0,0,37,0,0,0,0,0,0,37,0,0,0,0,0,0,37],[2,39,0,0,0,0,23,39,0,0,0,0,0,0,5,5,28,28,0,0,36,5,13,28,0,0,13,13,36,36,0,0,28,13,5,36],[1,23,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,40],[9,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0] >;
C5×SD16⋊C4 in GAP, Magma, Sage, TeX
C_5\times {\rm SD}_{16}\rtimes C_4 % in TeX
G:=Group("C5xSD16:C4"); // GroupNames label
G:=SmallGroup(320,941);
// by ID
G=gap.SmallGroup(320,941);
# by ID
G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-2,560,589,3446,436,7004,3511,172]);
// Polycyclic
G:=Group<a,b,c,d|a^5=b^8=c^2=d^4=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^3,d*b*d^-1=b^5,c*d=d*c>;
// generators/relations