metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C5⋊2C8⋊3D4, C5⋊2(C8⋊D4), C4⋊C4.28D10, C4.167(D4×D5), D10⋊3Q8⋊2C2, Q8⋊C4⋊18D5, C4⋊D20.4C2, C20.125(C2×D4), (C2×C8).177D10, (C2×Q8).18D10, D20⋊5C4⋊25C2, C20.20(C4○D4), C4.33(C4○D20), C10.D8⋊10C2, (C2×Dic5).41D4, (C22×D5).27D4, C22.202(D4×D5), C10.24(C4⋊D4), C2.18(D40⋊C2), C10.64(C8⋊C22), (C2×C40).201C22, (C2×C20).252C23, (C2×D20).70C22, C4⋊Dic5.96C22, (Q8×C10).35C22, C2.27(D10⋊D4), C2.16(Q16⋊D5), C10.62(C8.C22), (C2×Q8⋊D5)⋊5C2, (C2×C8⋊D5)⋊20C2, (C2×C4×D5).29C22, (C5×Q8⋊C4)⋊23C2, (C2×C10).265(C2×D4), (C5×C4⋊C4).53C22, (C2×C5⋊2C8).43C22, (C2×C4).359(C22×D5), SmallGroup(320,439)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5⋊(C8⋊D4)
G = < a,b,c,d | a5=b8=c4=d2=1, bab-1=dad=a-1, ac=ca, cbc-1=b-1, dbd=b3, dcd=c-1 >
Subgroups: 582 in 120 conjugacy classes, 39 normal (37 characteristic)
C1, C2 [×3], C2 [×2], C4 [×2], C4 [×4], C22, C22 [×6], C5, C8 [×3], C2×C4, C2×C4 [×6], D4 [×4], Q8 [×2], C23 [×2], D5 [×2], C10 [×3], C22⋊C4 [×2], C4⋊C4, C4⋊C4 [×2], C2×C8, C2×C8, M4(2) [×2], SD16 [×2], C22×C4, C2×D4 [×2], C2×Q8, Dic5 [×2], C20 [×2], C20 [×2], D10 [×6], C2×C10, D4⋊C4, Q8⋊C4, C2.D8, C4⋊D4, C22⋊Q8, C2×M4(2), C2×SD16, C5⋊2C8 [×2], C40, C4×D5 [×2], D20 [×4], C2×Dic5, C2×Dic5, C2×C20, C2×C20 [×2], C5×Q8 [×2], C22×D5, C22×D5, C8⋊D4, C8⋊D5 [×2], C2×C5⋊2C8, C10.D4, C4⋊Dic5, D10⋊C4 [×2], Q8⋊D5 [×2], C5×C4⋊C4, C2×C40, C2×C4×D5, C2×D20, C2×D20, Q8×C10, C10.D8, D20⋊5C4, C5×Q8⋊C4, C4⋊D20, C2×C8⋊D5, C2×Q8⋊D5, D10⋊3Q8, C5⋊(C8⋊D4)
Quotients: C1, C2 [×7], C22 [×7], D4 [×4], C23, D5, C2×D4 [×2], C4○D4, D10 [×3], C4⋊D4, C8⋊C22, C8.C22, C22×D5, C8⋊D4, C4○D20, D4×D5 [×2], D10⋊D4, D40⋊C2, Q16⋊D5, C5⋊(C8⋊D4)
►On 160 points
Generators in S160
(1 151 103 21 107)(2 108 22 104 152)(3 145 97 23 109)(4 110 24 98 146)(5 147 99 17 111)(6 112 18 100 148)(7 149 101 19 105)(8 106 20 102 150)(9 143 59 51 91)(10 92 52 60 144)(11 137 61 53 93)(12 94 54 62 138)(13 139 63 55 95)(14 96 56 64 140)(15 141 57 49 89)(16 90 50 58 142)(25 36 115 67 128)(26 121 68 116 37)(27 38 117 69 122)(28 123 70 118 39)(29 40 119 71 124)(30 125 72 120 33)(31 34 113 65 126)(32 127 66 114 35)(41 155 136 77 88)(42 81 78 129 156)(43 157 130 79 82)(44 83 80 131 158)(45 159 132 73 84)(46 85 74 133 160)(47 153 134 75 86)(48 87 76 135 154)
(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 41 141 116)(2 48 142 115)(3 47 143 114)(4 46 144 113)(5 45 137 120)(6 44 138 119)(7 43 139 118)(8 42 140 117)(9 66 109 86)(10 65 110 85)(11 72 111 84)(12 71 112 83)(13 70 105 82)(14 69 106 81)(15 68 107 88)(16 67 108 87)(17 73 93 125)(18 80 94 124)(19 79 95 123)(20 78 96 122)(21 77 89 121)(22 76 90 128)(23 75 91 127)(24 74 92 126)(25 104 135 50)(26 103 136 49)(27 102 129 56)(28 101 130 55)(29 100 131 54)(30 99 132 53)(31 98 133 52)(32 97 134 51)(33 147 159 61)(34 146 160 60)(35 145 153 59)(36 152 154 58)(37 151 155 57)(38 150 156 64)(39 149 157 63)(40 148 158 62)
(1 141)(2 144)(3 139)(4 142)(5 137)(6 140)(7 143)(8 138)(9 149)(10 152)(11 147)(12 150)(13 145)(14 148)(15 151)(16 146)(17 53)(18 56)(19 51)(20 54)(21 49)(22 52)(23 55)(24 50)(25 126)(26 121)(27 124)(28 127)(29 122)(30 125)(31 128)(32 123)(33 72)(34 67)(35 70)(36 65)(37 68)(38 71)(39 66)(40 69)(42 44)(43 47)(46 48)(57 107)(58 110)(59 105)(60 108)(61 111)(62 106)(63 109)(64 112)(73 132)(74 135)(75 130)(76 133)(77 136)(78 131)(79 134)(80 129)(81 158)(82 153)(83 156)(84 159)(85 154)(86 157)(87 160)(88 155)(89 103)(90 98)(91 101)(92 104)(93 99)(94 102)(95 97)(96 100)(113 115)(114 118)(117 119)
G:=sub<Sym(160)| (1,151,103,21,107)(2,108,22,104,152)(3,145,97,23,109)(4,110,24,98,146)(5,147,99,17,111)(6,112,18,100,148)(7,149,101,19,105)(8,106,20,102,150)(9,143,59,51,91)(10,92,52,60,144)(11,137,61,53,93)(12,94,54,62,138)(13,139,63,55,95)(14,96,56,64,140)(15,141,57,49,89)(16,90,50,58,142)(25,36,115,67,128)(26,121,68,116,37)(27,38,117,69,122)(28,123,70,118,39)(29,40,119,71,124)(30,125,72,120,33)(31,34,113,65,126)(32,127,66,114,35)(41,155,136,77,88)(42,81,78,129,156)(43,157,130,79,82)(44,83,80,131,158)(45,159,132,73,84)(46,85,74,133,160)(47,153,134,75,86)(48,87,76,135,154), (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), (1,41,141,116)(2,48,142,115)(3,47,143,114)(4,46,144,113)(5,45,137,120)(6,44,138,119)(7,43,139,118)(8,42,140,117)(9,66,109,86)(10,65,110,85)(11,72,111,84)(12,71,112,83)(13,70,105,82)(14,69,106,81)(15,68,107,88)(16,67,108,87)(17,73,93,125)(18,80,94,124)(19,79,95,123)(20,78,96,122)(21,77,89,121)(22,76,90,128)(23,75,91,127)(24,74,92,126)(25,104,135,50)(26,103,136,49)(27,102,129,56)(28,101,130,55)(29,100,131,54)(30,99,132,53)(31,98,133,52)(32,97,134,51)(33,147,159,61)(34,146,160,60)(35,145,153,59)(36,152,154,58)(37,151,155,57)(38,150,156,64)(39,149,157,63)(40,148,158,62), (1,141)(2,144)(3,139)(4,142)(5,137)(6,140)(7,143)(8,138)(9,149)(10,152)(11,147)(12,150)(13,145)(14,148)(15,151)(16,146)(17,53)(18,56)(19,51)(20,54)(21,49)(22,52)(23,55)(24,50)(25,126)(26,121)(27,124)(28,127)(29,122)(30,125)(31,128)(32,123)(33,72)(34,67)(35,70)(36,65)(37,68)(38,71)(39,66)(40,69)(42,44)(43,47)(46,48)(57,107)(58,110)(59,105)(60,108)(61,111)(62,106)(63,109)(64,112)(73,132)(74,135)(75,130)(76,133)(77,136)(78,131)(79,134)(80,129)(81,158)(82,153)(83,156)(84,159)(85,154)(86,157)(87,160)(88,155)(89,103)(90,98)(91,101)(92,104)(93,99)(94,102)(95,97)(96,100)(113,115)(114,118)(117,119)>;
G:=Group( (1,151,103,21,107)(2,108,22,104,152)(3,145,97,23,109)(4,110,24,98,146)(5,147,99,17,111)(6,112,18,100,148)(7,149,101,19,105)(8,106,20,102,150)(9,143,59,51,91)(10,92,52,60,144)(11,137,61,53,93)(12,94,54,62,138)(13,139,63,55,95)(14,96,56,64,140)(15,141,57,49,89)(16,90,50,58,142)(25,36,115,67,128)(26,121,68,116,37)(27,38,117,69,122)(28,123,70,118,39)(29,40,119,71,124)(30,125,72,120,33)(31,34,113,65,126)(32,127,66,114,35)(41,155,136,77,88)(42,81,78,129,156)(43,157,130,79,82)(44,83,80,131,158)(45,159,132,73,84)(46,85,74,133,160)(47,153,134,75,86)(48,87,76,135,154), (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), (1,41,141,116)(2,48,142,115)(3,47,143,114)(4,46,144,113)(5,45,137,120)(6,44,138,119)(7,43,139,118)(8,42,140,117)(9,66,109,86)(10,65,110,85)(11,72,111,84)(12,71,112,83)(13,70,105,82)(14,69,106,81)(15,68,107,88)(16,67,108,87)(17,73,93,125)(18,80,94,124)(19,79,95,123)(20,78,96,122)(21,77,89,121)(22,76,90,128)(23,75,91,127)(24,74,92,126)(25,104,135,50)(26,103,136,49)(27,102,129,56)(28,101,130,55)(29,100,131,54)(30,99,132,53)(31,98,133,52)(32,97,134,51)(33,147,159,61)(34,146,160,60)(35,145,153,59)(36,152,154,58)(37,151,155,57)(38,150,156,64)(39,149,157,63)(40,148,158,62), (1,141)(2,144)(3,139)(4,142)(5,137)(6,140)(7,143)(8,138)(9,149)(10,152)(11,147)(12,150)(13,145)(14,148)(15,151)(16,146)(17,53)(18,56)(19,51)(20,54)(21,49)(22,52)(23,55)(24,50)(25,126)(26,121)(27,124)(28,127)(29,122)(30,125)(31,128)(32,123)(33,72)(34,67)(35,70)(36,65)(37,68)(38,71)(39,66)(40,69)(42,44)(43,47)(46,48)(57,107)(58,110)(59,105)(60,108)(61,111)(62,106)(63,109)(64,112)(73,132)(74,135)(75,130)(76,133)(77,136)(78,131)(79,134)(80,129)(81,158)(82,153)(83,156)(84,159)(85,154)(86,157)(87,160)(88,155)(89,103)(90,98)(91,101)(92,104)(93,99)(94,102)(95,97)(96,100)(113,115)(114,118)(117,119) );
G=PermutationGroup([(1,151,103,21,107),(2,108,22,104,152),(3,145,97,23,109),(4,110,24,98,146),(5,147,99,17,111),(6,112,18,100,148),(7,149,101,19,105),(8,106,20,102,150),(9,143,59,51,91),(10,92,52,60,144),(11,137,61,53,93),(12,94,54,62,138),(13,139,63,55,95),(14,96,56,64,140),(15,141,57,49,89),(16,90,50,58,142),(25,36,115,67,128),(26,121,68,116,37),(27,38,117,69,122),(28,123,70,118,39),(29,40,119,71,124),(30,125,72,120,33),(31,34,113,65,126),(32,127,66,114,35),(41,155,136,77,88),(42,81,78,129,156),(43,157,130,79,82),(44,83,80,131,158),(45,159,132,73,84),(46,85,74,133,160),(47,153,134,75,86),(48,87,76,135,154)], [(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)], [(1,41,141,116),(2,48,142,115),(3,47,143,114),(4,46,144,113),(5,45,137,120),(6,44,138,119),(7,43,139,118),(8,42,140,117),(9,66,109,86),(10,65,110,85),(11,72,111,84),(12,71,112,83),(13,70,105,82),(14,69,106,81),(15,68,107,88),(16,67,108,87),(17,73,93,125),(18,80,94,124),(19,79,95,123),(20,78,96,122),(21,77,89,121),(22,76,90,128),(23,75,91,127),(24,74,92,126),(25,104,135,50),(26,103,136,49),(27,102,129,56),(28,101,130,55),(29,100,131,54),(30,99,132,53),(31,98,133,52),(32,97,134,51),(33,147,159,61),(34,146,160,60),(35,145,153,59),(36,152,154,58),(37,151,155,57),(38,150,156,64),(39,149,157,63),(40,148,158,62)], [(1,141),(2,144),(3,139),(4,142),(5,137),(6,140),(7,143),(8,138),(9,149),(10,152),(11,147),(12,150),(13,145),(14,148),(15,151),(16,146),(17,53),(18,56),(19,51),(20,54),(21,49),(22,52),(23,55),(24,50),(25,126),(26,121),(27,124),(28,127),(29,122),(30,125),(31,128),(32,123),(33,72),(34,67),(35,70),(36,65),(37,68),(38,71),(39,66),(40,69),(42,44),(43,47),(46,48),(57,107),(58,110),(59,105),(60,108),(61,111),(62,106),(63,109),(64,112),(73,132),(74,135),(75,130),(76,133),(77,136),(78,131),(79,134),(80,129),(81,158),(82,153),(83,156),(84,159),(85,154),(86,157),(87,160),(88,155),(89,103),(90,98),(91,101),(92,104),(93,99),(94,102),(95,97),(96,100),(113,115),(114,118),(117,119)])
44 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 5A | 5B | 8A | 8B | 8C | 8D | 10A | ··· | 10F | 20A | 20B | 20C | 20D | 20E | ··· | 20L | 40A | ··· | 40H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 20 | 20 | 20 | 20 | 20 | ··· | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 20 | 40 | 2 | 2 | 8 | 8 | 20 | 40 | 2 | 2 | 4 | 4 | 20 | 20 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 8 | ··· | 8 | 4 | ··· | 4 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | D5 | C4○D4 | D10 | D10 | D10 | C4○D20 | C8⋊C22 | C8.C22 | D4×D5 | D4×D5 | D40⋊C2 | Q16⋊D5 |
| kernel | C5⋊(C8⋊D4) | C10.D8 | D20⋊5C4 | C5×Q8⋊C4 | C4⋊D20 | C2×C8⋊D5 | C2×Q8⋊D5 | D10⋊3Q8 | C5⋊2C8 | C2×Dic5 | C22×D5 | Q8⋊C4 | C20 | C4⋊C4 | C2×C8 | C2×Q8 | C4 | C10 | C10 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 8 | 1 | 1 | 2 | 2 | 4 | 4 |
Matrix representation of C5⋊(C8⋊D4) ►in GL6(𝔽41)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 40 | 6 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 40 | 6 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 20 | 36 | 10 | 36 |
| 0 | 0 | 33 | 21 | 14 | 31 |
| 0 | 0 | 31 | 5 | 20 | 36 |
| 0 | 0 | 27 | 10 | 33 | 21 |
| 26 | 37 | 0 | 0 | 0 | 0 |
| 36 | 15 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 35 | 2 |
| 0 | 0 | 0 | 16 | 39 | 6 |
| 0 | 0 | 35 | 2 | 25 | 0 |
| 0 | 0 | 39 | 6 | 0 | 25 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 28 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 6 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 35 | 1 |
G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,40,0,0,0,0,1,6,0,0,0,0,0,0,0,40,0,0,0,0,1,6],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,20,33,31,27,0,0,36,21,5,10,0,0,10,14,20,33,0,0,36,31,36,21],[26,36,0,0,0,0,37,15,0,0,0,0,0,0,16,0,35,39,0,0,0,16,2,6,0,0,35,39,25,0,0,0,2,6,0,25],[40,28,0,0,0,0,0,1,0,0,0,0,0,0,1,6,0,0,0,0,0,40,0,0,0,0,0,0,40,35,0,0,0,0,0,1] >;
C5⋊(C8⋊D4) in GAP, Magma, Sage, TeX
C_5\rtimes (C_8\rtimes D_4)
% in TeX
G:=Group("C5:(C8:D4)"); // GroupNames label
G:=SmallGroup(320,439);
// by ID
G=gap.SmallGroup(320,439);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,253,120,1094,135,184,570,297,136,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^5=b^8=c^4=d^2=1,b*a*b^-1=d*a*d=a^-1,a*c=c*a,c*b*c^-1=b^-1,d*b*d=b^3,d*c*d=c^-1>;
// generators/relations