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, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, Q8, C23, D5, C10, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, M4(2), SD16, C22×C4, C2×D4, C2×Q8, Dic5, C20, C20, D10, C2×C10, D4⋊C4, Q8⋊C4, C2.D8, C4⋊D4, C22⋊Q8, C2×M4(2), C2×SD16, C5⋊2C8, C40, C4×D5, D20, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C5×Q8, C22×D5, C22×D5, C8⋊D4, C8⋊D5, C2×C5⋊2C8, C10.D4, C4⋊Dic5, D10⋊C4, Q8⋊D5, 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, C22, D4, C23, D5, C2×D4, C4○D4, D10, C4⋊D4, C8⋊C22, C8.C22, C22×D5, C8⋊D4, C4○D20, D4×D5, D10⋊D4, D40⋊C2, Q16⋊D5, C5⋊(C8⋊D4)
►On 160 points
Generators in S160
(1 151 100 19 107)(2 108 20 101 152)(3 145 102 21 109)(4 110 22 103 146)(5 147 104 23 111)(6 112 24 97 148)(7 149 98 17 105)(8 106 18 99 150)(9 53 93 58 137)(10 138 59 94 54)(11 55 95 60 139)(12 140 61 96 56)(13 49 89 62 141)(14 142 63 90 50)(15 51 91 64 143)(16 144 57 92 52)(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 82 73 157 130)(42 131 158 74 83)(43 84 75 159 132)(44 133 160 76 85)(45 86 77 153 134)(46 135 154 78 87)(47 88 79 155 136)(48 129 156 80 81)
(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 79 141 116)(2 78 142 115)(3 77 143 114)(4 76 144 113)(5 75 137 120)(6 74 138 119)(7 73 139 118)(8 80 140 117)(9 33 147 159)(10 40 148 158)(11 39 149 157)(12 38 150 156)(13 37 151 155)(14 36 152 154)(15 35 145 153)(16 34 146 160)(17 41 95 123)(18 48 96 122)(19 47 89 121)(20 46 90 128)(21 45 91 127)(22 44 92 126)(23 43 93 125)(24 42 94 124)(25 101 135 50)(26 100 136 49)(27 99 129 56)(28 98 130 55)(29 97 131 54)(30 104 132 53)(31 103 133 52)(32 102 134 51)(57 65 110 85)(58 72 111 84)(59 71 112 83)(60 70 105 82)(61 69 106 81)(62 68 107 88)(63 67 108 87)(64 66 109 86)
(1 141)(2 144)(3 139)(4 142)(5 137)(6 140)(7 143)(8 138)(9 111)(10 106)(11 109)(12 112)(13 107)(14 110)(15 105)(16 108)(17 51)(18 54)(19 49)(20 52)(21 55)(22 50)(23 53)(24 56)(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)(41 134)(42 129)(43 132)(44 135)(45 130)(46 133)(47 136)(48 131)(57 152)(58 147)(59 150)(60 145)(61 148)(62 151)(63 146)(64 149)(73 77)(74 80)(76 78)(81 158)(82 153)(83 156)(84 159)(85 154)(86 157)(87 160)(88 155)(89 100)(90 103)(91 98)(92 101)(93 104)(94 99)(95 102)(96 97)(113 115)(114 118)(117 119)
G:=sub<Sym(160)| (1,151,100,19,107)(2,108,20,101,152)(3,145,102,21,109)(4,110,22,103,146)(5,147,104,23,111)(6,112,24,97,148)(7,149,98,17,105)(8,106,18,99,150)(9,53,93,58,137)(10,138,59,94,54)(11,55,95,60,139)(12,140,61,96,56)(13,49,89,62,141)(14,142,63,90,50)(15,51,91,64,143)(16,144,57,92,52)(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)(41,82,73,157,130)(42,131,158,74,83)(43,84,75,159,132)(44,133,160,76,85)(45,86,77,153,134)(46,135,154,78,87)(47,88,79,155,136)(48,129,156,80,81), (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), (1,79,141,116)(2,78,142,115)(3,77,143,114)(4,76,144,113)(5,75,137,120)(6,74,138,119)(7,73,139,118)(8,80,140,117)(9,33,147,159)(10,40,148,158)(11,39,149,157)(12,38,150,156)(13,37,151,155)(14,36,152,154)(15,35,145,153)(16,34,146,160)(17,41,95,123)(18,48,96,122)(19,47,89,121)(20,46,90,128)(21,45,91,127)(22,44,92,126)(23,43,93,125)(24,42,94,124)(25,101,135,50)(26,100,136,49)(27,99,129,56)(28,98,130,55)(29,97,131,54)(30,104,132,53)(31,103,133,52)(32,102,134,51)(57,65,110,85)(58,72,111,84)(59,71,112,83)(60,70,105,82)(61,69,106,81)(62,68,107,88)(63,67,108,87)(64,66,109,86), (1,141)(2,144)(3,139)(4,142)(5,137)(6,140)(7,143)(8,138)(9,111)(10,106)(11,109)(12,112)(13,107)(14,110)(15,105)(16,108)(17,51)(18,54)(19,49)(20,52)(21,55)(22,50)(23,53)(24,56)(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)(41,134)(42,129)(43,132)(44,135)(45,130)(46,133)(47,136)(48,131)(57,152)(58,147)(59,150)(60,145)(61,148)(62,151)(63,146)(64,149)(73,77)(74,80)(76,78)(81,158)(82,153)(83,156)(84,159)(85,154)(86,157)(87,160)(88,155)(89,100)(90,103)(91,98)(92,101)(93,104)(94,99)(95,102)(96,97)(113,115)(114,118)(117,119)>;
G:=Group( (1,151,100,19,107)(2,108,20,101,152)(3,145,102,21,109)(4,110,22,103,146)(5,147,104,23,111)(6,112,24,97,148)(7,149,98,17,105)(8,106,18,99,150)(9,53,93,58,137)(10,138,59,94,54)(11,55,95,60,139)(12,140,61,96,56)(13,49,89,62,141)(14,142,63,90,50)(15,51,91,64,143)(16,144,57,92,52)(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)(41,82,73,157,130)(42,131,158,74,83)(43,84,75,159,132)(44,133,160,76,85)(45,86,77,153,134)(46,135,154,78,87)(47,88,79,155,136)(48,129,156,80,81), (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), (1,79,141,116)(2,78,142,115)(3,77,143,114)(4,76,144,113)(5,75,137,120)(6,74,138,119)(7,73,139,118)(8,80,140,117)(9,33,147,159)(10,40,148,158)(11,39,149,157)(12,38,150,156)(13,37,151,155)(14,36,152,154)(15,35,145,153)(16,34,146,160)(17,41,95,123)(18,48,96,122)(19,47,89,121)(20,46,90,128)(21,45,91,127)(22,44,92,126)(23,43,93,125)(24,42,94,124)(25,101,135,50)(26,100,136,49)(27,99,129,56)(28,98,130,55)(29,97,131,54)(30,104,132,53)(31,103,133,52)(32,102,134,51)(57,65,110,85)(58,72,111,84)(59,71,112,83)(60,70,105,82)(61,69,106,81)(62,68,107,88)(63,67,108,87)(64,66,109,86), (1,141)(2,144)(3,139)(4,142)(5,137)(6,140)(7,143)(8,138)(9,111)(10,106)(11,109)(12,112)(13,107)(14,110)(15,105)(16,108)(17,51)(18,54)(19,49)(20,52)(21,55)(22,50)(23,53)(24,56)(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)(41,134)(42,129)(43,132)(44,135)(45,130)(46,133)(47,136)(48,131)(57,152)(58,147)(59,150)(60,145)(61,148)(62,151)(63,146)(64,149)(73,77)(74,80)(76,78)(81,158)(82,153)(83,156)(84,159)(85,154)(86,157)(87,160)(88,155)(89,100)(90,103)(91,98)(92,101)(93,104)(94,99)(95,102)(96,97)(113,115)(114,118)(117,119) );
G=PermutationGroup([[(1,151,100,19,107),(2,108,20,101,152),(3,145,102,21,109),(4,110,22,103,146),(5,147,104,23,111),(6,112,24,97,148),(7,149,98,17,105),(8,106,18,99,150),(9,53,93,58,137),(10,138,59,94,54),(11,55,95,60,139),(12,140,61,96,56),(13,49,89,62,141),(14,142,63,90,50),(15,51,91,64,143),(16,144,57,92,52),(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),(41,82,73,157,130),(42,131,158,74,83),(43,84,75,159,132),(44,133,160,76,85),(45,86,77,153,134),(46,135,154,78,87),(47,88,79,155,136),(48,129,156,80,81)], [(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)], [(1,79,141,116),(2,78,142,115),(3,77,143,114),(4,76,144,113),(5,75,137,120),(6,74,138,119),(7,73,139,118),(8,80,140,117),(9,33,147,159),(10,40,148,158),(11,39,149,157),(12,38,150,156),(13,37,151,155),(14,36,152,154),(15,35,145,153),(16,34,146,160),(17,41,95,123),(18,48,96,122),(19,47,89,121),(20,46,90,128),(21,45,91,127),(22,44,92,126),(23,43,93,125),(24,42,94,124),(25,101,135,50),(26,100,136,49),(27,99,129,56),(28,98,130,55),(29,97,131,54),(30,104,132,53),(31,103,133,52),(32,102,134,51),(57,65,110,85),(58,72,111,84),(59,71,112,83),(60,70,105,82),(61,69,106,81),(62,68,107,88),(63,67,108,87),(64,66,109,86)], [(1,141),(2,144),(3,139),(4,142),(5,137),(6,140),(7,143),(8,138),(9,111),(10,106),(11,109),(12,112),(13,107),(14,110),(15,105),(16,108),(17,51),(18,54),(19,49),(20,52),(21,55),(22,50),(23,53),(24,56),(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),(41,134),(42,129),(43,132),(44,135),(45,130),(46,133),(47,136),(48,131),(57,152),(58,147),(59,150),(60,145),(61,148),(62,151),(63,146),(64,149),(73,77),(74,80),(76,78),(81,158),(82,153),(83,156),(84,159),(85,154),(86,157),(87,160),(88,155),(89,100),(90,103),(91,98),(92,101),(93,104),(94,99),(95,102),(96,97),(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