direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C5×C8⋊6D4, C40⋊28D4, C20⋊12M4(2), C8⋊6(C5×D4), C4⋊C8⋊16C10, (C4×C40)⋊31C2, (C4×C8)⋊15C10, C4⋊C4.8C20, (C2×D4).9C20, (C4×D4).3C10, C4.81(D4×C10), C2.11(D4×C20), C4⋊1(C5×M4(2)), C22⋊C8⋊14C10, (D4×C20).18C2, (D4×C10).34C4, C20.486(C2×D4), C10.143(C4×D4), C22⋊C4.5C20, C10.74(C8○D4), C42.69(C2×C10), C23.12(C2×C20), C20.355(C4○D4), (C2×M4(2))⋊15C10, (C10×M4(2))⋊33C2, (C4×C20).354C22, (C2×C20).992C23, (C2×C40).328C22, C2.10(C10×M4(2)), C10.88(C2×M4(2)), C22.48(C22×C20), (C22×C20).418C22, (C5×C4⋊C8)⋊35C2, C2.8(C5×C8○D4), (C5×C4⋊C4).33C4, C4.53(C5×C4○D4), (C5×C22⋊C8)⋊31C2, (C2×C8).53(C2×C10), (C2×C4).29(C2×C20), (C2×C20).386(C2×C4), (C5×C22⋊C4).20C4, (C22×C4).36(C2×C10), (C22×C10).92(C2×C4), (C2×C4).160(C22×C10), (C2×C10).343(C22×C4), SmallGroup(320,937)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5×C8⋊6D4
G = < a,b,c,d | a5=b8=c4=d2=1, ab=ba, ac=ca, ad=da, bc=cb, dbd=b5, dcd=c-1 >
Subgroups: 178 in 122 conjugacy classes, 74 normal (38 characteristic)
C1, C2 [×3], C2 [×2], C4 [×2], C4 [×2], C4 [×3], C22, C22 [×6], C5, C8 [×2], C8 [×3], C2×C4 [×3], C2×C4 [×2], C2×C4 [×4], D4 [×2], C23 [×2], C10 [×3], C10 [×2], C42, C22⋊C4 [×2], C4⋊C4, C2×C8 [×2], C2×C8 [×2], M4(2) [×4], C22×C4 [×2], C2×D4, C20 [×2], C20 [×2], C20 [×3], C2×C10, C2×C10 [×6], C4×C8, C22⋊C8 [×2], C4⋊C8, C4×D4, C2×M4(2) [×2], C40 [×2], C40 [×3], C2×C20 [×3], C2×C20 [×2], C2×C20 [×4], C5×D4 [×2], C22×C10 [×2], C8⋊6D4, C4×C20, C5×C22⋊C4 [×2], C5×C4⋊C4, C2×C40 [×2], C2×C40 [×2], C5×M4(2) [×4], C22×C20 [×2], D4×C10, C4×C40, C5×C22⋊C8 [×2], C5×C4⋊C8, D4×C20, C10×M4(2) [×2], C5×C8⋊6D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C5, C2×C4 [×6], D4 [×2], C23, C10 [×7], M4(2) [×2], C22×C4, C2×D4, C4○D4, C20 [×4], C2×C10 [×7], C4×D4, C2×M4(2), C8○D4, C2×C20 [×6], C5×D4 [×2], C22×C10, C8⋊6D4, C5×M4(2) [×2], C22×C20, D4×C10, C5×C4○D4, D4×C20, C10×M4(2), C5×C8○D4, C5×C8⋊6D4
►On 160 points
Generators in S160
(1 51 14 57 17)(2 52 15 58 18)(3 53 16 59 19)(4 54 9 60 20)(5 55 10 61 21)(6 56 11 62 22)(7 49 12 63 23)(8 50 13 64 24)(25 88 73 33 65)(26 81 74 34 66)(27 82 75 35 67)(28 83 76 36 68)(29 84 77 37 69)(30 85 78 38 70)(31 86 79 39 71)(32 87 80 40 72)(41 157 116 149 108)(42 158 117 150 109)(43 159 118 151 110)(44 160 119 152 111)(45 153 120 145 112)(46 154 113 146 105)(47 155 114 147 106)(48 156 115 148 107)(89 122 142 101 129)(90 123 143 102 130)(91 124 144 103 131)(92 125 137 104 132)(93 126 138 97 133)(94 127 139 98 134)(95 128 140 99 135)(96 121 141 100 136)
(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)
(1 105 27 101)(2 106 28 102)(3 107 29 103)(4 108 30 104)(5 109 31 97)(6 110 32 98)(7 111 25 99)(8 112 26 100)(9 157 78 92)(10 158 79 93)(11 159 80 94)(12 160 73 95)(13 153 74 96)(14 154 75 89)(15 155 76 90)(16 156 77 91)(17 146 67 142)(18 147 68 143)(19 148 69 144)(20 149 70 137)(21 150 71 138)(22 151 72 139)(23 152 65 140)(24 145 66 141)(33 128 63 119)(34 121 64 120)(35 122 57 113)(36 123 58 114)(37 124 59 115)(38 125 60 116)(39 126 61 117)(40 127 62 118)(41 85 132 54)(42 86 133 55)(43 87 134 56)(44 88 135 49)(45 81 136 50)(46 82 129 51)(47 83 130 52)(48 84 131 53)
(1 101)(2 98)(3 103)(4 100)(5 97)(6 102)(7 99)(8 104)(9 96)(10 93)(11 90)(12 95)(13 92)(14 89)(15 94)(16 91)(17 142)(18 139)(19 144)(20 141)(21 138)(22 143)(23 140)(24 137)(25 111)(26 108)(27 105)(28 110)(29 107)(30 112)(31 109)(32 106)(33 119)(34 116)(35 113)(36 118)(37 115)(38 120)(39 117)(40 114)(41 81)(42 86)(43 83)(44 88)(45 85)(46 82)(47 87)(48 84)(49 135)(50 132)(51 129)(52 134)(53 131)(54 136)(55 133)(56 130)(57 122)(58 127)(59 124)(60 121)(61 126)(62 123)(63 128)(64 125)(65 152)(66 149)(67 146)(68 151)(69 148)(70 145)(71 150)(72 147)(73 160)(74 157)(75 154)(76 159)(77 156)(78 153)(79 158)(80 155)
G:=sub<Sym(160)| (1,51,14,57,17)(2,52,15,58,18)(3,53,16,59,19)(4,54,9,60,20)(5,55,10,61,21)(6,56,11,62,22)(7,49,12,63,23)(8,50,13,64,24)(25,88,73,33,65)(26,81,74,34,66)(27,82,75,35,67)(28,83,76,36,68)(29,84,77,37,69)(30,85,78,38,70)(31,86,79,39,71)(32,87,80,40,72)(41,157,116,149,108)(42,158,117,150,109)(43,159,118,151,110)(44,160,119,152,111)(45,153,120,145,112)(46,154,113,146,105)(47,155,114,147,106)(48,156,115,148,107)(89,122,142,101,129)(90,123,143,102,130)(91,124,144,103,131)(92,125,137,104,132)(93,126,138,97,133)(94,127,139,98,134)(95,128,140,99,135)(96,121,141,100,136), (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), (1,105,27,101)(2,106,28,102)(3,107,29,103)(4,108,30,104)(5,109,31,97)(6,110,32,98)(7,111,25,99)(8,112,26,100)(9,157,78,92)(10,158,79,93)(11,159,80,94)(12,160,73,95)(13,153,74,96)(14,154,75,89)(15,155,76,90)(16,156,77,91)(17,146,67,142)(18,147,68,143)(19,148,69,144)(20,149,70,137)(21,150,71,138)(22,151,72,139)(23,152,65,140)(24,145,66,141)(33,128,63,119)(34,121,64,120)(35,122,57,113)(36,123,58,114)(37,124,59,115)(38,125,60,116)(39,126,61,117)(40,127,62,118)(41,85,132,54)(42,86,133,55)(43,87,134,56)(44,88,135,49)(45,81,136,50)(46,82,129,51)(47,83,130,52)(48,84,131,53), (1,101)(2,98)(3,103)(4,100)(5,97)(6,102)(7,99)(8,104)(9,96)(10,93)(11,90)(12,95)(13,92)(14,89)(15,94)(16,91)(17,142)(18,139)(19,144)(20,141)(21,138)(22,143)(23,140)(24,137)(25,111)(26,108)(27,105)(28,110)(29,107)(30,112)(31,109)(32,106)(33,119)(34,116)(35,113)(36,118)(37,115)(38,120)(39,117)(40,114)(41,81)(42,86)(43,83)(44,88)(45,85)(46,82)(47,87)(48,84)(49,135)(50,132)(51,129)(52,134)(53,131)(54,136)(55,133)(56,130)(57,122)(58,127)(59,124)(60,121)(61,126)(62,123)(63,128)(64,125)(65,152)(66,149)(67,146)(68,151)(69,148)(70,145)(71,150)(72,147)(73,160)(74,157)(75,154)(76,159)(77,156)(78,153)(79,158)(80,155)>;
G:=Group( (1,51,14,57,17)(2,52,15,58,18)(3,53,16,59,19)(4,54,9,60,20)(5,55,10,61,21)(6,56,11,62,22)(7,49,12,63,23)(8,50,13,64,24)(25,88,73,33,65)(26,81,74,34,66)(27,82,75,35,67)(28,83,76,36,68)(29,84,77,37,69)(30,85,78,38,70)(31,86,79,39,71)(32,87,80,40,72)(41,157,116,149,108)(42,158,117,150,109)(43,159,118,151,110)(44,160,119,152,111)(45,153,120,145,112)(46,154,113,146,105)(47,155,114,147,106)(48,156,115,148,107)(89,122,142,101,129)(90,123,143,102,130)(91,124,144,103,131)(92,125,137,104,132)(93,126,138,97,133)(94,127,139,98,134)(95,128,140,99,135)(96,121,141,100,136), (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), (1,105,27,101)(2,106,28,102)(3,107,29,103)(4,108,30,104)(5,109,31,97)(6,110,32,98)(7,111,25,99)(8,112,26,100)(9,157,78,92)(10,158,79,93)(11,159,80,94)(12,160,73,95)(13,153,74,96)(14,154,75,89)(15,155,76,90)(16,156,77,91)(17,146,67,142)(18,147,68,143)(19,148,69,144)(20,149,70,137)(21,150,71,138)(22,151,72,139)(23,152,65,140)(24,145,66,141)(33,128,63,119)(34,121,64,120)(35,122,57,113)(36,123,58,114)(37,124,59,115)(38,125,60,116)(39,126,61,117)(40,127,62,118)(41,85,132,54)(42,86,133,55)(43,87,134,56)(44,88,135,49)(45,81,136,50)(46,82,129,51)(47,83,130,52)(48,84,131,53), (1,101)(2,98)(3,103)(4,100)(5,97)(6,102)(7,99)(8,104)(9,96)(10,93)(11,90)(12,95)(13,92)(14,89)(15,94)(16,91)(17,142)(18,139)(19,144)(20,141)(21,138)(22,143)(23,140)(24,137)(25,111)(26,108)(27,105)(28,110)(29,107)(30,112)(31,109)(32,106)(33,119)(34,116)(35,113)(36,118)(37,115)(38,120)(39,117)(40,114)(41,81)(42,86)(43,83)(44,88)(45,85)(46,82)(47,87)(48,84)(49,135)(50,132)(51,129)(52,134)(53,131)(54,136)(55,133)(56,130)(57,122)(58,127)(59,124)(60,121)(61,126)(62,123)(63,128)(64,125)(65,152)(66,149)(67,146)(68,151)(69,148)(70,145)(71,150)(72,147)(73,160)(74,157)(75,154)(76,159)(77,156)(78,153)(79,158)(80,155) );
G=PermutationGroup([(1,51,14,57,17),(2,52,15,58,18),(3,53,16,59,19),(4,54,9,60,20),(5,55,10,61,21),(6,56,11,62,22),(7,49,12,63,23),(8,50,13,64,24),(25,88,73,33,65),(26,81,74,34,66),(27,82,75,35,67),(28,83,76,36,68),(29,84,77,37,69),(30,85,78,38,70),(31,86,79,39,71),(32,87,80,40,72),(41,157,116,149,108),(42,158,117,150,109),(43,159,118,151,110),(44,160,119,152,111),(45,153,120,145,112),(46,154,113,146,105),(47,155,114,147,106),(48,156,115,148,107),(89,122,142,101,129),(90,123,143,102,130),(91,124,144,103,131),(92,125,137,104,132),(93,126,138,97,133),(94,127,139,98,134),(95,128,140,99,135),(96,121,141,100,136)], [(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)], [(1,105,27,101),(2,106,28,102),(3,107,29,103),(4,108,30,104),(5,109,31,97),(6,110,32,98),(7,111,25,99),(8,112,26,100),(9,157,78,92),(10,158,79,93),(11,159,80,94),(12,160,73,95),(13,153,74,96),(14,154,75,89),(15,155,76,90),(16,156,77,91),(17,146,67,142),(18,147,68,143),(19,148,69,144),(20,149,70,137),(21,150,71,138),(22,151,72,139),(23,152,65,140),(24,145,66,141),(33,128,63,119),(34,121,64,120),(35,122,57,113),(36,123,58,114),(37,124,59,115),(38,125,60,116),(39,126,61,117),(40,127,62,118),(41,85,132,54),(42,86,133,55),(43,87,134,56),(44,88,135,49),(45,81,136,50),(46,82,129,51),(47,83,130,52),(48,84,131,53)], [(1,101),(2,98),(3,103),(4,100),(5,97),(6,102),(7,99),(8,104),(9,96),(10,93),(11,90),(12,95),(13,92),(14,89),(15,94),(16,91),(17,142),(18,139),(19,144),(20,141),(21,138),(22,143),(23,140),(24,137),(25,111),(26,108),(27,105),(28,110),(29,107),(30,112),(31,109),(32,106),(33,119),(34,116),(35,113),(36,118),(37,115),(38,120),(39,117),(40,114),(41,81),(42,86),(43,83),(44,88),(45,85),(46,82),(47,87),(48,84),(49,135),(50,132),(51,129),(52,134),(53,131),(54,136),(55,133),(56,130),(57,122),(58,127),(59,124),(60,121),(61,126),(62,123),(63,128),(64,125),(65,152),(66,149),(67,146),(68,151),(69,148),(70,145),(71,150),(72,147),(73,160),(74,157),(75,154),(76,159),(77,156),(78,153),(79,158),(80,155)])
140 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 5A | 5B | 5C | 5D | 8A | ··· | 8H | 8I | 8J | 8K | 8L | 10A | ··· | 10L | 10M | ··· | 10T | 20A | ··· | 20P | 20Q | ··· | 20AF | 20AG | ··· | 20AN | 40A | ··· | 40AF | 40AG | ··· | 40AV |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 8 | ··· | 8 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 | 20 | ··· | 20 | 40 | ··· | 40 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 1 | ··· | 1 | 4 | ··· | 4 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
140 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 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | |||||||||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | C5 | C10 | C10 | C10 | C10 | C10 | C20 | C20 | C20 | D4 | M4(2) | C4○D4 | C8○D4 | C5×D4 | C5×M4(2) | C5×C4○D4 | C5×C8○D4 |
| kernel | C5×C8⋊6D4 | C4×C40 | C5×C22⋊C8 | C5×C4⋊C8 | D4×C20 | C10×M4(2) | C5×C22⋊C4 | C5×C4⋊C4 | D4×C10 | C8⋊6D4 | C4×C8 | C22⋊C8 | C4⋊C8 | C4×D4 | C2×M4(2) | C22⋊C4 | C4⋊C4 | C2×D4 | C40 | C20 | C20 | C10 | C8 | C4 | C4 | C2 |
| # reps | 1 | 1 | 2 | 1 | 1 | 2 | 4 | 2 | 2 | 4 | 4 | 8 | 4 | 4 | 8 | 16 | 8 | 8 | 2 | 4 | 2 | 4 | 8 | 16 | 8 | 16 |
Matrix representation of C5×C8⋊6D4 ►in GL4(𝔽41) generated by
| 18 | 0 | 0 | 0 |
| 0 | 18 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 16 | 39 | 0 | 0 |
| 21 | 25 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 9 | 0 |
| 34 | 6 | 0 | 0 |
| 19 | 7 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 34 | 6 | 0 | 0 |
| 33 | 7 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 40 |
G:=sub<GL(4,GF(41))| [18,0,0,0,0,18,0,0,0,0,1,0,0,0,0,1],[16,21,0,0,39,25,0,0,0,0,0,9,0,0,1,0],[34,19,0,0,6,7,0,0,0,0,1,0,0,0,0,1],[34,33,0,0,6,7,0,0,0,0,1,0,0,0,0,40] >;
C5×C8⋊6D4 in GAP, Magma, Sage, TeX
C_5\times C_8\rtimes_6D_4
% in TeX
G:=Group("C5xC8:6D4"); // GroupNames label
G:=SmallGroup(320,937);
// by ID
G=gap.SmallGroup(320,937);
# by ID
G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-2,560,589,3446,1731,436,124]);
// Polycyclic
G:=Group<a,b,c,d|a^5=b^8=c^4=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d=b^5,d*c*d=c^-1>;
// generators/relations