metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C8⋊3(C4×D5), C40⋊17(C2×C4), C4.Q8⋊2D5, C8⋊D5⋊4C4, (C4×D5).1Q8, C4.25(Q8×D5), C40⋊5C4⋊25C2, (C2×C8).60D10, C20.14(C2×Q8), C4⋊C4.162D10, C22.85(D4×D5), D10.15(C4⋊C4), C10.D8⋊15C2, C2.5(D40⋊C2), C20.Q8⋊15C2, C10.68(C8⋊C22), C5⋊2(M4(2)⋊C4), Dic5.16(C4⋊C4), C20.103(C22×C4), (C2×C40).109C22, (C2×C20).277C23, (C2×Dic5).218D4, C2.6(SD16⋊D5), (C22×D5).118D4, C10.41(C8.C22), C4⋊Dic5.109C22, C4.78(C2×C4×D5), C5⋊2C8⋊4(C2×C4), (D5×C4⋊C4).5C2, C2.13(D5×C4⋊C4), (C5×C4.Q8)⋊2C2, C10.35(C2×C4⋊C4), (C4×D5).6(C2×C4), C4⋊C4⋊7D5.5C2, (C2×C8⋊D5).2C2, (C2×C4×D5).34C22, (C2×C10).282(C2×D4), (C5×C4⋊C4).70C22, (C2×C5⋊2C8).55C22, (C2×C4).380(C22×D5), SmallGroup(320,488)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C8⋊(C4×D5)
G = < a,b,c,d | a8=b4=c5=d2=1, bab-1=a3, ac=ca, dad=a5, bc=cb, bd=db, dcd=c-1 >
Subgroups: 430 in 118 conjugacy classes, 55 normal (37 characteristic)
C1, C2 [×3], C2 [×2], C4 [×2], C4 [×6], C22, C22 [×4], C5, C8 [×2], C8 [×2], C2×C4, C2×C4 [×11], C23, D5 [×2], C10 [×3], C42, C22⋊C4, C4⋊C4 [×2], C4⋊C4 [×3], C2×C8, C2×C8, M4(2) [×4], C22×C4 [×2], Dic5 [×2], Dic5 [×2], C20 [×2], C20 [×2], D10 [×2], D10 [×2], C2×C10, C4.Q8, C4.Q8, C2.D8 [×2], C2×C4⋊C4, C42⋊C2, C2×M4(2), C5⋊2C8 [×2], C40 [×2], C4×D5 [×4], C4×D5 [×2], C2×Dic5, C2×Dic5 [×2], C2×C20, C2×C20 [×2], C22×D5, M4(2)⋊C4, C8⋊D5 [×4], C2×C5⋊2C8, C4×Dic5, C10.D4, C4⋊Dic5 [×2], D10⋊C4, C5×C4⋊C4 [×2], C2×C40, C2×C4×D5, C2×C4×D5, C10.D8, C20.Q8, C40⋊5C4, C5×C4.Q8, D5×C4⋊C4, C4⋊C4⋊7D5, C2×C8⋊D5, C8⋊(C4×D5)
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×2], Q8 [×2], C23, D5, C4⋊C4 [×4], C22×C4, C2×D4, C2×Q8, D10 [×3], C2×C4⋊C4, C8⋊C22, C8.C22, C4×D5 [×2], C22×D5, M4(2)⋊C4, C2×C4×D5, D4×D5, Q8×D5, D5×C4⋊C4, D40⋊C2, SD16⋊D5, C8⋊(C4×D5)
►On 160 points
Generators in S160
(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 55 28 155)(2 50 29 158)(3 53 30 153)(4 56 31 156)(5 51 32 159)(6 54 25 154)(7 49 26 157)(8 52 27 160)(9 63 102 23)(10 58 103 18)(11 61 104 21)(12 64 97 24)(13 59 98 19)(14 62 99 22)(15 57 100 17)(16 60 101 20)(33 86 150 139)(34 81 151 142)(35 84 152 137)(36 87 145 140)(37 82 146 143)(38 85 147 138)(39 88 148 141)(40 83 149 144)(41 109 79 131)(42 112 80 134)(43 107 73 129)(44 110 74 132)(45 105 75 135)(46 108 76 130)(47 111 77 133)(48 106 78 136)(65 124 117 95)(66 127 118 90)(67 122 119 93)(68 125 120 96)(69 128 113 91)(70 123 114 94)(71 126 115 89)(72 121 116 92)
(1 39 133 23 113)(2 40 134 24 114)(3 33 135 17 115)(4 34 136 18 116)(5 35 129 19 117)(6 36 130 20 118)(7 37 131 21 119)(8 38 132 22 120)(9 91 55 88 47)(10 92 56 81 48)(11 93 49 82 41)(12 94 50 83 42)(13 95 51 84 43)(14 96 52 85 44)(15 89 53 86 45)(16 90 54 87 46)(25 145 108 60 66)(26 146 109 61 67)(27 147 110 62 68)(28 148 111 63 69)(29 149 112 64 70)(30 150 105 57 71)(31 151 106 58 72)(32 152 107 59 65)(73 98 124 159 137)(74 99 125 160 138)(75 100 126 153 139)(76 101 127 154 140)(77 102 128 155 141)(78 103 121 156 142)(79 104 122 157 143)(80 97 123 158 144)
(1 113)(2 118)(3 115)(4 120)(5 117)(6 114)(7 119)(8 116)(9 88)(10 85)(11 82)(12 87)(13 84)(14 81)(15 86)(16 83)(17 33)(18 38)(19 35)(20 40)(21 37)(22 34)(23 39)(24 36)(25 70)(26 67)(27 72)(28 69)(29 66)(30 71)(31 68)(32 65)(42 46)(44 48)(49 93)(50 90)(51 95)(52 92)(53 89)(54 94)(55 91)(56 96)(57 150)(58 147)(59 152)(60 149)(61 146)(62 151)(63 148)(64 145)(74 78)(76 80)(97 140)(98 137)(99 142)(100 139)(101 144)(102 141)(103 138)(104 143)(106 110)(108 112)(121 160)(122 157)(123 154)(124 159)(125 156)(126 153)(127 158)(128 155)(130 134)(132 136)
G:=sub<Sym(160)| (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), (1,55,28,155)(2,50,29,158)(3,53,30,153)(4,56,31,156)(5,51,32,159)(6,54,25,154)(7,49,26,157)(8,52,27,160)(9,63,102,23)(10,58,103,18)(11,61,104,21)(12,64,97,24)(13,59,98,19)(14,62,99,22)(15,57,100,17)(16,60,101,20)(33,86,150,139)(34,81,151,142)(35,84,152,137)(36,87,145,140)(37,82,146,143)(38,85,147,138)(39,88,148,141)(40,83,149,144)(41,109,79,131)(42,112,80,134)(43,107,73,129)(44,110,74,132)(45,105,75,135)(46,108,76,130)(47,111,77,133)(48,106,78,136)(65,124,117,95)(66,127,118,90)(67,122,119,93)(68,125,120,96)(69,128,113,91)(70,123,114,94)(71,126,115,89)(72,121,116,92), (1,39,133,23,113)(2,40,134,24,114)(3,33,135,17,115)(4,34,136,18,116)(5,35,129,19,117)(6,36,130,20,118)(7,37,131,21,119)(8,38,132,22,120)(9,91,55,88,47)(10,92,56,81,48)(11,93,49,82,41)(12,94,50,83,42)(13,95,51,84,43)(14,96,52,85,44)(15,89,53,86,45)(16,90,54,87,46)(25,145,108,60,66)(26,146,109,61,67)(27,147,110,62,68)(28,148,111,63,69)(29,149,112,64,70)(30,150,105,57,71)(31,151,106,58,72)(32,152,107,59,65)(73,98,124,159,137)(74,99,125,160,138)(75,100,126,153,139)(76,101,127,154,140)(77,102,128,155,141)(78,103,121,156,142)(79,104,122,157,143)(80,97,123,158,144), (1,113)(2,118)(3,115)(4,120)(5,117)(6,114)(7,119)(8,116)(9,88)(10,85)(11,82)(12,87)(13,84)(14,81)(15,86)(16,83)(17,33)(18,38)(19,35)(20,40)(21,37)(22,34)(23,39)(24,36)(25,70)(26,67)(27,72)(28,69)(29,66)(30,71)(31,68)(32,65)(42,46)(44,48)(49,93)(50,90)(51,95)(52,92)(53,89)(54,94)(55,91)(56,96)(57,150)(58,147)(59,152)(60,149)(61,146)(62,151)(63,148)(64,145)(74,78)(76,80)(97,140)(98,137)(99,142)(100,139)(101,144)(102,141)(103,138)(104,143)(106,110)(108,112)(121,160)(122,157)(123,154)(124,159)(125,156)(126,153)(127,158)(128,155)(130,134)(132,136)>;
G:=Group( (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), (1,55,28,155)(2,50,29,158)(3,53,30,153)(4,56,31,156)(5,51,32,159)(6,54,25,154)(7,49,26,157)(8,52,27,160)(9,63,102,23)(10,58,103,18)(11,61,104,21)(12,64,97,24)(13,59,98,19)(14,62,99,22)(15,57,100,17)(16,60,101,20)(33,86,150,139)(34,81,151,142)(35,84,152,137)(36,87,145,140)(37,82,146,143)(38,85,147,138)(39,88,148,141)(40,83,149,144)(41,109,79,131)(42,112,80,134)(43,107,73,129)(44,110,74,132)(45,105,75,135)(46,108,76,130)(47,111,77,133)(48,106,78,136)(65,124,117,95)(66,127,118,90)(67,122,119,93)(68,125,120,96)(69,128,113,91)(70,123,114,94)(71,126,115,89)(72,121,116,92), (1,39,133,23,113)(2,40,134,24,114)(3,33,135,17,115)(4,34,136,18,116)(5,35,129,19,117)(6,36,130,20,118)(7,37,131,21,119)(8,38,132,22,120)(9,91,55,88,47)(10,92,56,81,48)(11,93,49,82,41)(12,94,50,83,42)(13,95,51,84,43)(14,96,52,85,44)(15,89,53,86,45)(16,90,54,87,46)(25,145,108,60,66)(26,146,109,61,67)(27,147,110,62,68)(28,148,111,63,69)(29,149,112,64,70)(30,150,105,57,71)(31,151,106,58,72)(32,152,107,59,65)(73,98,124,159,137)(74,99,125,160,138)(75,100,126,153,139)(76,101,127,154,140)(77,102,128,155,141)(78,103,121,156,142)(79,104,122,157,143)(80,97,123,158,144), (1,113)(2,118)(3,115)(4,120)(5,117)(6,114)(7,119)(8,116)(9,88)(10,85)(11,82)(12,87)(13,84)(14,81)(15,86)(16,83)(17,33)(18,38)(19,35)(20,40)(21,37)(22,34)(23,39)(24,36)(25,70)(26,67)(27,72)(28,69)(29,66)(30,71)(31,68)(32,65)(42,46)(44,48)(49,93)(50,90)(51,95)(52,92)(53,89)(54,94)(55,91)(56,96)(57,150)(58,147)(59,152)(60,149)(61,146)(62,151)(63,148)(64,145)(74,78)(76,80)(97,140)(98,137)(99,142)(100,139)(101,144)(102,141)(103,138)(104,143)(106,110)(108,112)(121,160)(122,157)(123,154)(124,159)(125,156)(126,153)(127,158)(128,155)(130,134)(132,136) );
G=PermutationGroup([(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)], [(1,55,28,155),(2,50,29,158),(3,53,30,153),(4,56,31,156),(5,51,32,159),(6,54,25,154),(7,49,26,157),(8,52,27,160),(9,63,102,23),(10,58,103,18),(11,61,104,21),(12,64,97,24),(13,59,98,19),(14,62,99,22),(15,57,100,17),(16,60,101,20),(33,86,150,139),(34,81,151,142),(35,84,152,137),(36,87,145,140),(37,82,146,143),(38,85,147,138),(39,88,148,141),(40,83,149,144),(41,109,79,131),(42,112,80,134),(43,107,73,129),(44,110,74,132),(45,105,75,135),(46,108,76,130),(47,111,77,133),(48,106,78,136),(65,124,117,95),(66,127,118,90),(67,122,119,93),(68,125,120,96),(69,128,113,91),(70,123,114,94),(71,126,115,89),(72,121,116,92)], [(1,39,133,23,113),(2,40,134,24,114),(3,33,135,17,115),(4,34,136,18,116),(5,35,129,19,117),(6,36,130,20,118),(7,37,131,21,119),(8,38,132,22,120),(9,91,55,88,47),(10,92,56,81,48),(11,93,49,82,41),(12,94,50,83,42),(13,95,51,84,43),(14,96,52,85,44),(15,89,53,86,45),(16,90,54,87,46),(25,145,108,60,66),(26,146,109,61,67),(27,147,110,62,68),(28,148,111,63,69),(29,149,112,64,70),(30,150,105,57,71),(31,151,106,58,72),(32,152,107,59,65),(73,98,124,159,137),(74,99,125,160,138),(75,100,126,153,139),(76,101,127,154,140),(77,102,128,155,141),(78,103,121,156,142),(79,104,122,157,143),(80,97,123,158,144)], [(1,113),(2,118),(3,115),(4,120),(5,117),(6,114),(7,119),(8,116),(9,88),(10,85),(11,82),(12,87),(13,84),(14,81),(15,86),(16,83),(17,33),(18,38),(19,35),(20,40),(21,37),(22,34),(23,39),(24,36),(25,70),(26,67),(27,72),(28,69),(29,66),(30,71),(31,68),(32,65),(42,46),(44,48),(49,93),(50,90),(51,95),(52,92),(53,89),(54,94),(55,91),(56,96),(57,150),(58,147),(59,152),(60,149),(61,146),(62,151),(63,148),(64,145),(74,78),(76,80),(97,140),(98,137),(99,142),(100,139),(101,144),(102,141),(103,138),(104,143),(106,110),(108,112),(121,160),(122,157),(123,154),(124,159),(125,156),(126,153),(127,158),(128,155),(130,134),(132,136)])
50 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 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 | 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 | 10 | 10 | 2 | 2 | 4 | 4 | 4 | 4 | 10 | 10 | 20 | 20 | 20 | 20 | 2 | 2 | 4 | 4 | 20 | 20 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 8 | ··· | 8 | 4 | ··· | 4 |
50 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | - | + | + | + | + | + | + | - | - | + | + | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | Q8 | D4 | D4 | D5 | D10 | D10 | C4×D5 | C8⋊C22 | C8.C22 | Q8×D5 | D4×D5 | D40⋊C2 | SD16⋊D5 |
| kernel | C8⋊(C4×D5) | C10.D8 | C20.Q8 | C40⋊5C4 | C5×C4.Q8 | D5×C4⋊C4 | C4⋊C4⋊7D5 | C2×C8⋊D5 | C8⋊D5 | C4×D5 | C2×Dic5 | C22×D5 | C4.Q8 | C4⋊C4 | C2×C8 | C8 | C10 | C10 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 8 | 2 | 1 | 1 | 2 | 4 | 2 | 8 | 1 | 1 | 2 | 2 | 4 | 4 |
Matrix representation of C8⋊(C4×D5) ►in GL6(𝔽41)
| 35 | 2 | 0 | 0 | 0 | 0 |
| 2 | 6 | 0 | 0 | 0 | 0 |
| 0 | 0 | 32 | 15 | 9 | 26 |
| 0 | 0 | 26 | 9 | 15 | 32 |
| 0 | 0 | 32 | 15 | 32 | 15 |
| 0 | 0 | 26 | 9 | 26 | 9 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 27 | 0 | 7 | 0 |
| 0 | 0 | 0 | 27 | 0 | 7 |
| 0 | 0 | 7 | 0 | 14 | 0 |
| 0 | 0 | 0 | 7 | 0 | 14 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 34 | 40 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 34 | 40 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 34 | 40 | 0 | 0 |
| 0 | 0 | 7 | 7 | 0 | 0 |
| 0 | 0 | 0 | 0 | 34 | 40 |
| 0 | 0 | 0 | 0 | 7 | 7 |
G:=sub<GL(6,GF(41))| [35,2,0,0,0,0,2,6,0,0,0,0,0,0,32,26,32,26,0,0,15,9,15,9,0,0,9,15,32,26,0,0,26,32,15,9],[0,1,0,0,0,0,40,0,0,0,0,0,0,0,27,0,7,0,0,0,0,27,0,7,0,0,7,0,14,0,0,0,0,7,0,14],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,34,1,0,0,0,0,40,0,0,0,0,0,0,0,34,1,0,0,0,0,40,0],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,34,7,0,0,0,0,40,7,0,0,0,0,0,0,34,7,0,0,0,0,40,7] >;
C8⋊(C4×D5) in GAP, Magma, Sage, TeX
C_8\rtimes (C_4\times D_5)
% in TeX
G:=Group("C8:(C4xD5)"); // GroupNames label
G:=SmallGroup(320,488);
// by ID
G=gap.SmallGroup(320,488);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,477,120,555,58,438,102,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^8=b^4=c^5=d^2=1,b*a*b^-1=a^3,a*c=c*a,d*a*d=a^5,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations