metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C5⋊C8⋊4D4, C2.6(D4×F5), C5⋊1(C8⋊9D4), C20⋊C8⋊9C2, C10.2(C4×D4), D10⋊C8⋊7C2, C22⋊C4.2F5, C10.3(C8○D4), (C2×C10)⋊1M4(2), C2.6(D4.F5), C22⋊1(C4.F5), C23.25(C2×F5), D10⋊C4.1C4, Dic5.66(C2×D4), C10.D4.1C4, C23.2F5⋊4C2, C10.C42⋊9C2, Dic5⋊4D4.7C2, C10.10(C2×M4(2)), Dic5.51(C4○D4), C22.69(C22×F5), (C4×Dic5).240C22, (C2×Dic5).323C23, (C22×Dic5).178C22, (C22×C5⋊C8)⋊2C2, (C2×C4.F5)⋊9C2, C2.8(C2×C4.F5), (C2×C5⋊D4).4C4, (C2×C4).20(C2×F5), (C2×C20).78(C2×C4), (C5×C22⋊C4).2C4, (C2×C5⋊C8).21C22, (C2×C4×D5).273C22, (C22×C10).14(C2×C4), (C2×C10).31(C22×C4), (C2×Dic5).48(C2×C4), (C22×D5).40(C2×C4), SmallGroup(320,1031)
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=a3, cac-1=a-1, ad=da, cbc-1=b5, bd=db, dcd=c-1 >
Subgroups: 442 in 124 conjugacy classes, 48 normal (42 characteristic)
C1, C2, C2, C4, C22, C22, C22, C5, C8, C2×C4, C2×C4, D4, C23, C23, D5, C10, C10, C42, C22⋊C4, C22⋊C4, C4⋊C4, C2×C8, M4(2), C22×C4, C2×D4, Dic5, Dic5, C20, D10, C2×C10, C2×C10, C2×C10, C8⋊C4, C22⋊C8, C4⋊C8, C4×D4, C22×C8, C2×M4(2), C5⋊C8, C5⋊C8, C4×D5, C2×Dic5, C2×Dic5, C5⋊D4, C2×C20, C22×D5, C22×C10, C8⋊9D4, C4×Dic5, C10.D4, D10⋊C4, C5×C22⋊C4, C4.F5, C2×C5⋊C8, C2×C5⋊C8, C2×C4×D5, C22×Dic5, C2×C5⋊D4, C20⋊C8, C10.C42, D10⋊C8, C23.2F5, Dic5⋊4D4, C2×C4.F5, C22×C5⋊C8, C5⋊C8⋊D4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, M4(2), C22×C4, C2×D4, C4○D4, F5, C4×D4, C2×M4(2), C8○D4, C2×F5, C8⋊9D4, C4.F5, C22×F5, C2×C4.F5, D4.F5, D4×F5, C5⋊C8⋊D4
►On 160 points
Generators in S160
(1 65 83 36 159)(2 37 66 160 84)(3 153 38 85 67)(4 86 154 68 39)(5 69 87 40 155)(6 33 70 156 88)(7 157 34 81 71)(8 82 158 72 35)(9 133 122 54 103)(10 55 134 104 123)(11 97 56 124 135)(12 125 98 136 49)(13 129 126 50 99)(14 51 130 100 127)(15 101 52 128 131)(16 121 102 132 53)(17 138 114 90 63)(18 91 139 64 115)(19 57 92 116 140)(20 117 58 141 93)(21 142 118 94 59)(22 95 143 60 119)(23 61 96 120 144)(24 113 62 137 89)(25 47 74 150 112)(26 151 48 105 75)(27 106 152 76 41)(28 77 107 42 145)(29 43 78 146 108)(30 147 44 109 79)(31 110 148 80 45)(32 73 111 46 149)
(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 78 121 18)(2 75 122 23)(3 80 123 20)(4 77 124 17)(5 74 125 22)(6 79 126 19)(7 76 127 24)(8 73 128 21)(9 96 160 151)(10 93 153 148)(11 90 154 145)(12 95 155 150)(13 92 156 147)(14 89 157 152)(15 94 158 149)(16 91 159 146)(25 136 60 87)(26 133 61 84)(27 130 62 81)(28 135 63 86)(29 132 64 83)(30 129 57 88)(31 134 58 85)(32 131 59 82)(33 109 50 140)(34 106 51 137)(35 111 52 142)(36 108 53 139)(37 105 54 144)(38 110 55 141)(39 107 56 138)(40 112 49 143)(41 100 113 71)(42 97 114 68)(43 102 115 65)(44 99 116 70)(45 104 117 67)(46 101 118 72)(47 98 119 69)(48 103 120 66)
(1 18)(2 19)(3 20)(4 21)(5 22)(6 23)(7 24)(8 17)(9 44)(10 45)(11 46)(12 47)(13 48)(14 41)(15 42)(16 43)(25 49)(26 50)(27 51)(28 52)(29 53)(30 54)(31 55)(32 56)(33 61)(34 62)(35 63)(36 64)(37 57)(38 58)(39 59)(40 60)(65 91)(66 92)(67 93)(68 94)(69 95)(70 96)(71 89)(72 90)(73 124)(74 125)(75 126)(76 127)(77 128)(78 121)(79 122)(80 123)(81 137)(82 138)(83 139)(84 140)(85 141)(86 142)(87 143)(88 144)(97 149)(98 150)(99 151)(100 152)(101 145)(102 146)(103 147)(104 148)(105 129)(106 130)(107 131)(108 132)(109 133)(110 134)(111 135)(112 136)(113 157)(114 158)(115 159)(116 160)(117 153)(118 154)(119 155)(120 156)
G:=sub<Sym(160)| (1,65,83,36,159)(2,37,66,160,84)(3,153,38,85,67)(4,86,154,68,39)(5,69,87,40,155)(6,33,70,156,88)(7,157,34,81,71)(8,82,158,72,35)(9,133,122,54,103)(10,55,134,104,123)(11,97,56,124,135)(12,125,98,136,49)(13,129,126,50,99)(14,51,130,100,127)(15,101,52,128,131)(16,121,102,132,53)(17,138,114,90,63)(18,91,139,64,115)(19,57,92,116,140)(20,117,58,141,93)(21,142,118,94,59)(22,95,143,60,119)(23,61,96,120,144)(24,113,62,137,89)(25,47,74,150,112)(26,151,48,105,75)(27,106,152,76,41)(28,77,107,42,145)(29,43,78,146,108)(30,147,44,109,79)(31,110,148,80,45)(32,73,111,46,149), (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), (1,78,121,18)(2,75,122,23)(3,80,123,20)(4,77,124,17)(5,74,125,22)(6,79,126,19)(7,76,127,24)(8,73,128,21)(9,96,160,151)(10,93,153,148)(11,90,154,145)(12,95,155,150)(13,92,156,147)(14,89,157,152)(15,94,158,149)(16,91,159,146)(25,136,60,87)(26,133,61,84)(27,130,62,81)(28,135,63,86)(29,132,64,83)(30,129,57,88)(31,134,58,85)(32,131,59,82)(33,109,50,140)(34,106,51,137)(35,111,52,142)(36,108,53,139)(37,105,54,144)(38,110,55,141)(39,107,56,138)(40,112,49,143)(41,100,113,71)(42,97,114,68)(43,102,115,65)(44,99,116,70)(45,104,117,67)(46,101,118,72)(47,98,119,69)(48,103,120,66), (1,18)(2,19)(3,20)(4,21)(5,22)(6,23)(7,24)(8,17)(9,44)(10,45)(11,46)(12,47)(13,48)(14,41)(15,42)(16,43)(25,49)(26,50)(27,51)(28,52)(29,53)(30,54)(31,55)(32,56)(33,61)(34,62)(35,63)(36,64)(37,57)(38,58)(39,59)(40,60)(65,91)(66,92)(67,93)(68,94)(69,95)(70,96)(71,89)(72,90)(73,124)(74,125)(75,126)(76,127)(77,128)(78,121)(79,122)(80,123)(81,137)(82,138)(83,139)(84,140)(85,141)(86,142)(87,143)(88,144)(97,149)(98,150)(99,151)(100,152)(101,145)(102,146)(103,147)(104,148)(105,129)(106,130)(107,131)(108,132)(109,133)(110,134)(111,135)(112,136)(113,157)(114,158)(115,159)(116,160)(117,153)(118,154)(119,155)(120,156)>;
G:=Group( (1,65,83,36,159)(2,37,66,160,84)(3,153,38,85,67)(4,86,154,68,39)(5,69,87,40,155)(6,33,70,156,88)(7,157,34,81,71)(8,82,158,72,35)(9,133,122,54,103)(10,55,134,104,123)(11,97,56,124,135)(12,125,98,136,49)(13,129,126,50,99)(14,51,130,100,127)(15,101,52,128,131)(16,121,102,132,53)(17,138,114,90,63)(18,91,139,64,115)(19,57,92,116,140)(20,117,58,141,93)(21,142,118,94,59)(22,95,143,60,119)(23,61,96,120,144)(24,113,62,137,89)(25,47,74,150,112)(26,151,48,105,75)(27,106,152,76,41)(28,77,107,42,145)(29,43,78,146,108)(30,147,44,109,79)(31,110,148,80,45)(32,73,111,46,149), (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), (1,78,121,18)(2,75,122,23)(3,80,123,20)(4,77,124,17)(5,74,125,22)(6,79,126,19)(7,76,127,24)(8,73,128,21)(9,96,160,151)(10,93,153,148)(11,90,154,145)(12,95,155,150)(13,92,156,147)(14,89,157,152)(15,94,158,149)(16,91,159,146)(25,136,60,87)(26,133,61,84)(27,130,62,81)(28,135,63,86)(29,132,64,83)(30,129,57,88)(31,134,58,85)(32,131,59,82)(33,109,50,140)(34,106,51,137)(35,111,52,142)(36,108,53,139)(37,105,54,144)(38,110,55,141)(39,107,56,138)(40,112,49,143)(41,100,113,71)(42,97,114,68)(43,102,115,65)(44,99,116,70)(45,104,117,67)(46,101,118,72)(47,98,119,69)(48,103,120,66), (1,18)(2,19)(3,20)(4,21)(5,22)(6,23)(7,24)(8,17)(9,44)(10,45)(11,46)(12,47)(13,48)(14,41)(15,42)(16,43)(25,49)(26,50)(27,51)(28,52)(29,53)(30,54)(31,55)(32,56)(33,61)(34,62)(35,63)(36,64)(37,57)(38,58)(39,59)(40,60)(65,91)(66,92)(67,93)(68,94)(69,95)(70,96)(71,89)(72,90)(73,124)(74,125)(75,126)(76,127)(77,128)(78,121)(79,122)(80,123)(81,137)(82,138)(83,139)(84,140)(85,141)(86,142)(87,143)(88,144)(97,149)(98,150)(99,151)(100,152)(101,145)(102,146)(103,147)(104,148)(105,129)(106,130)(107,131)(108,132)(109,133)(110,134)(111,135)(112,136)(113,157)(114,158)(115,159)(116,160)(117,153)(118,154)(119,155)(120,156) );
G=PermutationGroup([[(1,65,83,36,159),(2,37,66,160,84),(3,153,38,85,67),(4,86,154,68,39),(5,69,87,40,155),(6,33,70,156,88),(7,157,34,81,71),(8,82,158,72,35),(9,133,122,54,103),(10,55,134,104,123),(11,97,56,124,135),(12,125,98,136,49),(13,129,126,50,99),(14,51,130,100,127),(15,101,52,128,131),(16,121,102,132,53),(17,138,114,90,63),(18,91,139,64,115),(19,57,92,116,140),(20,117,58,141,93),(21,142,118,94,59),(22,95,143,60,119),(23,61,96,120,144),(24,113,62,137,89),(25,47,74,150,112),(26,151,48,105,75),(27,106,152,76,41),(28,77,107,42,145),(29,43,78,146,108),(30,147,44,109,79),(31,110,148,80,45),(32,73,111,46,149)], [(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)], [(1,78,121,18),(2,75,122,23),(3,80,123,20),(4,77,124,17),(5,74,125,22),(6,79,126,19),(7,76,127,24),(8,73,128,21),(9,96,160,151),(10,93,153,148),(11,90,154,145),(12,95,155,150),(13,92,156,147),(14,89,157,152),(15,94,158,149),(16,91,159,146),(25,136,60,87),(26,133,61,84),(27,130,62,81),(28,135,63,86),(29,132,64,83),(30,129,57,88),(31,134,58,85),(32,131,59,82),(33,109,50,140),(34,106,51,137),(35,111,52,142),(36,108,53,139),(37,105,54,144),(38,110,55,141),(39,107,56,138),(40,112,49,143),(41,100,113,71),(42,97,114,68),(43,102,115,65),(44,99,116,70),(45,104,117,67),(46,101,118,72),(47,98,119,69),(48,103,120,66)], [(1,18),(2,19),(3,20),(4,21),(5,22),(6,23),(7,24),(8,17),(9,44),(10,45),(11,46),(12,47),(13,48),(14,41),(15,42),(16,43),(25,49),(26,50),(27,51),(28,52),(29,53),(30,54),(31,55),(32,56),(33,61),(34,62),(35,63),(36,64),(37,57),(38,58),(39,59),(40,60),(65,91),(66,92),(67,93),(68,94),(69,95),(70,96),(71,89),(72,90),(73,124),(74,125),(75,126),(76,127),(77,128),(78,121),(79,122),(80,123),(81,137),(82,138),(83,139),(84,140),(85,141),(86,142),(87,143),(88,144),(97,149),(98,150),(99,151),(100,152),(101,145),(102,146),(103,147),(104,148),(105,129),(106,130),(107,131),(108,132),(109,133),(110,134),(111,135),(112,136),(113,157),(114,158),(115,159),(116,160),(117,153),(118,154),(119,155),(120,156)]])
38 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 5 | 8A | ··· | 8H | 8I | 8J | 8K | 8L | 10A | 10B | 10C | 10D | 10E | 20A | 20B | 20C | 20D |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 8 | ··· | 8 | 8 | 8 | 8 | 8 | 10 | 10 | 10 | 10 | 10 | 20 | 20 | 20 | 20 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 20 | 4 | 4 | 5 | 5 | 5 | 5 | 10 | 10 | 20 | 4 | 10 | ··· | 10 | 20 | 20 | 20 | 20 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 8 | 8 |
38 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 8 | 8 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | - | + | ||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | C4 | D4 | C4○D4 | M4(2) | C8○D4 | F5 | C2×F5 | C2×F5 | C4.F5 | D4.F5 | D4×F5 |
| kernel | C5⋊C8⋊D4 | C20⋊C8 | C10.C42 | D10⋊C8 | C23.2F5 | Dic5⋊4D4 | C2×C4.F5 | C22×C5⋊C8 | C10.D4 | D10⋊C4 | C5×C22⋊C4 | C2×C5⋊D4 | C5⋊C8 | Dic5 | C2×C10 | C10 | C22⋊C4 | C2×C4 | C23 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 1 | 2 | 1 | 4 | 1 | 1 |
Matrix representation of C5⋊C8⋊D4 ►in GL8(𝔽41)
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 40 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 40 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 40 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 40 |
| 38 | 2 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 3 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 9 | 32 | 0 | 0 |
| 0 | 0 | 0 | 0 | 9 | 32 | 0 | 9 |
| 0 | 0 | 0 | 0 | 9 | 0 | 32 | 9 |
| 0 | 0 | 0 | 0 | 0 | 0 | 32 | 9 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 3 | 40 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 40 |
| 0 | 0 | 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 40 |
G:=sub<GL(8,GF(41))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,40,40,40,40],[38,0,0,0,0,0,0,0,2,3,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,9,9,9,0,0,0,0,0,32,32,0,0,0,0,0,0,0,0,32,32,0,0,0,0,0,9,9,9],[1,3,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40] >;
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(320,1031);
// by ID
G=gap.SmallGroup(320,1031);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,758,219,184,136,6278,1595]);
// Polycyclic
G:=Group<a,b,c,d|a^5=b^8=c^4=d^2=1,b*a*b^-1=a^3,c*a*c^-1=a^-1,a*d=d*a,c*b*c^-1=b^5,b*d=d*b,d*c*d=c^-1>;
// generators/relations