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), (C2×Dic5).323C23, (C4×Dic5).240C22, (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, (C2×C10).31(C22×C4), (C22×C10).14(C2×C4), (C2×Dic5).48(C2×C4), (C22×D5).40(C2×C4), SmallGroup(320,1031)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 442 in 124 conjugacy classes, 48 normal (42 characteristic)
C1, C2 [×3], C2 [×3], C4 [×6], C22, C22 [×2], C22 [×5], C5, C8 [×5], C2×C4 [×2], C2×C4 [×7], D4 [×2], C23, C23, D5, C10 [×3], C10 [×2], C42, C22⋊C4, C22⋊C4, C4⋊C4, C2×C8 [×6], M4(2) [×2], C22×C4 [×2], C2×D4, Dic5 [×2], Dic5 [×2], C20 [×2], D10 [×3], C2×C10, C2×C10 [×2], C2×C10 [×2], C8⋊C4, C22⋊C8 [×2], C4⋊C8, C4×D4, C22×C8, C2×M4(2), C5⋊C8 [×2], C5⋊C8 [×3], C4×D5 [×2], C2×Dic5 [×3], C2×Dic5 [×2], C5⋊D4 [×2], C2×C20 [×2], C22×D5, C22×C10, C8⋊9D4, C4×Dic5, C10.D4, D10⋊C4, C5×C22⋊C4, C4.F5 [×2], C2×C5⋊C8 [×4], C2×C5⋊C8 [×2], 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 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×2], C23, M4(2) [×2], C22×C4, C2×D4, C4○D4, F5, C4×D4, C2×M4(2), C8○D4, C2×F5 [×3], C8⋊9D4, C4.F5 [×2], C22×F5, C2×C4.F5, D4.F5, D4×F5, C5⋊C8⋊D4
Generators and relations
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 >
►On 160 points
Generators in S160
(1 65 86 36 159)(2 37 66 160 87)(3 153 38 88 67)(4 81 154 68 39)(5 69 82 40 155)(6 33 70 156 83)(7 157 34 84 71)(8 85 158 72 35)(9 131 60 99 43)(10 100 132 44 61)(11 45 101 62 133)(12 63 46 134 102)(13 135 64 103 47)(14 104 136 48 57)(15 41 97 58 129)(16 59 42 130 98)(17 116 121 89 144)(18 90 117 137 122)(19 138 91 123 118)(20 124 139 119 92)(21 120 125 93 140)(22 94 113 141 126)(23 142 95 127 114)(24 128 143 115 96)(25 52 74 150 112)(26 151 53 105 75)(27 106 152 76 54)(28 77 107 55 145)(29 56 78 146 108)(30 147 49 109 79)(31 110 148 80 50)(32 73 111 51 149)
(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 16 96)(2 75 9 93)(3 80 10 90)(4 77 11 95)(5 74 12 92)(6 79 13 89)(7 76 14 94)(8 73 15 91)(17 156 147 103)(18 153 148 100)(19 158 149 97)(20 155 150 102)(21 160 151 99)(22 157 152 104)(23 154 145 101)(24 159 146 98)(25 46 139 82)(26 43 140 87)(27 48 141 84)(28 45 142 81)(29 42 143 86)(30 47 144 83)(31 44 137 88)(32 41 138 85)(33 109 135 121)(34 106 136 126)(35 111 129 123)(36 108 130 128)(37 105 131 125)(38 110 132 122)(39 107 133 127)(40 112 134 124)(49 64 116 70)(50 61 117 67)(51 58 118 72)(52 63 119 69)(53 60 120 66)(54 57 113 71)(55 62 114 68)(56 59 115 65)
(1 96)(2 89)(3 90)(4 91)(5 92)(6 93)(7 94)(8 95)(9 79)(10 80)(11 73)(12 74)(13 75)(14 76)(15 77)(16 78)(17 66)(18 67)(19 68)(20 69)(21 70)(22 71)(23 72)(24 65)(25 134)(26 135)(27 136)(28 129)(29 130)(30 131)(31 132)(32 133)(33 140)(34 141)(35 142)(36 143)(37 144)(38 137)(39 138)(40 139)(41 107)(42 108)(43 109)(44 110)(45 111)(46 112)(47 105)(48 106)(49 99)(50 100)(51 101)(52 102)(53 103)(54 104)(55 97)(56 98)(57 152)(58 145)(59 146)(60 147)(61 148)(62 149)(63 150)(64 151)(81 123)(82 124)(83 125)(84 126)(85 127)(86 128)(87 121)(88 122)(113 157)(114 158)(115 159)(116 160)(117 153)(118 154)(119 155)(120 156)
G:=sub<Sym(160)| (1,65,86,36,159)(2,37,66,160,87)(3,153,38,88,67)(4,81,154,68,39)(5,69,82,40,155)(6,33,70,156,83)(7,157,34,84,71)(8,85,158,72,35)(9,131,60,99,43)(10,100,132,44,61)(11,45,101,62,133)(12,63,46,134,102)(13,135,64,103,47)(14,104,136,48,57)(15,41,97,58,129)(16,59,42,130,98)(17,116,121,89,144)(18,90,117,137,122)(19,138,91,123,118)(20,124,139,119,92)(21,120,125,93,140)(22,94,113,141,126)(23,142,95,127,114)(24,128,143,115,96)(25,52,74,150,112)(26,151,53,105,75)(27,106,152,76,54)(28,77,107,55,145)(29,56,78,146,108)(30,147,49,109,79)(31,110,148,80,50)(32,73,111,51,149), (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), (1,78,16,96)(2,75,9,93)(3,80,10,90)(4,77,11,95)(5,74,12,92)(6,79,13,89)(7,76,14,94)(8,73,15,91)(17,156,147,103)(18,153,148,100)(19,158,149,97)(20,155,150,102)(21,160,151,99)(22,157,152,104)(23,154,145,101)(24,159,146,98)(25,46,139,82)(26,43,140,87)(27,48,141,84)(28,45,142,81)(29,42,143,86)(30,47,144,83)(31,44,137,88)(32,41,138,85)(33,109,135,121)(34,106,136,126)(35,111,129,123)(36,108,130,128)(37,105,131,125)(38,110,132,122)(39,107,133,127)(40,112,134,124)(49,64,116,70)(50,61,117,67)(51,58,118,72)(52,63,119,69)(53,60,120,66)(54,57,113,71)(55,62,114,68)(56,59,115,65), (1,96)(2,89)(3,90)(4,91)(5,92)(6,93)(7,94)(8,95)(9,79)(10,80)(11,73)(12,74)(13,75)(14,76)(15,77)(16,78)(17,66)(18,67)(19,68)(20,69)(21,70)(22,71)(23,72)(24,65)(25,134)(26,135)(27,136)(28,129)(29,130)(30,131)(31,132)(32,133)(33,140)(34,141)(35,142)(36,143)(37,144)(38,137)(39,138)(40,139)(41,107)(42,108)(43,109)(44,110)(45,111)(46,112)(47,105)(48,106)(49,99)(50,100)(51,101)(52,102)(53,103)(54,104)(55,97)(56,98)(57,152)(58,145)(59,146)(60,147)(61,148)(62,149)(63,150)(64,151)(81,123)(82,124)(83,125)(84,126)(85,127)(86,128)(87,121)(88,122)(113,157)(114,158)(115,159)(116,160)(117,153)(118,154)(119,155)(120,156)>;
G:=Group( (1,65,86,36,159)(2,37,66,160,87)(3,153,38,88,67)(4,81,154,68,39)(5,69,82,40,155)(6,33,70,156,83)(7,157,34,84,71)(8,85,158,72,35)(9,131,60,99,43)(10,100,132,44,61)(11,45,101,62,133)(12,63,46,134,102)(13,135,64,103,47)(14,104,136,48,57)(15,41,97,58,129)(16,59,42,130,98)(17,116,121,89,144)(18,90,117,137,122)(19,138,91,123,118)(20,124,139,119,92)(21,120,125,93,140)(22,94,113,141,126)(23,142,95,127,114)(24,128,143,115,96)(25,52,74,150,112)(26,151,53,105,75)(27,106,152,76,54)(28,77,107,55,145)(29,56,78,146,108)(30,147,49,109,79)(31,110,148,80,50)(32,73,111,51,149), (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), (1,78,16,96)(2,75,9,93)(3,80,10,90)(4,77,11,95)(5,74,12,92)(6,79,13,89)(7,76,14,94)(8,73,15,91)(17,156,147,103)(18,153,148,100)(19,158,149,97)(20,155,150,102)(21,160,151,99)(22,157,152,104)(23,154,145,101)(24,159,146,98)(25,46,139,82)(26,43,140,87)(27,48,141,84)(28,45,142,81)(29,42,143,86)(30,47,144,83)(31,44,137,88)(32,41,138,85)(33,109,135,121)(34,106,136,126)(35,111,129,123)(36,108,130,128)(37,105,131,125)(38,110,132,122)(39,107,133,127)(40,112,134,124)(49,64,116,70)(50,61,117,67)(51,58,118,72)(52,63,119,69)(53,60,120,66)(54,57,113,71)(55,62,114,68)(56,59,115,65), (1,96)(2,89)(3,90)(4,91)(5,92)(6,93)(7,94)(8,95)(9,79)(10,80)(11,73)(12,74)(13,75)(14,76)(15,77)(16,78)(17,66)(18,67)(19,68)(20,69)(21,70)(22,71)(23,72)(24,65)(25,134)(26,135)(27,136)(28,129)(29,130)(30,131)(31,132)(32,133)(33,140)(34,141)(35,142)(36,143)(37,144)(38,137)(39,138)(40,139)(41,107)(42,108)(43,109)(44,110)(45,111)(46,112)(47,105)(48,106)(49,99)(50,100)(51,101)(52,102)(53,103)(54,104)(55,97)(56,98)(57,152)(58,145)(59,146)(60,147)(61,148)(62,149)(63,150)(64,151)(81,123)(82,124)(83,125)(84,126)(85,127)(86,128)(87,121)(88,122)(113,157)(114,158)(115,159)(116,160)(117,153)(118,154)(119,155)(120,156) );
G=PermutationGroup([(1,65,86,36,159),(2,37,66,160,87),(3,153,38,88,67),(4,81,154,68,39),(5,69,82,40,155),(6,33,70,156,83),(7,157,34,84,71),(8,85,158,72,35),(9,131,60,99,43),(10,100,132,44,61),(11,45,101,62,133),(12,63,46,134,102),(13,135,64,103,47),(14,104,136,48,57),(15,41,97,58,129),(16,59,42,130,98),(17,116,121,89,144),(18,90,117,137,122),(19,138,91,123,118),(20,124,139,119,92),(21,120,125,93,140),(22,94,113,141,126),(23,142,95,127,114),(24,128,143,115,96),(25,52,74,150,112),(26,151,53,105,75),(27,106,152,76,54),(28,77,107,55,145),(29,56,78,146,108),(30,147,49,109,79),(31,110,148,80,50),(32,73,111,51,149)], [(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)], [(1,78,16,96),(2,75,9,93),(3,80,10,90),(4,77,11,95),(5,74,12,92),(6,79,13,89),(7,76,14,94),(8,73,15,91),(17,156,147,103),(18,153,148,100),(19,158,149,97),(20,155,150,102),(21,160,151,99),(22,157,152,104),(23,154,145,101),(24,159,146,98),(25,46,139,82),(26,43,140,87),(27,48,141,84),(28,45,142,81),(29,42,143,86),(30,47,144,83),(31,44,137,88),(32,41,138,85),(33,109,135,121),(34,106,136,126),(35,111,129,123),(36,108,130,128),(37,105,131,125),(38,110,132,122),(39,107,133,127),(40,112,134,124),(49,64,116,70),(50,61,117,67),(51,58,118,72),(52,63,119,69),(53,60,120,66),(54,57,113,71),(55,62,114,68),(56,59,115,65)], [(1,96),(2,89),(3,90),(4,91),(5,92),(6,93),(7,94),(8,95),(9,79),(10,80),(11,73),(12,74),(13,75),(14,76),(15,77),(16,78),(17,66),(18,67),(19,68),(20,69),(21,70),(22,71),(23,72),(24,65),(25,134),(26,135),(27,136),(28,129),(29,130),(30,131),(31,132),(32,133),(33,140),(34,141),(35,142),(36,143),(37,144),(38,137),(39,138),(40,139),(41,107),(42,108),(43,109),(44,110),(45,111),(46,112),(47,105),(48,106),(49,99),(50,100),(51,101),(52,102),(53,103),(54,104),(55,97),(56,98),(57,152),(58,145),(59,146),(60,147),(61,148),(62,149),(63,150),(64,151),(81,123),(82,124),(83,125),(84,126),(85,127),(86,128),(87,121),(88,122),(113,157),(114,158),(115,159),(116,160),(117,153),(118,154),(119,155),(120,156)])
Matrix representation ►G ⊆ 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] >;
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 |
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
×
⋊
ℤ
𝔽
○
≀