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, C2, C4, C4, C22, C22, C5, C8, C8, C2×C4, C2×C4, C23, D5, C10, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, M4(2), C22×C4, Dic5, Dic5, C20, C20, D10, D10, C2×C10, C4.Q8, C4.Q8, C2.D8, C2×C4⋊C4, C42⋊C2, C2×M4(2), C5⋊2C8, C40, C4×D5, C4×D5, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C22×D5, M4(2)⋊C4, C8⋊D5, C2×C5⋊2C8, C4×Dic5, C10.D4, C4⋊Dic5, D10⋊C4, C5×C4⋊C4, 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, C4, C22, C2×C4, D4, Q8, C23, D5, C4⋊C4, C22×C4, C2×D4, C2×Q8, D10, C2×C4⋊C4, C8⋊C22, C8.C22, C4×D5, 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 25 155)(2 50 26 158)(3 53 27 153)(4 56 28 156)(5 51 29 159)(6 54 30 154)(7 49 31 157)(8 52 32 160)(9 143 63 21)(10 138 64 24)(11 141 57 19)(12 144 58 22)(13 139 59 17)(14 142 60 20)(15 137 61 23)(16 140 62 18)(33 89 66 126)(34 92 67 121)(35 95 68 124)(36 90 69 127)(37 93 70 122)(38 96 71 125)(39 91 72 128)(40 94 65 123)(41 108 76 130)(42 111 77 133)(43 106 78 136)(44 109 79 131)(45 112 80 134)(46 107 73 129)(47 110 74 132)(48 105 75 135)(81 152 113 99)(82 147 114 102)(83 150 115 97)(84 145 116 100)(85 148 117 103)(86 151 118 98)(87 146 119 101)(88 149 120 104)
(1 103 133 21 39)(2 104 134 22 40)(3 97 135 23 33)(4 98 136 24 34)(5 99 129 17 35)(6 100 130 18 36)(7 101 131 19 37)(8 102 132 20 38)(9 91 55 85 42)(10 92 56 86 43)(11 93 49 87 44)(12 94 50 88 45)(13 95 51 81 46)(14 96 52 82 47)(15 89 53 83 48)(16 90 54 84 41)(25 148 111 143 72)(26 149 112 144 65)(27 150 105 137 66)(28 151 106 138 67)(29 152 107 139 68)(30 145 108 140 69)(31 146 109 141 70)(32 147 110 142 71)(57 122 157 119 79)(58 123 158 120 80)(59 124 159 113 73)(60 125 160 114 74)(61 126 153 115 75)(62 127 154 116 76)(63 128 155 117 77)(64 121 156 118 78)
(1 39)(2 36)(3 33)(4 38)(5 35)(6 40)(7 37)(8 34)(9 85)(10 82)(11 87)(12 84)(13 81)(14 86)(15 83)(16 88)(17 99)(18 104)(19 101)(20 98)(21 103)(22 100)(23 97)(24 102)(25 72)(26 69)(27 66)(28 71)(29 68)(30 65)(31 70)(32 67)(41 45)(43 47)(49 93)(50 90)(51 95)(52 92)(53 89)(54 94)(55 91)(56 96)(57 119)(58 116)(59 113)(60 118)(61 115)(62 120)(63 117)(64 114)(74 78)(76 80)(106 110)(108 112)(121 160)(122 157)(123 154)(124 159)(125 156)(126 153)(127 158)(128 155)(130 134)(132 136)(137 150)(138 147)(139 152)(140 149)(141 146)(142 151)(143 148)(144 145)
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), (1,55,25,155)(2,50,26,158)(3,53,27,153)(4,56,28,156)(5,51,29,159)(6,54,30,154)(7,49,31,157)(8,52,32,160)(9,143,63,21)(10,138,64,24)(11,141,57,19)(12,144,58,22)(13,139,59,17)(14,142,60,20)(15,137,61,23)(16,140,62,18)(33,89,66,126)(34,92,67,121)(35,95,68,124)(36,90,69,127)(37,93,70,122)(38,96,71,125)(39,91,72,128)(40,94,65,123)(41,108,76,130)(42,111,77,133)(43,106,78,136)(44,109,79,131)(45,112,80,134)(46,107,73,129)(47,110,74,132)(48,105,75,135)(81,152,113,99)(82,147,114,102)(83,150,115,97)(84,145,116,100)(85,148,117,103)(86,151,118,98)(87,146,119,101)(88,149,120,104), (1,103,133,21,39)(2,104,134,22,40)(3,97,135,23,33)(4,98,136,24,34)(5,99,129,17,35)(6,100,130,18,36)(7,101,131,19,37)(8,102,132,20,38)(9,91,55,85,42)(10,92,56,86,43)(11,93,49,87,44)(12,94,50,88,45)(13,95,51,81,46)(14,96,52,82,47)(15,89,53,83,48)(16,90,54,84,41)(25,148,111,143,72)(26,149,112,144,65)(27,150,105,137,66)(28,151,106,138,67)(29,152,107,139,68)(30,145,108,140,69)(31,146,109,141,70)(32,147,110,142,71)(57,122,157,119,79)(58,123,158,120,80)(59,124,159,113,73)(60,125,160,114,74)(61,126,153,115,75)(62,127,154,116,76)(63,128,155,117,77)(64,121,156,118,78), (1,39)(2,36)(3,33)(4,38)(5,35)(6,40)(7,37)(8,34)(9,85)(10,82)(11,87)(12,84)(13,81)(14,86)(15,83)(16,88)(17,99)(18,104)(19,101)(20,98)(21,103)(22,100)(23,97)(24,102)(25,72)(26,69)(27,66)(28,71)(29,68)(30,65)(31,70)(32,67)(41,45)(43,47)(49,93)(50,90)(51,95)(52,92)(53,89)(54,94)(55,91)(56,96)(57,119)(58,116)(59,113)(60,118)(61,115)(62,120)(63,117)(64,114)(74,78)(76,80)(106,110)(108,112)(121,160)(122,157)(123,154)(124,159)(125,156)(126,153)(127,158)(128,155)(130,134)(132,136)(137,150)(138,147)(139,152)(140,149)(141,146)(142,151)(143,148)(144,145)>;
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), (1,55,25,155)(2,50,26,158)(3,53,27,153)(4,56,28,156)(5,51,29,159)(6,54,30,154)(7,49,31,157)(8,52,32,160)(9,143,63,21)(10,138,64,24)(11,141,57,19)(12,144,58,22)(13,139,59,17)(14,142,60,20)(15,137,61,23)(16,140,62,18)(33,89,66,126)(34,92,67,121)(35,95,68,124)(36,90,69,127)(37,93,70,122)(38,96,71,125)(39,91,72,128)(40,94,65,123)(41,108,76,130)(42,111,77,133)(43,106,78,136)(44,109,79,131)(45,112,80,134)(46,107,73,129)(47,110,74,132)(48,105,75,135)(81,152,113,99)(82,147,114,102)(83,150,115,97)(84,145,116,100)(85,148,117,103)(86,151,118,98)(87,146,119,101)(88,149,120,104), (1,103,133,21,39)(2,104,134,22,40)(3,97,135,23,33)(4,98,136,24,34)(5,99,129,17,35)(6,100,130,18,36)(7,101,131,19,37)(8,102,132,20,38)(9,91,55,85,42)(10,92,56,86,43)(11,93,49,87,44)(12,94,50,88,45)(13,95,51,81,46)(14,96,52,82,47)(15,89,53,83,48)(16,90,54,84,41)(25,148,111,143,72)(26,149,112,144,65)(27,150,105,137,66)(28,151,106,138,67)(29,152,107,139,68)(30,145,108,140,69)(31,146,109,141,70)(32,147,110,142,71)(57,122,157,119,79)(58,123,158,120,80)(59,124,159,113,73)(60,125,160,114,74)(61,126,153,115,75)(62,127,154,116,76)(63,128,155,117,77)(64,121,156,118,78), (1,39)(2,36)(3,33)(4,38)(5,35)(6,40)(7,37)(8,34)(9,85)(10,82)(11,87)(12,84)(13,81)(14,86)(15,83)(16,88)(17,99)(18,104)(19,101)(20,98)(21,103)(22,100)(23,97)(24,102)(25,72)(26,69)(27,66)(28,71)(29,68)(30,65)(31,70)(32,67)(41,45)(43,47)(49,93)(50,90)(51,95)(52,92)(53,89)(54,94)(55,91)(56,96)(57,119)(58,116)(59,113)(60,118)(61,115)(62,120)(63,117)(64,114)(74,78)(76,80)(106,110)(108,112)(121,160)(122,157)(123,154)(124,159)(125,156)(126,153)(127,158)(128,155)(130,134)(132,136)(137,150)(138,147)(139,152)(140,149)(141,146)(142,151)(143,148)(144,145) );
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)], [(1,55,25,155),(2,50,26,158),(3,53,27,153),(4,56,28,156),(5,51,29,159),(6,54,30,154),(7,49,31,157),(8,52,32,160),(9,143,63,21),(10,138,64,24),(11,141,57,19),(12,144,58,22),(13,139,59,17),(14,142,60,20),(15,137,61,23),(16,140,62,18),(33,89,66,126),(34,92,67,121),(35,95,68,124),(36,90,69,127),(37,93,70,122),(38,96,71,125),(39,91,72,128),(40,94,65,123),(41,108,76,130),(42,111,77,133),(43,106,78,136),(44,109,79,131),(45,112,80,134),(46,107,73,129),(47,110,74,132),(48,105,75,135),(81,152,113,99),(82,147,114,102),(83,150,115,97),(84,145,116,100),(85,148,117,103),(86,151,118,98),(87,146,119,101),(88,149,120,104)], [(1,103,133,21,39),(2,104,134,22,40),(3,97,135,23,33),(4,98,136,24,34),(5,99,129,17,35),(6,100,130,18,36),(7,101,131,19,37),(8,102,132,20,38),(9,91,55,85,42),(10,92,56,86,43),(11,93,49,87,44),(12,94,50,88,45),(13,95,51,81,46),(14,96,52,82,47),(15,89,53,83,48),(16,90,54,84,41),(25,148,111,143,72),(26,149,112,144,65),(27,150,105,137,66),(28,151,106,138,67),(29,152,107,139,68),(30,145,108,140,69),(31,146,109,141,70),(32,147,110,142,71),(57,122,157,119,79),(58,123,158,120,80),(59,124,159,113,73),(60,125,160,114,74),(61,126,153,115,75),(62,127,154,116,76),(63,128,155,117,77),(64,121,156,118,78)], [(1,39),(2,36),(3,33),(4,38),(5,35),(6,40),(7,37),(8,34),(9,85),(10,82),(11,87),(12,84),(13,81),(14,86),(15,83),(16,88),(17,99),(18,104),(19,101),(20,98),(21,103),(22,100),(23,97),(24,102),(25,72),(26,69),(27,66),(28,71),(29,68),(30,65),(31,70),(32,67),(41,45),(43,47),(49,93),(50,90),(51,95),(52,92),(53,89),(54,94),(55,91),(56,96),(57,119),(58,116),(59,113),(60,118),(61,115),(62,120),(63,117),(64,114),(74,78),(76,80),(106,110),(108,112),(121,160),(122,157),(123,154),(124,159),(125,156),(126,153),(127,158),(128,155),(130,134),(132,136),(137,150),(138,147),(139,152),(140,149),(141,146),(142,151),(143,148),(144,145)]])
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