metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C5⋊2C8⋊2D4, C5⋊1(C8⋊2D4), C20⋊2D4⋊2C2, C4⋊D20⋊4C2, C4⋊C4.13D10, C4.161(D4×D5), D4⋊C4⋊19D5, (C2×D4).29D10, (C2×C8).170D10, C20.111(C2×D4), C20.Q8⋊6C2, D20⋊5C4⋊20C2, C4.27(C4○D20), C20.10(C4○D4), (C2×Dic5).31D4, (C22×D5).22D4, C22.180(D4×D5), C2.18(D8⋊D5), C10.18(C4⋊D4), C2.11(D40⋊C2), C10.36(C8⋊C22), (C2×C20).222C23, (C2×C40).187C22, (C2×D20).56C22, (D4×C10).43C22, C4⋊Dic5.75C22, C2.21(D10⋊D4), (C2×D4⋊D5)⋊5C2, (C2×C8⋊D5)⋊18C2, (C5×D4⋊C4)⋊25C2, (C2×C4×D5).18C22, (C2×C10).235(C2×D4), (C5×C4⋊C4).23C22, (C2×C5⋊2C8).20C22, (C2×C4).329(C22×D5), SmallGroup(320,409)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5⋊(C8⋊2D4)
G = < a,b,c,d | a5=b8=c4=d2=1, bab-1=dad=a-1, ac=ca, cbc-1=b3, dbd=b-1, dcd=c-1 >
Subgroups: 662 in 130 conjugacy classes, 39 normal (37 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, C23, D5, C10, C10, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, M4(2), D8, C22×C4, C2×D4, C2×D4, Dic5, C20, C20, D10, C2×C10, C2×C10, D4⋊C4, D4⋊C4, C4.Q8, C4⋊D4, C2×M4(2), C2×D8, C5⋊2C8, C40, C4×D5, D20, C2×Dic5, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C5×D4, C22×D5, C22×D5, C22×C10, C8⋊2D4, C8⋊D5, C2×C5⋊2C8, C4⋊Dic5, D10⋊C4, D4⋊D5, C23.D5, C5×C4⋊C4, C2×C40, C2×C4×D5, C2×D20, C2×D20, C2×C5⋊D4, D4×C10, C20.Q8, D20⋊5C4, C5×D4⋊C4, C4⋊D20, C2×C8⋊D5, C2×D4⋊D5, C20⋊2D4, C5⋊(C8⋊2D4)
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, C4○D4, D10, C4⋊D4, C8⋊C22, C22×D5, C8⋊2D4, C4○D20, D4×D5, D10⋊D4, D8⋊D5, D40⋊C2, C5⋊(C8⋊2D4)
►On 160 points
Generators in S160
(1 74 14 143 43)(2 44 144 15 75)(3 76 16 137 45)(4 46 138 9 77)(5 78 10 139 47)(6 48 140 11 79)(7 80 12 141 41)(8 42 142 13 73)(17 108 117 149 85)(18 86 150 118 109)(19 110 119 151 87)(20 88 152 120 111)(21 112 113 145 81)(22 82 146 114 105)(23 106 115 147 83)(24 84 148 116 107)(25 133 155 56 91)(26 92 49 156 134)(27 135 157 50 93)(28 94 51 158 136)(29 129 159 52 95)(30 96 53 160 130)(31 131 153 54 89)(32 90 55 154 132)(33 69 98 62 121)(34 122 63 99 70)(35 71 100 64 123)(36 124 57 101 72)(37 65 102 58 125)(38 126 59 103 66)(39 67 104 60 127)(40 128 61 97 68)
(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)
(1 125 119 55)(2 128 120 50)(3 123 113 53)(4 126 114 56)(5 121 115 51)(6 124 116 54)(7 127 117 49)(8 122 118 52)(9 66 82 133)(10 69 83 136)(11 72 84 131)(12 67 85 134)(13 70 86 129)(14 65 87 132)(15 68 88 135)(16 71 81 130)(17 26 141 104)(18 29 142 99)(19 32 143 102)(20 27 144 97)(21 30 137 100)(22 25 138 103)(23 28 139 98)(24 31 140 101)(33 147 158 78)(34 150 159 73)(35 145 160 76)(36 148 153 79)(37 151 154 74)(38 146 155 77)(39 149 156 80)(40 152 157 75)(41 60 108 92)(42 63 109 95)(43 58 110 90)(44 61 111 93)(45 64 112 96)(46 59 105 91)(47 62 106 94)(48 57 107 89)
(1 115)(2 114)(3 113)(4 120)(5 119)(6 118)(7 117)(8 116)(9 20)(10 19)(11 18)(12 17)(13 24)(14 23)(15 22)(16 21)(25 135)(26 134)(27 133)(28 132)(29 131)(30 130)(31 129)(32 136)(33 58)(34 57)(35 64)(36 63)(37 62)(38 61)(39 60)(40 59)(41 149)(42 148)(43 147)(44 146)(45 145)(46 152)(47 151)(48 150)(50 56)(51 55)(52 54)(65 98)(66 97)(67 104)(68 103)(69 102)(70 101)(71 100)(72 99)(73 107)(74 106)(75 105)(76 112)(77 111)(78 110)(79 109)(80 108)(81 137)(82 144)(83 143)(84 142)(85 141)(86 140)(87 139)(88 138)(89 159)(90 158)(91 157)(92 156)(93 155)(94 154)(95 153)(96 160)(121 125)(122 124)(126 128)
G:=sub<Sym(160)| (1,74,14,143,43)(2,44,144,15,75)(3,76,16,137,45)(4,46,138,9,77)(5,78,10,139,47)(6,48,140,11,79)(7,80,12,141,41)(8,42,142,13,73)(17,108,117,149,85)(18,86,150,118,109)(19,110,119,151,87)(20,88,152,120,111)(21,112,113,145,81)(22,82,146,114,105)(23,106,115,147,83)(24,84,148,116,107)(25,133,155,56,91)(26,92,49,156,134)(27,135,157,50,93)(28,94,51,158,136)(29,129,159,52,95)(30,96,53,160,130)(31,131,153,54,89)(32,90,55,154,132)(33,69,98,62,121)(34,122,63,99,70)(35,71,100,64,123)(36,124,57,101,72)(37,65,102,58,125)(38,126,59,103,66)(39,67,104,60,127)(40,128,61,97,68), (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), (1,125,119,55)(2,128,120,50)(3,123,113,53)(4,126,114,56)(5,121,115,51)(6,124,116,54)(7,127,117,49)(8,122,118,52)(9,66,82,133)(10,69,83,136)(11,72,84,131)(12,67,85,134)(13,70,86,129)(14,65,87,132)(15,68,88,135)(16,71,81,130)(17,26,141,104)(18,29,142,99)(19,32,143,102)(20,27,144,97)(21,30,137,100)(22,25,138,103)(23,28,139,98)(24,31,140,101)(33,147,158,78)(34,150,159,73)(35,145,160,76)(36,148,153,79)(37,151,154,74)(38,146,155,77)(39,149,156,80)(40,152,157,75)(41,60,108,92)(42,63,109,95)(43,58,110,90)(44,61,111,93)(45,64,112,96)(46,59,105,91)(47,62,106,94)(48,57,107,89), (1,115)(2,114)(3,113)(4,120)(5,119)(6,118)(7,117)(8,116)(9,20)(10,19)(11,18)(12,17)(13,24)(14,23)(15,22)(16,21)(25,135)(26,134)(27,133)(28,132)(29,131)(30,130)(31,129)(32,136)(33,58)(34,57)(35,64)(36,63)(37,62)(38,61)(39,60)(40,59)(41,149)(42,148)(43,147)(44,146)(45,145)(46,152)(47,151)(48,150)(50,56)(51,55)(52,54)(65,98)(66,97)(67,104)(68,103)(69,102)(70,101)(71,100)(72,99)(73,107)(74,106)(75,105)(76,112)(77,111)(78,110)(79,109)(80,108)(81,137)(82,144)(83,143)(84,142)(85,141)(86,140)(87,139)(88,138)(89,159)(90,158)(91,157)(92,156)(93,155)(94,154)(95,153)(96,160)(121,125)(122,124)(126,128)>;
G:=Group( (1,74,14,143,43)(2,44,144,15,75)(3,76,16,137,45)(4,46,138,9,77)(5,78,10,139,47)(6,48,140,11,79)(7,80,12,141,41)(8,42,142,13,73)(17,108,117,149,85)(18,86,150,118,109)(19,110,119,151,87)(20,88,152,120,111)(21,112,113,145,81)(22,82,146,114,105)(23,106,115,147,83)(24,84,148,116,107)(25,133,155,56,91)(26,92,49,156,134)(27,135,157,50,93)(28,94,51,158,136)(29,129,159,52,95)(30,96,53,160,130)(31,131,153,54,89)(32,90,55,154,132)(33,69,98,62,121)(34,122,63,99,70)(35,71,100,64,123)(36,124,57,101,72)(37,65,102,58,125)(38,126,59,103,66)(39,67,104,60,127)(40,128,61,97,68), (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), (1,125,119,55)(2,128,120,50)(3,123,113,53)(4,126,114,56)(5,121,115,51)(6,124,116,54)(7,127,117,49)(8,122,118,52)(9,66,82,133)(10,69,83,136)(11,72,84,131)(12,67,85,134)(13,70,86,129)(14,65,87,132)(15,68,88,135)(16,71,81,130)(17,26,141,104)(18,29,142,99)(19,32,143,102)(20,27,144,97)(21,30,137,100)(22,25,138,103)(23,28,139,98)(24,31,140,101)(33,147,158,78)(34,150,159,73)(35,145,160,76)(36,148,153,79)(37,151,154,74)(38,146,155,77)(39,149,156,80)(40,152,157,75)(41,60,108,92)(42,63,109,95)(43,58,110,90)(44,61,111,93)(45,64,112,96)(46,59,105,91)(47,62,106,94)(48,57,107,89), (1,115)(2,114)(3,113)(4,120)(5,119)(6,118)(7,117)(8,116)(9,20)(10,19)(11,18)(12,17)(13,24)(14,23)(15,22)(16,21)(25,135)(26,134)(27,133)(28,132)(29,131)(30,130)(31,129)(32,136)(33,58)(34,57)(35,64)(36,63)(37,62)(38,61)(39,60)(40,59)(41,149)(42,148)(43,147)(44,146)(45,145)(46,152)(47,151)(48,150)(50,56)(51,55)(52,54)(65,98)(66,97)(67,104)(68,103)(69,102)(70,101)(71,100)(72,99)(73,107)(74,106)(75,105)(76,112)(77,111)(78,110)(79,109)(80,108)(81,137)(82,144)(83,143)(84,142)(85,141)(86,140)(87,139)(88,138)(89,159)(90,158)(91,157)(92,156)(93,155)(94,154)(95,153)(96,160)(121,125)(122,124)(126,128) );
G=PermutationGroup([[(1,74,14,143,43),(2,44,144,15,75),(3,76,16,137,45),(4,46,138,9,77),(5,78,10,139,47),(6,48,140,11,79),(7,80,12,141,41),(8,42,142,13,73),(17,108,117,149,85),(18,86,150,118,109),(19,110,119,151,87),(20,88,152,120,111),(21,112,113,145,81),(22,82,146,114,105),(23,106,115,147,83),(24,84,148,116,107),(25,133,155,56,91),(26,92,49,156,134),(27,135,157,50,93),(28,94,51,158,136),(29,129,159,52,95),(30,96,53,160,130),(31,131,153,54,89),(32,90,55,154,132),(33,69,98,62,121),(34,122,63,99,70),(35,71,100,64,123),(36,124,57,101,72),(37,65,102,58,125),(38,126,59,103,66),(39,67,104,60,127),(40,128,61,97,68)], [(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)], [(1,125,119,55),(2,128,120,50),(3,123,113,53),(4,126,114,56),(5,121,115,51),(6,124,116,54),(7,127,117,49),(8,122,118,52),(9,66,82,133),(10,69,83,136),(11,72,84,131),(12,67,85,134),(13,70,86,129),(14,65,87,132),(15,68,88,135),(16,71,81,130),(17,26,141,104),(18,29,142,99),(19,32,143,102),(20,27,144,97),(21,30,137,100),(22,25,138,103),(23,28,139,98),(24,31,140,101),(33,147,158,78),(34,150,159,73),(35,145,160,76),(36,148,153,79),(37,151,154,74),(38,146,155,77),(39,149,156,80),(40,152,157,75),(41,60,108,92),(42,63,109,95),(43,58,110,90),(44,61,111,93),(45,64,112,96),(46,59,105,91),(47,62,106,94),(48,57,107,89)], [(1,115),(2,114),(3,113),(4,120),(5,119),(6,118),(7,117),(8,116),(9,20),(10,19),(11,18),(12,17),(13,24),(14,23),(15,22),(16,21),(25,135),(26,134),(27,133),(28,132),(29,131),(30,130),(31,129),(32,136),(33,58),(34,57),(35,64),(36,63),(37,62),(38,61),(39,60),(40,59),(41,149),(42,148),(43,147),(44,146),(45,145),(46,152),(47,151),(48,150),(50,56),(51,55),(52,54),(65,98),(66,97),(67,104),(68,103),(69,102),(70,101),(71,100),(72,99),(73,107),(74,106),(75,105),(76,112),(77,111),(78,110),(79,109),(80,108),(81,137),(82,144),(83,143),(84,142),(85,141),(86,140),(87,139),(88,138),(89,159),(90,158),(91,157),(92,156),(93,155),(94,154),(95,153),(96,160),(121,125),(122,124),(126,128)]])
44 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | 4B | 4C | 4D | 4E | 5A | 5B | 8A | 8B | 8C | 8D | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 20A | 20B | 20C | 20D | 20E | 20F | 20G | 20H | 40A | ··· | 40H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 8 | 20 | 40 | 2 | 2 | 8 | 20 | 40 | 2 | 2 | 4 | 4 | 20 | 20 | 2 | ··· | 2 | 8 | 8 | 8 | 8 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 4 | ··· | 4 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | D5 | C4○D4 | D10 | D10 | D10 | C4○D20 | C8⋊C22 | D4×D5 | D4×D5 | D8⋊D5 | D40⋊C2 |
| kernel | C5⋊(C8⋊2D4) | C20.Q8 | D20⋊5C4 | C5×D4⋊C4 | C4⋊D20 | C2×C8⋊D5 | C2×D4⋊D5 | C20⋊2D4 | C5⋊2C8 | C2×Dic5 | C22×D5 | D4⋊C4 | C20 | C4⋊C4 | C2×C8 | C2×D4 | C4 | C10 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 8 | 2 | 2 | 2 | 4 | 4 |
Matrix representation of C5⋊(C8⋊2D4) ►in GL6(𝔽41)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 40 | 6 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 40 | 6 |
| 37 | 5 | 0 | 0 | 0 | 0 |
| 13 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 30 | 0 | 6 | 29 |
| 0 | 0 | 16 | 11 | 24 | 35 |
| 0 | 0 | 38 | 6 | 36 | 29 |
| 0 | 0 | 29 | 3 | 40 | 5 |
| 5 | 4 | 0 | 0 | 0 | 0 |
| 14 | 36 | 0 | 0 | 0 | 0 |
| 0 | 0 | 7 | 35 | 22 | 0 |
| 0 | 0 | 6 | 12 | 0 | 22 |
| 0 | 0 | 18 | 35 | 34 | 6 |
| 0 | 0 | 6 | 23 | 35 | 29 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 23 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 6 | 40 | 0 | 0 |
| 0 | 0 | 1 | 0 | 40 | 0 |
| 0 | 0 | 6 | 40 | 35 | 1 |
G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,40,0,0,0,0,1,6,0,0,0,0,0,0,0,40,0,0,0,0,1,6],[37,13,0,0,0,0,5,4,0,0,0,0,0,0,30,16,38,29,0,0,0,11,6,3,0,0,6,24,36,40,0,0,29,35,29,5],[5,14,0,0,0,0,4,36,0,0,0,0,0,0,7,6,18,6,0,0,35,12,35,23,0,0,22,0,34,35,0,0,0,22,6,29],[40,23,0,0,0,0,0,1,0,0,0,0,0,0,1,6,1,6,0,0,0,40,0,40,0,0,0,0,40,35,0,0,0,0,0,1] >;
C5⋊(C8⋊2D4) in GAP, Magma, Sage, TeX
C_5\rtimes (C_8\rtimes_2D_4)
% in TeX
G:=Group("C5:(C8:2D4)"); // GroupNames label
G:=SmallGroup(320,409);
// by ID
G=gap.SmallGroup(320,409);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,253,120,1094,135,570,297,136,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^5=b^8=c^4=d^2=1,b*a*b^-1=d*a*d=a^-1,a*c=c*a,c*b*c^-1=b^3,d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations