metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C5⋊2C8.1D4, C4⋊C4.30D10, C4.169(D4×D5), C5⋊1(C8.D4), Q8⋊C4⋊19D5, C20.127(C2×D4), (C2×C8).178D10, (C2×Q8).22D10, C20.22(C4○D4), C4.35(C4○D20), D10⋊2Q8.4C2, D10⋊3Q8.7C2, C20.Q8⋊13C2, (C2×Dic5).42D4, (C22×D5).30D4, C22.206(D4×D5), C20.44D4⋊25C2, C10.26(C4⋊D4), (C2×C20).256C23, (C2×C40).203C22, (Q8×C10).39C22, C2.29(D10⋊D4), C2.18(Q16⋊D5), C2.19(SD16⋊D5), C10.64(C8.C22), C4⋊Dic5.100C22, (C2×Dic10).78C22, (C2×C5⋊Q16)⋊6C2, (C2×C4×D5).32C22, (C5×Q8⋊C4)⋊25C2, (C2×C10).269(C2×D4), (C2×C8⋊D5).11C2, (C5×C4⋊C4).57C22, (C2×C5⋊2C8).46C22, (C2×C4).363(C22×D5), SmallGroup(320,443)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5⋊2C8.D4
G = < a,b,c,d | a5=b8=c4=d2=1, bab-1=dad=a-1, ac=ca, cbc-1=b3, dbd=b5, dcd=b4c-1 >
Subgroups: 438 in 110 conjugacy classes, 39 normal (37 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, Q8, C23, D5, C10, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, M4(2), Q16, C22×C4, C2×Q8, C2×Q8, Dic5, C20, C20, D10, C2×C10, Q8⋊C4, Q8⋊C4, C4.Q8, C22⋊Q8, C2×M4(2), C2×Q16, C5⋊2C8, C40, Dic10, C4×D5, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C5×Q8, C22×D5, C8.D4, C8⋊D5, C2×C5⋊2C8, C10.D4, C4⋊Dic5, C4⋊Dic5, D10⋊C4, C5⋊Q16, C5×C4⋊C4, C2×C40, C2×Dic10, C2×C4×D5, Q8×C10, C20.Q8, C20.44D4, C5×Q8⋊C4, D10⋊2Q8, C2×C8⋊D5, C2×C5⋊Q16, D10⋊3Q8, C5⋊2C8.D4
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, C4○D4, D10, C4⋊D4, C8.C22, C22×D5, C8.D4, C4○D20, D4×D5, D10⋊D4, SD16⋊D5, Q16⋊D5, C5⋊2C8.D4
►On 160 points
Generators in S160
(1 52 35 149 110)(2 111 150 36 53)(3 54 37 151 112)(4 105 152 38 55)(5 56 39 145 106)(6 107 146 40 49)(7 50 33 147 108)(8 109 148 34 51)(9 95 124 71 77)(10 78 72 125 96)(11 89 126 65 79)(12 80 66 127 90)(13 91 128 67 73)(14 74 68 121 92)(15 93 122 69 75)(16 76 70 123 94)(17 45 159 83 113)(18 114 84 160 46)(19 47 153 85 115)(20 116 86 154 48)(21 41 155 87 117)(22 118 88 156 42)(23 43 157 81 119)(24 120 82 158 44)(25 97 143 61 134)(26 135 62 144 98)(27 99 137 63 136)(28 129 64 138 100)(29 101 139 57 130)(30 131 58 140 102)(31 103 141 59 132)(32 133 60 142 104)
(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 12 58 48)(2 15 59 43)(3 10 60 46)(4 13 61 41)(5 16 62 44)(6 11 63 47)(7 14 64 42)(8 9 57 45)(17 51 77 139)(18 54 78 142)(19 49 79 137)(20 52 80 140)(21 55 73 143)(22 50 74 138)(23 53 75 141)(24 56 76 144)(25 87 152 128)(26 82 145 123)(27 85 146 126)(28 88 147 121)(29 83 148 124)(30 86 149 127)(31 81 150 122)(32 84 151 125)(33 68 100 118)(34 71 101 113)(35 66 102 116)(36 69 103 119)(37 72 104 114)(38 67 97 117)(39 70 98 120)(40 65 99 115)(89 136 153 107)(90 131 154 110)(91 134 155 105)(92 129 156 108)(93 132 157 111)(94 135 158 106)(95 130 159 109)(96 133 160 112)
(1 62)(2 59)(3 64)(4 61)(5 58)(6 63)(7 60)(8 57)(9 13)(11 15)(17 155)(18 160)(19 157)(20 154)(21 159)(22 156)(23 153)(24 158)(25 38)(26 35)(27 40)(28 37)(29 34)(30 39)(31 36)(32 33)(41 45)(43 47)(49 136)(50 133)(51 130)(52 135)(53 132)(54 129)(55 134)(56 131)(65 122)(66 127)(67 124)(68 121)(69 126)(70 123)(71 128)(72 125)(73 95)(74 92)(75 89)(76 94)(77 91)(78 96)(79 93)(80 90)(81 115)(82 120)(83 117)(84 114)(85 119)(86 116)(87 113)(88 118)(97 152)(98 149)(99 146)(100 151)(101 148)(102 145)(103 150)(104 147)(105 143)(106 140)(107 137)(108 142)(109 139)(110 144)(111 141)(112 138)
G:=sub<Sym(160)| (1,52,35,149,110)(2,111,150,36,53)(3,54,37,151,112)(4,105,152,38,55)(5,56,39,145,106)(6,107,146,40,49)(7,50,33,147,108)(8,109,148,34,51)(9,95,124,71,77)(10,78,72,125,96)(11,89,126,65,79)(12,80,66,127,90)(13,91,128,67,73)(14,74,68,121,92)(15,93,122,69,75)(16,76,70,123,94)(17,45,159,83,113)(18,114,84,160,46)(19,47,153,85,115)(20,116,86,154,48)(21,41,155,87,117)(22,118,88,156,42)(23,43,157,81,119)(24,120,82,158,44)(25,97,143,61,134)(26,135,62,144,98)(27,99,137,63,136)(28,129,64,138,100)(29,101,139,57,130)(30,131,58,140,102)(31,103,141,59,132)(32,133,60,142,104), (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), (1,12,58,48)(2,15,59,43)(3,10,60,46)(4,13,61,41)(5,16,62,44)(6,11,63,47)(7,14,64,42)(8,9,57,45)(17,51,77,139)(18,54,78,142)(19,49,79,137)(20,52,80,140)(21,55,73,143)(22,50,74,138)(23,53,75,141)(24,56,76,144)(25,87,152,128)(26,82,145,123)(27,85,146,126)(28,88,147,121)(29,83,148,124)(30,86,149,127)(31,81,150,122)(32,84,151,125)(33,68,100,118)(34,71,101,113)(35,66,102,116)(36,69,103,119)(37,72,104,114)(38,67,97,117)(39,70,98,120)(40,65,99,115)(89,136,153,107)(90,131,154,110)(91,134,155,105)(92,129,156,108)(93,132,157,111)(94,135,158,106)(95,130,159,109)(96,133,160,112), (1,62)(2,59)(3,64)(4,61)(5,58)(6,63)(7,60)(8,57)(9,13)(11,15)(17,155)(18,160)(19,157)(20,154)(21,159)(22,156)(23,153)(24,158)(25,38)(26,35)(27,40)(28,37)(29,34)(30,39)(31,36)(32,33)(41,45)(43,47)(49,136)(50,133)(51,130)(52,135)(53,132)(54,129)(55,134)(56,131)(65,122)(66,127)(67,124)(68,121)(69,126)(70,123)(71,128)(72,125)(73,95)(74,92)(75,89)(76,94)(77,91)(78,96)(79,93)(80,90)(81,115)(82,120)(83,117)(84,114)(85,119)(86,116)(87,113)(88,118)(97,152)(98,149)(99,146)(100,151)(101,148)(102,145)(103,150)(104,147)(105,143)(106,140)(107,137)(108,142)(109,139)(110,144)(111,141)(112,138)>;
G:=Group( (1,52,35,149,110)(2,111,150,36,53)(3,54,37,151,112)(4,105,152,38,55)(5,56,39,145,106)(6,107,146,40,49)(7,50,33,147,108)(8,109,148,34,51)(9,95,124,71,77)(10,78,72,125,96)(11,89,126,65,79)(12,80,66,127,90)(13,91,128,67,73)(14,74,68,121,92)(15,93,122,69,75)(16,76,70,123,94)(17,45,159,83,113)(18,114,84,160,46)(19,47,153,85,115)(20,116,86,154,48)(21,41,155,87,117)(22,118,88,156,42)(23,43,157,81,119)(24,120,82,158,44)(25,97,143,61,134)(26,135,62,144,98)(27,99,137,63,136)(28,129,64,138,100)(29,101,139,57,130)(30,131,58,140,102)(31,103,141,59,132)(32,133,60,142,104), (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), (1,12,58,48)(2,15,59,43)(3,10,60,46)(4,13,61,41)(5,16,62,44)(6,11,63,47)(7,14,64,42)(8,9,57,45)(17,51,77,139)(18,54,78,142)(19,49,79,137)(20,52,80,140)(21,55,73,143)(22,50,74,138)(23,53,75,141)(24,56,76,144)(25,87,152,128)(26,82,145,123)(27,85,146,126)(28,88,147,121)(29,83,148,124)(30,86,149,127)(31,81,150,122)(32,84,151,125)(33,68,100,118)(34,71,101,113)(35,66,102,116)(36,69,103,119)(37,72,104,114)(38,67,97,117)(39,70,98,120)(40,65,99,115)(89,136,153,107)(90,131,154,110)(91,134,155,105)(92,129,156,108)(93,132,157,111)(94,135,158,106)(95,130,159,109)(96,133,160,112), (1,62)(2,59)(3,64)(4,61)(5,58)(6,63)(7,60)(8,57)(9,13)(11,15)(17,155)(18,160)(19,157)(20,154)(21,159)(22,156)(23,153)(24,158)(25,38)(26,35)(27,40)(28,37)(29,34)(30,39)(31,36)(32,33)(41,45)(43,47)(49,136)(50,133)(51,130)(52,135)(53,132)(54,129)(55,134)(56,131)(65,122)(66,127)(67,124)(68,121)(69,126)(70,123)(71,128)(72,125)(73,95)(74,92)(75,89)(76,94)(77,91)(78,96)(79,93)(80,90)(81,115)(82,120)(83,117)(84,114)(85,119)(86,116)(87,113)(88,118)(97,152)(98,149)(99,146)(100,151)(101,148)(102,145)(103,150)(104,147)(105,143)(106,140)(107,137)(108,142)(109,139)(110,144)(111,141)(112,138) );
G=PermutationGroup([[(1,52,35,149,110),(2,111,150,36,53),(3,54,37,151,112),(4,105,152,38,55),(5,56,39,145,106),(6,107,146,40,49),(7,50,33,147,108),(8,109,148,34,51),(9,95,124,71,77),(10,78,72,125,96),(11,89,126,65,79),(12,80,66,127,90),(13,91,128,67,73),(14,74,68,121,92),(15,93,122,69,75),(16,76,70,123,94),(17,45,159,83,113),(18,114,84,160,46),(19,47,153,85,115),(20,116,86,154,48),(21,41,155,87,117),(22,118,88,156,42),(23,43,157,81,119),(24,120,82,158,44),(25,97,143,61,134),(26,135,62,144,98),(27,99,137,63,136),(28,129,64,138,100),(29,101,139,57,130),(30,131,58,140,102),(31,103,141,59,132),(32,133,60,142,104)], [(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)], [(1,12,58,48),(2,15,59,43),(3,10,60,46),(4,13,61,41),(5,16,62,44),(6,11,63,47),(7,14,64,42),(8,9,57,45),(17,51,77,139),(18,54,78,142),(19,49,79,137),(20,52,80,140),(21,55,73,143),(22,50,74,138),(23,53,75,141),(24,56,76,144),(25,87,152,128),(26,82,145,123),(27,85,146,126),(28,88,147,121),(29,83,148,124),(30,86,149,127),(31,81,150,122),(32,84,151,125),(33,68,100,118),(34,71,101,113),(35,66,102,116),(36,69,103,119),(37,72,104,114),(38,67,97,117),(39,70,98,120),(40,65,99,115),(89,136,153,107),(90,131,154,110),(91,134,155,105),(92,129,156,108),(93,132,157,111),(94,135,158,106),(95,130,159,109),(96,133,160,112)], [(1,62),(2,59),(3,64),(4,61),(5,58),(6,63),(7,60),(8,57),(9,13),(11,15),(17,155),(18,160),(19,157),(20,154),(21,159),(22,156),(23,153),(24,158),(25,38),(26,35),(27,40),(28,37),(29,34),(30,39),(31,36),(32,33),(41,45),(43,47),(49,136),(50,133),(51,130),(52,135),(53,132),(54,129),(55,134),(56,131),(65,122),(66,127),(67,124),(68,121),(69,126),(70,123),(71,128),(72,125),(73,95),(74,92),(75,89),(76,94),(77,91),(78,96),(79,93),(80,90),(81,115),(82,120),(83,117),(84,114),(85,119),(86,116),(87,113),(88,118),(97,152),(98,149),(99,146),(100,151),(101,148),(102,145),(103,150),(104,147),(105,143),(106,140),(107,137),(108,142),(109,139),(110,144),(111,141),(112,138)]])
44 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 5A | 5B | 8A | 8B | 8C | 8D | 10A | ··· | 10F | 20A | 20B | 20C | 20D | 20E | ··· | 20L | 40A | ··· | 40H |
| order | 1 | 2 | 2 | 2 | 2 | 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 | 20 | 2 | 2 | 8 | 8 | 20 | 40 | 40 | 2 | 2 | 4 | 4 | 20 | 20 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 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 | SD16⋊D5 | Q16⋊D5 |
| kernel | C5⋊2C8.D4 | C20.Q8 | C20.44D4 | C5×Q8⋊C4 | D10⋊2Q8 | C2×C8⋊D5 | C2×C5⋊Q16 | D10⋊3Q8 | C5⋊2C8 | C2×Dic5 | C22×D5 | Q8⋊C4 | C20 | C4⋊C4 | C2×C8 | C2×Q8 | 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⋊2C8.D4 ►in GL6(𝔽41)
| 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 | 18 | 0 | 6 | 40 |
| 0 | 0 | 0 | 23 | 1 | 0 |
| 32 | 0 | 0 | 0 | 0 | 0 |
| 6 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 31 | 31 | 0 | 16 |
| 0 | 0 | 9 | 9 | 16 | 27 |
| 0 | 0 | 26 | 18 | 29 | 12 |
| 0 | 0 | 32 | 14 | 22 | 13 |
| 15 | 4 | 0 | 0 | 0 | 0 |
| 5 | 26 | 0 | 0 | 0 | 0 |
| 0 | 0 | 29 | 6 | 27 | 29 |
| 0 | 0 | 0 | 23 | 14 | 27 |
| 0 | 0 | 4 | 13 | 18 | 35 |
| 0 | 0 | 4 | 4 | 0 | 12 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 28 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 6 | 35 | 0 | 0 |
| 0 | 0 | 40 | 35 | 0 | 0 |
| 0 | 0 | 23 | 38 | 1 | 0 |
| 0 | 0 | 0 | 15 | 6 | 40 |
G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,18,0,0,0,40,6,0,23,0,0,0,0,6,1,0,0,0,0,40,0],[32,6,0,0,0,0,0,9,0,0,0,0,0,0,31,9,26,32,0,0,31,9,18,14,0,0,0,16,29,22,0,0,16,27,12,13],[15,5,0,0,0,0,4,26,0,0,0,0,0,0,29,0,4,4,0,0,6,23,13,4,0,0,27,14,18,0,0,0,29,27,35,12],[40,28,0,0,0,0,0,1,0,0,0,0,0,0,6,40,23,0,0,0,35,35,38,15,0,0,0,0,1,6,0,0,0,0,0,40] >;
C5⋊2C8.D4 in GAP, Magma, Sage, TeX
C_5\rtimes_2C_8.D_4
% in TeX
G:=Group("C5:2C8.D4"); // GroupNames label
G:=SmallGroup(320,443);
// by ID
G=gap.SmallGroup(320,443);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,253,344,1094,135,184,570,297,136,12550]);
// 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^5,d*c*d=b^4*c^-1>;
// generators/relations