metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: (C8×D5)⋊4C4, C8.30(C4×D5), C4.24(Q8×D5), C4.Q8⋊14D5, C40.75(C2×C4), C40⋊6C4⋊25C2, (C4×D5).14Q8, C20.13(C2×Q8), C4⋊C4.161D10, (C2×C8).259D10, C22.84(D4×D5), D10.14(C4⋊C4), C10.54(C4○D8), C10.D8⋊14C2, (C22×D5).83D4, Dic5.38(C4⋊C4), (C2×C20).276C23, C20.102(C22×C4), (C2×C40).160C22, (C2×Dic5).274D4, C5⋊2(C23.25D4), C2.6(SD16⋊3D5), C4⋊Dic5.108C22, (D5×C2×C8).8C2, C4.77(C2×C4×D5), C2.12(D5×C4⋊C4), (C5×C4.Q8)⋊8C2, C10.34(C2×C4⋊C4), C5⋊2C8.38(C2×C4), C4⋊C4⋊7D5.4C2, (C4×D5).74(C2×C4), (C2×C10).281(C2×D4), (C5×C4⋊C4).69C22, (C2×C4×D5).302C22, (C2×C4).379(C22×D5), (C2×C5⋊2C8).235C22, SmallGroup(320,487)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for (C8×D5)⋊C4
G = < a,b,c,d | a8=b5=c2=d4=1, ab=ba, ac=ca, dad-1=a3, cbc=b-1, bd=db, dcd-1=a4c >
Subgroups: 382 in 114 conjugacy classes, 55 normal (23 characteristic)
C1, C2, C2 [×2], C2 [×2], C4 [×2], C4 [×6], C22, C22 [×4], C5, C8 [×2], C8 [×2], C2×C4, C2×C4 [×9], C23, D5 [×2], C10, C10 [×2], C42 [×2], C22⋊C4 [×2], C4⋊C4 [×2], C4⋊C4 [×2], C2×C8, C2×C8 [×5], C22×C4, Dic5 [×2], Dic5 [×2], C20 [×2], C20 [×2], D10 [×2], D10 [×2], C2×C10, C4.Q8, C4.Q8, C2.D8 [×2], C42⋊C2 [×2], C22×C8, C5⋊2C8 [×2], C40 [×2], C4×D5 [×4], C2×Dic5, C2×Dic5 [×2], C2×C20, C2×C20 [×2], C22×D5, C23.25D4, C8×D5 [×4], C2×C5⋊2C8, C4×Dic5 [×2], C4⋊Dic5 [×2], D10⋊C4 [×2], C5×C4⋊C4 [×2], C2×C40, C2×C4×D5, C10.D8 [×2], C40⋊6C4, C5×C4.Q8, C4⋊C4⋊7D5 [×2], D5×C2×C8, (C8×D5)⋊C4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×2], Q8 [×2], C23, D5, C4⋊C4 [×4], C22×C4, C2×D4, C2×Q8, D10 [×3], C2×C4⋊C4, C4○D8 [×2], C4×D5 [×2], C22×D5, C23.25D4, C2×C4×D5, D4×D5, Q8×D5, D5×C4⋊C4, SD16⋊3D5 [×2], (C8×D5)⋊C4
►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 99 133 21 113)(2 100 134 22 114)(3 101 135 23 115)(4 102 136 24 116)(5 103 129 17 117)(6 104 130 18 118)(7 97 131 19 119)(8 98 132 20 120)(9 124 81 93 49)(10 125 82 94 50)(11 126 83 95 51)(12 127 84 96 52)(13 128 85 89 53)(14 121 86 90 54)(15 122 87 91 55)(16 123 88 92 56)(25 137 146 107 70)(26 138 147 108 71)(27 139 148 109 72)(28 140 149 110 65)(29 141 150 111 66)(30 142 151 112 67)(31 143 152 105 68)(32 144 145 106 69)(33 45 160 58 74)(34 46 153 59 75)(35 47 154 60 76)(36 48 155 61 77)(37 41 156 62 78)(38 42 157 63 79)(39 43 158 64 80)(40 44 159 57 73)
(1 148)(2 149)(3 150)(4 151)(5 152)(6 145)(7 146)(8 147)(9 38)(10 39)(11 40)(12 33)(13 34)(14 35)(15 36)(16 37)(17 68)(18 69)(19 70)(20 71)(21 72)(22 65)(23 66)(24 67)(25 131)(26 132)(27 133)(28 134)(29 135)(30 136)(31 129)(32 130)(41 56)(42 49)(43 50)(44 51)(45 52)(46 53)(47 54)(48 55)(57 83)(58 84)(59 85)(60 86)(61 87)(62 88)(63 81)(64 82)(73 126)(74 127)(75 128)(76 121)(77 122)(78 123)(79 124)(80 125)(89 153)(90 154)(91 155)(92 156)(93 157)(94 158)(95 159)(96 160)(97 137)(98 138)(99 139)(100 140)(101 141)(102 142)(103 143)(104 144)(105 117)(106 118)(107 119)(108 120)(109 113)(110 114)(111 115)(112 116)
(1 55 105 155)(2 50 106 158)(3 53 107 153)(4 56 108 156)(5 51 109 159)(6 54 110 154)(7 49 111 157)(8 52 112 160)(9 66 63 97)(10 69 64 100)(11 72 57 103)(12 67 58 98)(13 70 59 101)(14 65 60 104)(15 68 61 99)(16 71 62 102)(17 83 139 40)(18 86 140 35)(19 81 141 38)(20 84 142 33)(21 87 143 36)(22 82 144 39)(23 85 137 34)(24 88 138 37)(25 75 135 128)(26 78 136 123)(27 73 129 126)(28 76 130 121)(29 79 131 124)(30 74 132 127)(31 77 133 122)(32 80 134 125)(41 116 92 147)(42 119 93 150)(43 114 94 145)(44 117 95 148)(45 120 96 151)(46 115 89 146)(47 118 90 149)(48 113 91 152)
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), (1,99,133,21,113)(2,100,134,22,114)(3,101,135,23,115)(4,102,136,24,116)(5,103,129,17,117)(6,104,130,18,118)(7,97,131,19,119)(8,98,132,20,120)(9,124,81,93,49)(10,125,82,94,50)(11,126,83,95,51)(12,127,84,96,52)(13,128,85,89,53)(14,121,86,90,54)(15,122,87,91,55)(16,123,88,92,56)(25,137,146,107,70)(26,138,147,108,71)(27,139,148,109,72)(28,140,149,110,65)(29,141,150,111,66)(30,142,151,112,67)(31,143,152,105,68)(32,144,145,106,69)(33,45,160,58,74)(34,46,153,59,75)(35,47,154,60,76)(36,48,155,61,77)(37,41,156,62,78)(38,42,157,63,79)(39,43,158,64,80)(40,44,159,57,73), (1,148)(2,149)(3,150)(4,151)(5,152)(6,145)(7,146)(8,147)(9,38)(10,39)(11,40)(12,33)(13,34)(14,35)(15,36)(16,37)(17,68)(18,69)(19,70)(20,71)(21,72)(22,65)(23,66)(24,67)(25,131)(26,132)(27,133)(28,134)(29,135)(30,136)(31,129)(32,130)(41,56)(42,49)(43,50)(44,51)(45,52)(46,53)(47,54)(48,55)(57,83)(58,84)(59,85)(60,86)(61,87)(62,88)(63,81)(64,82)(73,126)(74,127)(75,128)(76,121)(77,122)(78,123)(79,124)(80,125)(89,153)(90,154)(91,155)(92,156)(93,157)(94,158)(95,159)(96,160)(97,137)(98,138)(99,139)(100,140)(101,141)(102,142)(103,143)(104,144)(105,117)(106,118)(107,119)(108,120)(109,113)(110,114)(111,115)(112,116), (1,55,105,155)(2,50,106,158)(3,53,107,153)(4,56,108,156)(5,51,109,159)(6,54,110,154)(7,49,111,157)(8,52,112,160)(9,66,63,97)(10,69,64,100)(11,72,57,103)(12,67,58,98)(13,70,59,101)(14,65,60,104)(15,68,61,99)(16,71,62,102)(17,83,139,40)(18,86,140,35)(19,81,141,38)(20,84,142,33)(21,87,143,36)(22,82,144,39)(23,85,137,34)(24,88,138,37)(25,75,135,128)(26,78,136,123)(27,73,129,126)(28,76,130,121)(29,79,131,124)(30,74,132,127)(31,77,133,122)(32,80,134,125)(41,116,92,147)(42,119,93,150)(43,114,94,145)(44,117,95,148)(45,120,96,151)(46,115,89,146)(47,118,90,149)(48,113,91,152)>;
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), (1,99,133,21,113)(2,100,134,22,114)(3,101,135,23,115)(4,102,136,24,116)(5,103,129,17,117)(6,104,130,18,118)(7,97,131,19,119)(8,98,132,20,120)(9,124,81,93,49)(10,125,82,94,50)(11,126,83,95,51)(12,127,84,96,52)(13,128,85,89,53)(14,121,86,90,54)(15,122,87,91,55)(16,123,88,92,56)(25,137,146,107,70)(26,138,147,108,71)(27,139,148,109,72)(28,140,149,110,65)(29,141,150,111,66)(30,142,151,112,67)(31,143,152,105,68)(32,144,145,106,69)(33,45,160,58,74)(34,46,153,59,75)(35,47,154,60,76)(36,48,155,61,77)(37,41,156,62,78)(38,42,157,63,79)(39,43,158,64,80)(40,44,159,57,73), (1,148)(2,149)(3,150)(4,151)(5,152)(6,145)(7,146)(8,147)(9,38)(10,39)(11,40)(12,33)(13,34)(14,35)(15,36)(16,37)(17,68)(18,69)(19,70)(20,71)(21,72)(22,65)(23,66)(24,67)(25,131)(26,132)(27,133)(28,134)(29,135)(30,136)(31,129)(32,130)(41,56)(42,49)(43,50)(44,51)(45,52)(46,53)(47,54)(48,55)(57,83)(58,84)(59,85)(60,86)(61,87)(62,88)(63,81)(64,82)(73,126)(74,127)(75,128)(76,121)(77,122)(78,123)(79,124)(80,125)(89,153)(90,154)(91,155)(92,156)(93,157)(94,158)(95,159)(96,160)(97,137)(98,138)(99,139)(100,140)(101,141)(102,142)(103,143)(104,144)(105,117)(106,118)(107,119)(108,120)(109,113)(110,114)(111,115)(112,116), (1,55,105,155)(2,50,106,158)(3,53,107,153)(4,56,108,156)(5,51,109,159)(6,54,110,154)(7,49,111,157)(8,52,112,160)(9,66,63,97)(10,69,64,100)(11,72,57,103)(12,67,58,98)(13,70,59,101)(14,65,60,104)(15,68,61,99)(16,71,62,102)(17,83,139,40)(18,86,140,35)(19,81,141,38)(20,84,142,33)(21,87,143,36)(22,82,144,39)(23,85,137,34)(24,88,138,37)(25,75,135,128)(26,78,136,123)(27,73,129,126)(28,76,130,121)(29,79,131,124)(30,74,132,127)(31,77,133,122)(32,80,134,125)(41,116,92,147)(42,119,93,150)(43,114,94,145)(44,117,95,148)(45,120,96,151)(46,115,89,146)(47,118,90,149)(48,113,91,152) );
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)], [(1,99,133,21,113),(2,100,134,22,114),(3,101,135,23,115),(4,102,136,24,116),(5,103,129,17,117),(6,104,130,18,118),(7,97,131,19,119),(8,98,132,20,120),(9,124,81,93,49),(10,125,82,94,50),(11,126,83,95,51),(12,127,84,96,52),(13,128,85,89,53),(14,121,86,90,54),(15,122,87,91,55),(16,123,88,92,56),(25,137,146,107,70),(26,138,147,108,71),(27,139,148,109,72),(28,140,149,110,65),(29,141,150,111,66),(30,142,151,112,67),(31,143,152,105,68),(32,144,145,106,69),(33,45,160,58,74),(34,46,153,59,75),(35,47,154,60,76),(36,48,155,61,77),(37,41,156,62,78),(38,42,157,63,79),(39,43,158,64,80),(40,44,159,57,73)], [(1,148),(2,149),(3,150),(4,151),(5,152),(6,145),(7,146),(8,147),(9,38),(10,39),(11,40),(12,33),(13,34),(14,35),(15,36),(16,37),(17,68),(18,69),(19,70),(20,71),(21,72),(22,65),(23,66),(24,67),(25,131),(26,132),(27,133),(28,134),(29,135),(30,136),(31,129),(32,130),(41,56),(42,49),(43,50),(44,51),(45,52),(46,53),(47,54),(48,55),(57,83),(58,84),(59,85),(60,86),(61,87),(62,88),(63,81),(64,82),(73,126),(74,127),(75,128),(76,121),(77,122),(78,123),(79,124),(80,125),(89,153),(90,154),(91,155),(92,156),(93,157),(94,158),(95,159),(96,160),(97,137),(98,138),(99,139),(100,140),(101,141),(102,142),(103,143),(104,144),(105,117),(106,118),(107,119),(108,120),(109,113),(110,114),(111,115),(112,116)], [(1,55,105,155),(2,50,106,158),(3,53,107,153),(4,56,108,156),(5,51,109,159),(6,54,110,154),(7,49,111,157),(8,52,112,160),(9,66,63,97),(10,69,64,100),(11,72,57,103),(12,67,58,98),(13,70,59,101),(14,65,60,104),(15,68,61,99),(16,71,62,102),(17,83,139,40),(18,86,140,35),(19,81,141,38),(20,84,142,33),(21,87,143,36),(22,82,144,39),(23,85,137,34),(24,88,138,37),(25,75,135,128),(26,78,136,123),(27,73,129,126),(28,76,130,121),(29,79,131,124),(30,74,132,127),(31,77,133,122),(32,80,134,125),(41,116,92,147),(42,119,93,150),(43,114,94,145),(44,117,95,148),(45,120,96,151),(46,115,89,146),(47,118,90,149),(48,113,91,152)])
56 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 5A | 5B | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 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 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 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 | 5 | 5 | 5 | 5 | 20 | 20 | 20 | 20 | 2 | 2 | 2 | 2 | 2 | 2 | 10 | 10 | 10 | 10 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 8 | ··· | 8 | 4 | ··· | 4 |
56 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | - | + | + | + | + | + | - | + | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | Q8 | D4 | D4 | D5 | D10 | D10 | C4○D8 | C4×D5 | Q8×D5 | D4×D5 | SD16⋊3D5 |
| kernel | (C8×D5)⋊C4 | C10.D8 | C40⋊6C4 | C5×C4.Q8 | C4⋊C4⋊7D5 | D5×C2×C8 | C8×D5 | C4×D5 | C2×Dic5 | C22×D5 | C4.Q8 | C4⋊C4 | C2×C8 | C10 | C8 | C4 | C22 | C2 |
| # reps | 1 | 2 | 1 | 1 | 2 | 1 | 8 | 2 | 1 | 1 | 2 | 4 | 2 | 8 | 8 | 2 | 2 | 8 |
Matrix representation of (C8×D5)⋊C4 ►in GL4(𝔽41) generated by
| 40 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 27 | 0 |
| 0 | 0 | 29 | 3 |
| 35 | 40 | 0 | 0 |
| 36 | 40 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 7 | 0 | 0 |
| 6 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 1 | 1 |
| 32 | 0 | 0 | 0 |
| 0 | 32 | 0 | 0 |
| 0 | 0 | 40 | 39 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(41))| [40,0,0,0,0,40,0,0,0,0,27,29,0,0,0,3],[35,36,0,0,40,40,0,0,0,0,1,0,0,0,0,1],[0,6,0,0,7,0,0,0,0,0,40,1,0,0,0,1],[32,0,0,0,0,32,0,0,0,0,40,0,0,0,39,1] >;
(C8×D5)⋊C4 in GAP, Magma, Sage, TeX
(C_8\times D_5)\rtimes C_4
% in TeX
G:=Group("(C8xD5):C4"); // GroupNames label
G:=SmallGroup(320,487);
// by ID
G=gap.SmallGroup(320,487);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,120,758,555,58,438,102,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^8=b^5=c^2=d^4=1,a*b=b*a,a*c=c*a,d*a*d^-1=a^3,c*b*c=b^-1,b*d=d*b,d*c*d^-1=a^4*c>;
// generators/relations