direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C5×C8⋊5D4, C40⋊27D4, C20⋊9SD16, C8⋊5(C5×D4), C4⋊Q8⋊7C10, (C4×C8)⋊12C10, (C4×C40)⋊28C2, C4⋊1(C5×SD16), C4.1(D4×C10), C20.308(C2×D4), C4⋊1D4.6C10, (C2×C20).421D4, (C10×SD16)⋊31C2, (C2×SD16)⋊14C10, C42.79(C2×C10), C10.96(C2×SD16), C2.16(C10×SD16), C10.42(C4⋊1D4), (C4×C20).363C22, (C2×C40).438C22, (C2×C20).948C23, C22.113(D4×C10), (D4×C10).202C22, (Q8×C10).176C22, (C5×C4⋊Q8)⋊28C2, (C2×C4).77(C5×D4), C2.5(C5×C4⋊1D4), (C2×C8).94(C2×C10), (C2×D4).25(C2×C10), (C5×C4⋊1D4).13C2, (C2×C10).669(C2×D4), (C2×Q8).20(C2×C10), (C2×C4).123(C22×C10), SmallGroup(320,993)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5×C8⋊5D4
G = < a,b,c,d | a5=b8=c4=d2=1, ab=ba, ac=ca, ad=da, bc=cb, dbd=b3, dcd=c-1 >
Subgroups: 290 in 142 conjugacy classes, 66 normal (18 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C10, C10, C10, C42, C4⋊C4, C2×C8, SD16, C2×D4, C2×D4, C2×Q8, C20, C20, C2×C10, C2×C10, C4×C8, C4⋊1D4, C4⋊Q8, C2×SD16, C40, C2×C20, C2×C20, C2×C20, C5×D4, C5×Q8, C22×C10, C8⋊5D4, C4×C20, C5×C4⋊C4, C2×C40, C5×SD16, D4×C10, D4×C10, Q8×C10, C4×C40, C5×C4⋊1D4, C5×C4⋊Q8, C10×SD16, C5×C8⋊5D4
Quotients: C1, C2, C22, C5, D4, C23, C10, SD16, C2×D4, C2×C10, C4⋊1D4, C2×SD16, C5×D4, C22×C10, C8⋊5D4, C5×SD16, D4×C10, C5×C4⋊1D4, C10×SD16, C5×C8⋊5D4
►On 160 points
Generators in S160
(1 94 54 86 46)(2 95 55 87 47)(3 96 56 88 48)(4 89 49 81 41)(5 90 50 82 42)(6 91 51 83 43)(7 92 52 84 44)(8 93 53 85 45)(9 115 139 17 131)(10 116 140 18 132)(11 117 141 19 133)(12 118 142 20 134)(13 119 143 21 135)(14 120 144 22 136)(15 113 137 23 129)(16 114 138 24 130)(25 121 155 33 147)(26 122 156 34 148)(27 123 157 35 149)(28 124 158 36 150)(29 125 159 37 151)(30 126 160 38 152)(31 127 153 39 145)(32 128 154 40 146)(57 75 107 67 98)(58 76 108 68 99)(59 77 109 69 100)(60 78 110 70 101)(61 79 111 71 102)(62 80 112 72 103)(63 73 105 65 104)(64 74 106 66 97)
(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 71 29 119)(2 72 30 120)(3 65 31 113)(4 66 32 114)(5 67 25 115)(6 68 26 116)(7 69 27 117)(8 70 28 118)(9 42 107 147)(10 43 108 148)(11 44 109 149)(12 45 110 150)(13 46 111 151)(14 47 112 152)(15 48 105 145)(16 41 106 146)(17 50 57 155)(18 51 58 156)(19 52 59 157)(20 53 60 158)(21 54 61 159)(22 55 62 160)(23 56 63 153)(24 49 64 154)(33 131 82 75)(34 132 83 76)(35 133 84 77)(36 134 85 78)(37 135 86 79)(38 136 87 80)(39 129 88 73)(40 130 81 74)(89 97 128 138)(90 98 121 139)(91 99 122 140)(92 100 123 141)(93 101 124 142)(94 102 125 143)(95 103 126 144)(96 104 127 137)
(1 119)(2 114)(3 117)(4 120)(5 115)(6 118)(7 113)(8 116)(9 42)(10 45)(11 48)(12 43)(13 46)(14 41)(15 44)(16 47)(17 50)(18 53)(19 56)(20 51)(21 54)(22 49)(23 52)(24 55)(25 67)(26 70)(27 65)(28 68)(29 71)(30 66)(31 69)(32 72)(33 75)(34 78)(35 73)(36 76)(37 79)(38 74)(39 77)(40 80)(57 155)(58 158)(59 153)(60 156)(61 159)(62 154)(63 157)(64 160)(81 136)(82 131)(83 134)(84 129)(85 132)(86 135)(87 130)(88 133)(89 144)(90 139)(91 142)(92 137)(93 140)(94 143)(95 138)(96 141)(97 126)(98 121)(99 124)(100 127)(101 122)(102 125)(103 128)(104 123)(105 149)(106 152)(107 147)(108 150)(109 145)(110 148)(111 151)(112 146)
G:=sub<Sym(160)| (1,94,54,86,46)(2,95,55,87,47)(3,96,56,88,48)(4,89,49,81,41)(5,90,50,82,42)(6,91,51,83,43)(7,92,52,84,44)(8,93,53,85,45)(9,115,139,17,131)(10,116,140,18,132)(11,117,141,19,133)(12,118,142,20,134)(13,119,143,21,135)(14,120,144,22,136)(15,113,137,23,129)(16,114,138,24,130)(25,121,155,33,147)(26,122,156,34,148)(27,123,157,35,149)(28,124,158,36,150)(29,125,159,37,151)(30,126,160,38,152)(31,127,153,39,145)(32,128,154,40,146)(57,75,107,67,98)(58,76,108,68,99)(59,77,109,69,100)(60,78,110,70,101)(61,79,111,71,102)(62,80,112,72,103)(63,73,105,65,104)(64,74,106,66,97), (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), (1,71,29,119)(2,72,30,120)(3,65,31,113)(4,66,32,114)(5,67,25,115)(6,68,26,116)(7,69,27,117)(8,70,28,118)(9,42,107,147)(10,43,108,148)(11,44,109,149)(12,45,110,150)(13,46,111,151)(14,47,112,152)(15,48,105,145)(16,41,106,146)(17,50,57,155)(18,51,58,156)(19,52,59,157)(20,53,60,158)(21,54,61,159)(22,55,62,160)(23,56,63,153)(24,49,64,154)(33,131,82,75)(34,132,83,76)(35,133,84,77)(36,134,85,78)(37,135,86,79)(38,136,87,80)(39,129,88,73)(40,130,81,74)(89,97,128,138)(90,98,121,139)(91,99,122,140)(92,100,123,141)(93,101,124,142)(94,102,125,143)(95,103,126,144)(96,104,127,137), (1,119)(2,114)(3,117)(4,120)(5,115)(6,118)(7,113)(8,116)(9,42)(10,45)(11,48)(12,43)(13,46)(14,41)(15,44)(16,47)(17,50)(18,53)(19,56)(20,51)(21,54)(22,49)(23,52)(24,55)(25,67)(26,70)(27,65)(28,68)(29,71)(30,66)(31,69)(32,72)(33,75)(34,78)(35,73)(36,76)(37,79)(38,74)(39,77)(40,80)(57,155)(58,158)(59,153)(60,156)(61,159)(62,154)(63,157)(64,160)(81,136)(82,131)(83,134)(84,129)(85,132)(86,135)(87,130)(88,133)(89,144)(90,139)(91,142)(92,137)(93,140)(94,143)(95,138)(96,141)(97,126)(98,121)(99,124)(100,127)(101,122)(102,125)(103,128)(104,123)(105,149)(106,152)(107,147)(108,150)(109,145)(110,148)(111,151)(112,146)>;
G:=Group( (1,94,54,86,46)(2,95,55,87,47)(3,96,56,88,48)(4,89,49,81,41)(5,90,50,82,42)(6,91,51,83,43)(7,92,52,84,44)(8,93,53,85,45)(9,115,139,17,131)(10,116,140,18,132)(11,117,141,19,133)(12,118,142,20,134)(13,119,143,21,135)(14,120,144,22,136)(15,113,137,23,129)(16,114,138,24,130)(25,121,155,33,147)(26,122,156,34,148)(27,123,157,35,149)(28,124,158,36,150)(29,125,159,37,151)(30,126,160,38,152)(31,127,153,39,145)(32,128,154,40,146)(57,75,107,67,98)(58,76,108,68,99)(59,77,109,69,100)(60,78,110,70,101)(61,79,111,71,102)(62,80,112,72,103)(63,73,105,65,104)(64,74,106,66,97), (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), (1,71,29,119)(2,72,30,120)(3,65,31,113)(4,66,32,114)(5,67,25,115)(6,68,26,116)(7,69,27,117)(8,70,28,118)(9,42,107,147)(10,43,108,148)(11,44,109,149)(12,45,110,150)(13,46,111,151)(14,47,112,152)(15,48,105,145)(16,41,106,146)(17,50,57,155)(18,51,58,156)(19,52,59,157)(20,53,60,158)(21,54,61,159)(22,55,62,160)(23,56,63,153)(24,49,64,154)(33,131,82,75)(34,132,83,76)(35,133,84,77)(36,134,85,78)(37,135,86,79)(38,136,87,80)(39,129,88,73)(40,130,81,74)(89,97,128,138)(90,98,121,139)(91,99,122,140)(92,100,123,141)(93,101,124,142)(94,102,125,143)(95,103,126,144)(96,104,127,137), (1,119)(2,114)(3,117)(4,120)(5,115)(6,118)(7,113)(8,116)(9,42)(10,45)(11,48)(12,43)(13,46)(14,41)(15,44)(16,47)(17,50)(18,53)(19,56)(20,51)(21,54)(22,49)(23,52)(24,55)(25,67)(26,70)(27,65)(28,68)(29,71)(30,66)(31,69)(32,72)(33,75)(34,78)(35,73)(36,76)(37,79)(38,74)(39,77)(40,80)(57,155)(58,158)(59,153)(60,156)(61,159)(62,154)(63,157)(64,160)(81,136)(82,131)(83,134)(84,129)(85,132)(86,135)(87,130)(88,133)(89,144)(90,139)(91,142)(92,137)(93,140)(94,143)(95,138)(96,141)(97,126)(98,121)(99,124)(100,127)(101,122)(102,125)(103,128)(104,123)(105,149)(106,152)(107,147)(108,150)(109,145)(110,148)(111,151)(112,146) );
G=PermutationGroup([[(1,94,54,86,46),(2,95,55,87,47),(3,96,56,88,48),(4,89,49,81,41),(5,90,50,82,42),(6,91,51,83,43),(7,92,52,84,44),(8,93,53,85,45),(9,115,139,17,131),(10,116,140,18,132),(11,117,141,19,133),(12,118,142,20,134),(13,119,143,21,135),(14,120,144,22,136),(15,113,137,23,129),(16,114,138,24,130),(25,121,155,33,147),(26,122,156,34,148),(27,123,157,35,149),(28,124,158,36,150),(29,125,159,37,151),(30,126,160,38,152),(31,127,153,39,145),(32,128,154,40,146),(57,75,107,67,98),(58,76,108,68,99),(59,77,109,69,100),(60,78,110,70,101),(61,79,111,71,102),(62,80,112,72,103),(63,73,105,65,104),(64,74,106,66,97)], [(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)], [(1,71,29,119),(2,72,30,120),(3,65,31,113),(4,66,32,114),(5,67,25,115),(6,68,26,116),(7,69,27,117),(8,70,28,118),(9,42,107,147),(10,43,108,148),(11,44,109,149),(12,45,110,150),(13,46,111,151),(14,47,112,152),(15,48,105,145),(16,41,106,146),(17,50,57,155),(18,51,58,156),(19,52,59,157),(20,53,60,158),(21,54,61,159),(22,55,62,160),(23,56,63,153),(24,49,64,154),(33,131,82,75),(34,132,83,76),(35,133,84,77),(36,134,85,78),(37,135,86,79),(38,136,87,80),(39,129,88,73),(40,130,81,74),(89,97,128,138),(90,98,121,139),(91,99,122,140),(92,100,123,141),(93,101,124,142),(94,102,125,143),(95,103,126,144),(96,104,127,137)], [(1,119),(2,114),(3,117),(4,120),(5,115),(6,118),(7,113),(8,116),(9,42),(10,45),(11,48),(12,43),(13,46),(14,41),(15,44),(16,47),(17,50),(18,53),(19,56),(20,51),(21,54),(22,49),(23,52),(24,55),(25,67),(26,70),(27,65),(28,68),(29,71),(30,66),(31,69),(32,72),(33,75),(34,78),(35,73),(36,76),(37,79),(38,74),(39,77),(40,80),(57,155),(58,158),(59,153),(60,156),(61,159),(62,154),(63,157),(64,160),(81,136),(82,131),(83,134),(84,129),(85,132),(86,135),(87,130),(88,133),(89,144),(90,139),(91,142),(92,137),(93,140),(94,143),(95,138),(96,141),(97,126),(98,121),(99,124),(100,127),(101,122),(102,125),(103,128),(104,123),(105,149),(106,152),(107,147),(108,150),(109,145),(110,148),(111,151),(112,146)]])
110 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | ··· | 4F | 4G | 4H | 5A | 5B | 5C | 5D | 8A | ··· | 8H | 10A | ··· | 10L | 10M | ··· | 10T | 20A | ··· | 20X | 20Y | ··· | 20AF | 40A | ··· | 40AF |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 8 | ··· | 8 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 8 | 8 | 2 | ··· | 2 | 8 | 8 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 8 | ··· | 8 | 2 | ··· | 2 | 8 | ··· | 8 | 2 | ··· | 2 |
110 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | |||||||||
| image | C1 | C2 | C2 | C2 | C2 | C5 | C10 | C10 | C10 | C10 | D4 | D4 | SD16 | C5×D4 | C5×D4 | C5×SD16 |
| kernel | C5×C8⋊5D4 | C4×C40 | C5×C4⋊1D4 | C5×C4⋊Q8 | C10×SD16 | C8⋊5D4 | C4×C8 | C4⋊1D4 | C4⋊Q8 | C2×SD16 | C40 | C2×C20 | C20 | C8 | C2×C4 | C4 |
| # reps | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 4 | 4 | 16 | 4 | 2 | 8 | 16 | 8 | 32 |
Matrix representation of C5×C8⋊5D4 ►in GL4(𝔽41) generated by
| 37 | 0 | 0 | 0 |
| 0 | 37 | 0 | 0 |
| 0 | 0 | 18 | 0 |
| 0 | 0 | 0 | 18 |
| 0 | 30 | 0 | 0 |
| 26 | 30 | 0 | 0 |
| 0 | 0 | 26 | 26 |
| 0 | 0 | 15 | 26 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 40 | 0 |
| 1 | 0 | 0 | 0 |
| 1 | 40 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
G:=sub<GL(4,GF(41))| [37,0,0,0,0,37,0,0,0,0,18,0,0,0,0,18],[0,26,0,0,30,30,0,0,0,0,26,15,0,0,26,26],[1,0,0,0,0,1,0,0,0,0,0,40,0,0,1,0],[1,1,0,0,0,40,0,0,0,0,0,1,0,0,1,0] >;
C5×C8⋊5D4 in GAP, Magma, Sage, TeX
C_5\times C_8\rtimes_5D_4
% in TeX
G:=Group("C5xC8:5D4"); // GroupNames label
G:=SmallGroup(320,993);
// by ID
G=gap.SmallGroup(320,993);
# by ID
G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-2,589,288,1766,436,7004,172]);
// Polycyclic
G:=Group<a,b,c,d|a^5=b^8=c^4=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d=b^3,d*c*d=c^-1>;
// generators/relations