direct product, metabelian, nilpotent (class 4), monomial, 2-elementary
Aliases: C5×C4○D16, D16⋊3C10, Q32⋊3C10, C40.76D4, C20.71D8, SD32⋊3C10, C80.23C22, C40.75C23, (C2×C80)⋊13C2, (C2×C16)⋊6C10, C4○D8⋊1C10, (C5×D16)⋊7C2, (C5×Q32)⋊7C2, C4.20(C5×D8), C8.13(C5×D4), C16.6(C2×C10), (C5×SD32)⋊7C2, D8.2(C2×C10), C10.87(C2×D8), C4.10(D4×C10), C2.15(C10×D8), (C2×C10).12D8, C22.1(C5×D8), C20.317(C2×D4), (C2×C20).429D4, C8.6(C22×C10), Q16.2(C2×C10), (C5×D8).12C22, (C2×C40).434C22, (C5×Q16).14C22, (C5×C4○D8)⋊8C2, (C2×C4).85(C5×D4), (C2×C8).91(C2×C10), SmallGroup(320,1009)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5×C4○D16
G = < a,b,c,d | a5=b4=d2=1, c8=b2, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=b2c7 >
Subgroups: 178 in 84 conjugacy classes, 46 normal (30 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, Q8, C10, C10, C16, C2×C8, D8, SD16, Q16, C4○D4, C20, C20, C2×C10, C2×C10, C2×C16, D16, SD32, Q32, C4○D8, C40, C2×C20, C2×C20, C5×D4, C5×Q8, C4○D16, C80, C2×C40, C5×D8, C5×SD16, C5×Q16, C5×C4○D4, C2×C80, C5×D16, C5×SD32, C5×Q32, C5×C4○D8, C5×C4○D16
Quotients: C1, C2, C22, C5, D4, C23, C10, D8, C2×D4, C2×C10, C2×D8, C5×D4, C22×C10, C4○D16, C5×D8, D4×C10, C10×D8, C5×C4○D16
►On 160 points
Generators in S160
(1 151 128 142 56)(2 152 113 143 57)(3 153 114 144 58)(4 154 115 129 59)(5 155 116 130 60)(6 156 117 131 61)(7 157 118 132 62)(8 158 119 133 63)(9 159 120 134 64)(10 160 121 135 49)(11 145 122 136 50)(12 146 123 137 51)(13 147 124 138 52)(14 148 125 139 53)(15 149 126 140 54)(16 150 127 141 55)(17 74 90 35 101)(18 75 91 36 102)(19 76 92 37 103)(20 77 93 38 104)(21 78 94 39 105)(22 79 95 40 106)(23 80 96 41 107)(24 65 81 42 108)(25 66 82 43 109)(26 67 83 44 110)(27 68 84 45 111)(28 69 85 46 112)(29 70 86 47 97)(30 71 87 48 98)(31 72 88 33 99)(32 73 89 34 100)
(1 40 9 48)(2 41 10 33)(3 42 11 34)(4 43 12 35)(5 44 13 36)(6 45 14 37)(7 46 15 38)(8 47 16 39)(17 115 25 123)(18 116 26 124)(19 117 27 125)(20 118 28 126)(21 119 29 127)(22 120 30 128)(23 121 31 113)(24 122 32 114)(49 88 57 96)(50 89 58 81)(51 90 59 82)(52 91 60 83)(53 92 61 84)(54 93 62 85)(55 94 63 86)(56 95 64 87)(65 136 73 144)(66 137 74 129)(67 138 75 130)(68 139 76 131)(69 140 77 132)(70 141 78 133)(71 142 79 134)(72 143 80 135)(97 150 105 158)(98 151 106 159)(99 152 107 160)(100 153 108 145)(101 154 109 146)(102 155 110 147)(103 156 111 148)(104 157 112 149)
(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 16)(2 15)(3 14)(4 13)(5 12)(6 11)(7 10)(8 9)(17 26)(18 25)(19 24)(20 23)(21 22)(27 32)(28 31)(29 30)(33 46)(34 45)(35 44)(36 43)(37 42)(38 41)(39 40)(47 48)(49 62)(50 61)(51 60)(52 59)(53 58)(54 57)(55 56)(63 64)(65 76)(66 75)(67 74)(68 73)(69 72)(70 71)(77 80)(78 79)(81 92)(82 91)(83 90)(84 89)(85 88)(86 87)(93 96)(94 95)(97 98)(99 112)(100 111)(101 110)(102 109)(103 108)(104 107)(105 106)(113 126)(114 125)(115 124)(116 123)(117 122)(118 121)(119 120)(127 128)(129 138)(130 137)(131 136)(132 135)(133 134)(139 144)(140 143)(141 142)(145 156)(146 155)(147 154)(148 153)(149 152)(150 151)(157 160)(158 159)
G:=sub<Sym(160)| (1,151,128,142,56)(2,152,113,143,57)(3,153,114,144,58)(4,154,115,129,59)(5,155,116,130,60)(6,156,117,131,61)(7,157,118,132,62)(8,158,119,133,63)(9,159,120,134,64)(10,160,121,135,49)(11,145,122,136,50)(12,146,123,137,51)(13,147,124,138,52)(14,148,125,139,53)(15,149,126,140,54)(16,150,127,141,55)(17,74,90,35,101)(18,75,91,36,102)(19,76,92,37,103)(20,77,93,38,104)(21,78,94,39,105)(22,79,95,40,106)(23,80,96,41,107)(24,65,81,42,108)(25,66,82,43,109)(26,67,83,44,110)(27,68,84,45,111)(28,69,85,46,112)(29,70,86,47,97)(30,71,87,48,98)(31,72,88,33,99)(32,73,89,34,100), (1,40,9,48)(2,41,10,33)(3,42,11,34)(4,43,12,35)(5,44,13,36)(6,45,14,37)(7,46,15,38)(8,47,16,39)(17,115,25,123)(18,116,26,124)(19,117,27,125)(20,118,28,126)(21,119,29,127)(22,120,30,128)(23,121,31,113)(24,122,32,114)(49,88,57,96)(50,89,58,81)(51,90,59,82)(52,91,60,83)(53,92,61,84)(54,93,62,85)(55,94,63,86)(56,95,64,87)(65,136,73,144)(66,137,74,129)(67,138,75,130)(68,139,76,131)(69,140,77,132)(70,141,78,133)(71,142,79,134)(72,143,80,135)(97,150,105,158)(98,151,106,159)(99,152,107,160)(100,153,108,145)(101,154,109,146)(102,155,110,147)(103,156,111,148)(104,157,112,149), (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), (1,16)(2,15)(3,14)(4,13)(5,12)(6,11)(7,10)(8,9)(17,26)(18,25)(19,24)(20,23)(21,22)(27,32)(28,31)(29,30)(33,46)(34,45)(35,44)(36,43)(37,42)(38,41)(39,40)(47,48)(49,62)(50,61)(51,60)(52,59)(53,58)(54,57)(55,56)(63,64)(65,76)(66,75)(67,74)(68,73)(69,72)(70,71)(77,80)(78,79)(81,92)(82,91)(83,90)(84,89)(85,88)(86,87)(93,96)(94,95)(97,98)(99,112)(100,111)(101,110)(102,109)(103,108)(104,107)(105,106)(113,126)(114,125)(115,124)(116,123)(117,122)(118,121)(119,120)(127,128)(129,138)(130,137)(131,136)(132,135)(133,134)(139,144)(140,143)(141,142)(145,156)(146,155)(147,154)(148,153)(149,152)(150,151)(157,160)(158,159)>;
G:=Group( (1,151,128,142,56)(2,152,113,143,57)(3,153,114,144,58)(4,154,115,129,59)(5,155,116,130,60)(6,156,117,131,61)(7,157,118,132,62)(8,158,119,133,63)(9,159,120,134,64)(10,160,121,135,49)(11,145,122,136,50)(12,146,123,137,51)(13,147,124,138,52)(14,148,125,139,53)(15,149,126,140,54)(16,150,127,141,55)(17,74,90,35,101)(18,75,91,36,102)(19,76,92,37,103)(20,77,93,38,104)(21,78,94,39,105)(22,79,95,40,106)(23,80,96,41,107)(24,65,81,42,108)(25,66,82,43,109)(26,67,83,44,110)(27,68,84,45,111)(28,69,85,46,112)(29,70,86,47,97)(30,71,87,48,98)(31,72,88,33,99)(32,73,89,34,100), (1,40,9,48)(2,41,10,33)(3,42,11,34)(4,43,12,35)(5,44,13,36)(6,45,14,37)(7,46,15,38)(8,47,16,39)(17,115,25,123)(18,116,26,124)(19,117,27,125)(20,118,28,126)(21,119,29,127)(22,120,30,128)(23,121,31,113)(24,122,32,114)(49,88,57,96)(50,89,58,81)(51,90,59,82)(52,91,60,83)(53,92,61,84)(54,93,62,85)(55,94,63,86)(56,95,64,87)(65,136,73,144)(66,137,74,129)(67,138,75,130)(68,139,76,131)(69,140,77,132)(70,141,78,133)(71,142,79,134)(72,143,80,135)(97,150,105,158)(98,151,106,159)(99,152,107,160)(100,153,108,145)(101,154,109,146)(102,155,110,147)(103,156,111,148)(104,157,112,149), (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), (1,16)(2,15)(3,14)(4,13)(5,12)(6,11)(7,10)(8,9)(17,26)(18,25)(19,24)(20,23)(21,22)(27,32)(28,31)(29,30)(33,46)(34,45)(35,44)(36,43)(37,42)(38,41)(39,40)(47,48)(49,62)(50,61)(51,60)(52,59)(53,58)(54,57)(55,56)(63,64)(65,76)(66,75)(67,74)(68,73)(69,72)(70,71)(77,80)(78,79)(81,92)(82,91)(83,90)(84,89)(85,88)(86,87)(93,96)(94,95)(97,98)(99,112)(100,111)(101,110)(102,109)(103,108)(104,107)(105,106)(113,126)(114,125)(115,124)(116,123)(117,122)(118,121)(119,120)(127,128)(129,138)(130,137)(131,136)(132,135)(133,134)(139,144)(140,143)(141,142)(145,156)(146,155)(147,154)(148,153)(149,152)(150,151)(157,160)(158,159) );
G=PermutationGroup([[(1,151,128,142,56),(2,152,113,143,57),(3,153,114,144,58),(4,154,115,129,59),(5,155,116,130,60),(6,156,117,131,61),(7,157,118,132,62),(8,158,119,133,63),(9,159,120,134,64),(10,160,121,135,49),(11,145,122,136,50),(12,146,123,137,51),(13,147,124,138,52),(14,148,125,139,53),(15,149,126,140,54),(16,150,127,141,55),(17,74,90,35,101),(18,75,91,36,102),(19,76,92,37,103),(20,77,93,38,104),(21,78,94,39,105),(22,79,95,40,106),(23,80,96,41,107),(24,65,81,42,108),(25,66,82,43,109),(26,67,83,44,110),(27,68,84,45,111),(28,69,85,46,112),(29,70,86,47,97),(30,71,87,48,98),(31,72,88,33,99),(32,73,89,34,100)], [(1,40,9,48),(2,41,10,33),(3,42,11,34),(4,43,12,35),(5,44,13,36),(6,45,14,37),(7,46,15,38),(8,47,16,39),(17,115,25,123),(18,116,26,124),(19,117,27,125),(20,118,28,126),(21,119,29,127),(22,120,30,128),(23,121,31,113),(24,122,32,114),(49,88,57,96),(50,89,58,81),(51,90,59,82),(52,91,60,83),(53,92,61,84),(54,93,62,85),(55,94,63,86),(56,95,64,87),(65,136,73,144),(66,137,74,129),(67,138,75,130),(68,139,76,131),(69,140,77,132),(70,141,78,133),(71,142,79,134),(72,143,80,135),(97,150,105,158),(98,151,106,159),(99,152,107,160),(100,153,108,145),(101,154,109,146),(102,155,110,147),(103,156,111,148),(104,157,112,149)], [(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)], [(1,16),(2,15),(3,14),(4,13),(5,12),(6,11),(7,10),(8,9),(17,26),(18,25),(19,24),(20,23),(21,22),(27,32),(28,31),(29,30),(33,46),(34,45),(35,44),(36,43),(37,42),(38,41),(39,40),(47,48),(49,62),(50,61),(51,60),(52,59),(53,58),(54,57),(55,56),(63,64),(65,76),(66,75),(67,74),(68,73),(69,72),(70,71),(77,80),(78,79),(81,92),(82,91),(83,90),(84,89),(85,88),(86,87),(93,96),(94,95),(97,98),(99,112),(100,111),(101,110),(102,109),(103,108),(104,107),(105,106),(113,126),(114,125),(115,124),(116,123),(117,122),(118,121),(119,120),(127,128),(129,138),(130,137),(131,136),(132,135),(133,134),(139,144),(140,143),(141,142),(145,156),(146,155),(147,154),(148,153),(149,152),(150,151),(157,160),(158,159)]])
110 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | 4D | 4E | 5A | 5B | 5C | 5D | 8A | 8B | 8C | 8D | 10A | 10B | 10C | 10D | 10E | 10F | 10G | 10H | 10I | ··· | 10P | 16A | ··· | 16H | 20A | ··· | 20H | 20I | 20J | 20K | 20L | 20M | ··· | 20T | 40A | ··· | 40P | 80A | ··· | 80AF |
| order | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 8 | 8 | 8 | 8 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | ··· | 10 | 16 | ··· | 16 | 20 | ··· | 20 | 20 | 20 | 20 | 20 | 20 | ··· | 20 | 40 | ··· | 40 | 80 | ··· | 80 |
| size | 1 | 1 | 2 | 8 | 8 | 1 | 1 | 2 | 8 | 8 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 8 | ··· | 8 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 8 | ··· | 8 | 2 | ··· | 2 | 2 | ··· | 2 |
110 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | ||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C5 | C10 | C10 | C10 | C10 | C10 | D4 | D4 | D8 | D8 | C5×D4 | C5×D4 | C4○D16 | C5×D8 | C5×D8 | C5×C4○D16 |
| kernel | C5×C4○D16 | C2×C80 | C5×D16 | C5×SD32 | C5×Q32 | C5×C4○D8 | C4○D16 | C2×C16 | D16 | SD32 | Q32 | C4○D8 | C40 | C2×C20 | C20 | C2×C10 | C8 | C2×C4 | C5 | C4 | C22 | C1 |
| # reps | 1 | 1 | 1 | 2 | 1 | 2 | 4 | 4 | 4 | 8 | 4 | 8 | 1 | 1 | 2 | 2 | 4 | 4 | 8 | 8 | 8 | 32 |
Matrix representation of C5×C4○D16 ►in GL2(𝔽241) generated by
| 91 | 0 |
| 0 | 91 |
| 177 | 0 |
| 0 | 177 |
| 183 | 187 |
| 27 | 129 |
| 183 | 187 |
| 156 | 58 |
G:=sub<GL(2,GF(241))| [91,0,0,91],[177,0,0,177],[183,27,187,129],[183,156,187,58] >;
C5×C4○D16 in GAP, Magma, Sage, TeX
C_5\times C_4\circ D_{16} % in TeX
G:=Group("C5xC4oD16"); // GroupNames label
G:=SmallGroup(320,1009);
// by ID
G=gap.SmallGroup(320,1009);
# by ID
G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-2,589,856,4204,2111,242,10085,5052,124]);
// Polycyclic
G:=Group<a,b,c,d|a^5=b^4=d^2=1,c^8=b^2,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=b^2*c^7>;
// generators/relations