direct product, metacyclic, nilpotent (class 5), monomial, 2-elementary
Aliases: C5×SD64, C32⋊2C10, C160⋊4C2, D16.C10, Q32⋊1C10, C20.40D8, C40.69D4, C10.16D16, C80.20C22, C8.6(C5×D4), C4.2(C5×D8), (C5×Q32)⋊5C2, C2.4(C5×D16), C16.3(C2×C10), (C5×D16).2C2, SmallGroup(320,177)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5×SD64
G = < a,b,c | a5=b32=c2=1, ab=ba, ac=ca, cbc=b15 >
Smallest permutation representation of C5×SD64
►On 160 points
Generators in S160
(1 141 103 70 61)(2 142 104 71 62)(3 143 105 72 63)(4 144 106 73 64)(5 145 107 74 33)(6 146 108 75 34)(7 147 109 76 35)(8 148 110 77 36)(9 149 111 78 37)(10 150 112 79 38)(11 151 113 80 39)(12 152 114 81 40)(13 153 115 82 41)(14 154 116 83 42)(15 155 117 84 43)(16 156 118 85 44)(17 157 119 86 45)(18 158 120 87 46)(19 159 121 88 47)(20 160 122 89 48)(21 129 123 90 49)(22 130 124 91 50)(23 131 125 92 51)(24 132 126 93 52)(25 133 127 94 53)(26 134 128 95 54)(27 135 97 96 55)(28 136 98 65 56)(29 137 99 66 57)(30 138 100 67 58)(31 139 101 68 59)(32 140 102 69 60)
(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)
(2 16)(3 31)(4 14)(5 29)(6 12)(7 27)(8 10)(9 25)(11 23)(13 21)(15 19)(18 32)(20 30)(22 28)(24 26)(33 57)(34 40)(35 55)(36 38)(37 53)(39 51)(41 49)(42 64)(43 47)(44 62)(46 60)(48 58)(50 56)(52 54)(59 63)(65 91)(66 74)(67 89)(68 72)(69 87)(71 85)(73 83)(75 81)(76 96)(77 79)(78 94)(80 92)(82 90)(84 88)(93 95)(97 109)(98 124)(99 107)(100 122)(101 105)(102 120)(104 118)(106 116)(108 114)(110 112)(111 127)(113 125)(115 123)(117 121)(126 128)(129 153)(130 136)(131 151)(132 134)(133 149)(135 147)(137 145)(138 160)(139 143)(140 158)(142 156)(144 154)(146 152)(148 150)(155 159)
G:=sub<Sym(160)| (1,141,103,70,61)(2,142,104,71,62)(3,143,105,72,63)(4,144,106,73,64)(5,145,107,74,33)(6,146,108,75,34)(7,147,109,76,35)(8,148,110,77,36)(9,149,111,78,37)(10,150,112,79,38)(11,151,113,80,39)(12,152,114,81,40)(13,153,115,82,41)(14,154,116,83,42)(15,155,117,84,43)(16,156,118,85,44)(17,157,119,86,45)(18,158,120,87,46)(19,159,121,88,47)(20,160,122,89,48)(21,129,123,90,49)(22,130,124,91,50)(23,131,125,92,51)(24,132,126,93,52)(25,133,127,94,53)(26,134,128,95,54)(27,135,97,96,55)(28,136,98,65,56)(29,137,99,66,57)(30,138,100,67,58)(31,139,101,68,59)(32,140,102,69,60), (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), (2,16)(3,31)(4,14)(5,29)(6,12)(7,27)(8,10)(9,25)(11,23)(13,21)(15,19)(18,32)(20,30)(22,28)(24,26)(33,57)(34,40)(35,55)(36,38)(37,53)(39,51)(41,49)(42,64)(43,47)(44,62)(46,60)(48,58)(50,56)(52,54)(59,63)(65,91)(66,74)(67,89)(68,72)(69,87)(71,85)(73,83)(75,81)(76,96)(77,79)(78,94)(80,92)(82,90)(84,88)(93,95)(97,109)(98,124)(99,107)(100,122)(101,105)(102,120)(104,118)(106,116)(108,114)(110,112)(111,127)(113,125)(115,123)(117,121)(126,128)(129,153)(130,136)(131,151)(132,134)(133,149)(135,147)(137,145)(138,160)(139,143)(140,158)(142,156)(144,154)(146,152)(148,150)(155,159)>;
G:=Group( (1,141,103,70,61)(2,142,104,71,62)(3,143,105,72,63)(4,144,106,73,64)(5,145,107,74,33)(6,146,108,75,34)(7,147,109,76,35)(8,148,110,77,36)(9,149,111,78,37)(10,150,112,79,38)(11,151,113,80,39)(12,152,114,81,40)(13,153,115,82,41)(14,154,116,83,42)(15,155,117,84,43)(16,156,118,85,44)(17,157,119,86,45)(18,158,120,87,46)(19,159,121,88,47)(20,160,122,89,48)(21,129,123,90,49)(22,130,124,91,50)(23,131,125,92,51)(24,132,126,93,52)(25,133,127,94,53)(26,134,128,95,54)(27,135,97,96,55)(28,136,98,65,56)(29,137,99,66,57)(30,138,100,67,58)(31,139,101,68,59)(32,140,102,69,60), (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), (2,16)(3,31)(4,14)(5,29)(6,12)(7,27)(8,10)(9,25)(11,23)(13,21)(15,19)(18,32)(20,30)(22,28)(24,26)(33,57)(34,40)(35,55)(36,38)(37,53)(39,51)(41,49)(42,64)(43,47)(44,62)(46,60)(48,58)(50,56)(52,54)(59,63)(65,91)(66,74)(67,89)(68,72)(69,87)(71,85)(73,83)(75,81)(76,96)(77,79)(78,94)(80,92)(82,90)(84,88)(93,95)(97,109)(98,124)(99,107)(100,122)(101,105)(102,120)(104,118)(106,116)(108,114)(110,112)(111,127)(113,125)(115,123)(117,121)(126,128)(129,153)(130,136)(131,151)(132,134)(133,149)(135,147)(137,145)(138,160)(139,143)(140,158)(142,156)(144,154)(146,152)(148,150)(155,159) );
G=PermutationGroup([[(1,141,103,70,61),(2,142,104,71,62),(3,143,105,72,63),(4,144,106,73,64),(5,145,107,74,33),(6,146,108,75,34),(7,147,109,76,35),(8,148,110,77,36),(9,149,111,78,37),(10,150,112,79,38),(11,151,113,80,39),(12,152,114,81,40),(13,153,115,82,41),(14,154,116,83,42),(15,155,117,84,43),(16,156,118,85,44),(17,157,119,86,45),(18,158,120,87,46),(19,159,121,88,47),(20,160,122,89,48),(21,129,123,90,49),(22,130,124,91,50),(23,131,125,92,51),(24,132,126,93,52),(25,133,127,94,53),(26,134,128,95,54),(27,135,97,96,55),(28,136,98,65,56),(29,137,99,66,57),(30,138,100,67,58),(31,139,101,68,59),(32,140,102,69,60)], [(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)], [(2,16),(3,31),(4,14),(5,29),(6,12),(7,27),(8,10),(9,25),(11,23),(13,21),(15,19),(18,32),(20,30),(22,28),(24,26),(33,57),(34,40),(35,55),(36,38),(37,53),(39,51),(41,49),(42,64),(43,47),(44,62),(46,60),(48,58),(50,56),(52,54),(59,63),(65,91),(66,74),(67,89),(68,72),(69,87),(71,85),(73,83),(75,81),(76,96),(77,79),(78,94),(80,92),(82,90),(84,88),(93,95),(97,109),(98,124),(99,107),(100,122),(101,105),(102,120),(104,118),(106,116),(108,114),(110,112),(111,127),(113,125),(115,123),(117,121),(126,128),(129,153),(130,136),(131,151),(132,134),(133,149),(135,147),(137,145),(138,160),(139,143),(140,158),(142,156),(144,154),(146,152),(148,150),(155,159)]])
95 conjugacy classes
| class | 1 | 2A | 2B | 4A | 4B | 5A | 5B | 5C | 5D | 8A | 8B | 10A | 10B | 10C | 10D | 10E | 10F | 10G | 10H | 16A | 16B | 16C | 16D | 20A | 20B | 20C | 20D | 20E | 20F | 20G | 20H | 32A | ··· | 32H | 40A | ··· | 40H | 80A | ··· | 80P | 160A | ··· | 160AF |
| order | 1 | 2 | 2 | 4 | 4 | 5 | 5 | 5 | 5 | 8 | 8 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 16 | 16 | 16 | 16 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 32 | ··· | 32 | 40 | ··· | 40 | 80 | ··· | 80 | 160 | ··· | 160 |
| size | 1 | 1 | 16 | 2 | 16 | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 16 | 16 | 16 | 16 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 16 | 16 | 16 | 16 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
95 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | |||||||||
| image | C1 | C2 | C2 | C2 | C5 | C10 | C10 | C10 | D4 | D8 | D16 | C5×D4 | SD64 | C5×D8 | C5×D16 | C5×SD64 |
| kernel | C5×SD64 | C160 | C5×D16 | C5×Q32 | SD64 | C32 | D16 | Q32 | C40 | C20 | C10 | C8 | C5 | C4 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 4 | 1 | 2 | 4 | 4 | 8 | 8 | 16 | 32 |
Matrix representation of C5×SD64 ►in GL2(𝔽641) generated by
| 531 | 0 |
| 0 | 531 |
| 474 | 82 |
| 559 | 474 |
| 287 | 354 |
| 354 | 354 |
G:=sub<GL(2,GF(641))| [531,0,0,531],[474,559,82,474],[287,354,354,354] >;
C5×SD64 in GAP, Magma, Sage, TeX
C_5\times {\rm SD}_{64} % in TeX
G:=Group("C5xSD64"); // GroupNames label
G:=SmallGroup(320,177);
// by ID
G=gap.SmallGroup(320,177);
# by ID
G:=PCGroup([7,-2,-2,-5,-2,-2,-2,-2,1120,309,1683,850,192,4204,2111,242,10085,5052,124]);
// Polycyclic
G:=Group<a,b,c|a^5=b^32=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^15>;
// generators/relations
Export