direct product, metabelian, nilpotent (class 4), monomial, 2-elementary
Aliases: C5×Q32⋊C2, Q32⋊2C10, C20.66D8, C40.54D4, SD32⋊2C10, M5(2)⋊2C10, C40.77C23, C80.12C22, C16.(C2×C10), C8.4(C5×D4), (C5×Q32)⋊6C2, C4.15(C5×D8), (C5×SD32)⋊6C2, C4○D8.4C10, D8.3(C2×C10), C2.17(C10×D8), C10.89(C2×D8), (C2×C10).28D8, C4.12(D4×C10), C22.6(C5×D8), (C10×Q16)⋊24C2, (C2×Q16)⋊10C10, (C2×C20).347D4, C20.319(C2×D4), (C5×M5(2))⋊4C2, C8.8(C22×C10), Q16.3(C2×C10), (C5×D8).13C22, (C2×C40).279C22, (C5×Q16).15C22, (C5×C4○D8).9C2, (C2×C4).48(C5×D4), (C2×C8).31(C2×C10), SmallGroup(320,1011)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 162 in 82 conjugacy classes, 46 normal (30 characteristic)
C1, C2, C2 [×2], C4 [×2], C4 [×3], C22, C22, C5, C8 [×2], C2×C4, C2×C4 [×2], D4 [×2], Q8 [×4], C10, C10 [×2], C16 [×2], C2×C8, D8, SD16, Q16, Q16 [×2], Q16, C2×Q8, C4○D4, C20 [×2], C20 [×3], C2×C10, C2×C10, M5(2), SD32 [×2], Q32 [×2], C2×Q16, C4○D8, C40 [×2], C2×C20, C2×C20 [×2], C5×D4 [×2], C5×Q8 [×4], Q32⋊C2, C80 [×2], C2×C40, C5×D8, C5×SD16, C5×Q16, C5×Q16 [×2], C5×Q16, Q8×C10, C5×C4○D4, C5×M5(2), C5×SD32 [×2], C5×Q32 [×2], C10×Q16, C5×C4○D8, C5×Q32⋊C2
Quotients:
C1, C2 [×7], C22 [×7], C5, D4 [×2], C23, C10 [×7], D8 [×2], C2×D4, C2×C10 [×7], C2×D8, C5×D4 [×2], C22×C10, Q32⋊C2, C5×D8 [×2], D4×C10, C10×D8, C5×Q32⋊C2
Generators and relations
G = < a,b,c,d | a5=b16=d2=1, c2=b8, ab=ba, ac=ca, ad=da, cbc-1=b-1, dbd=b9, cd=dc >
►On 160 points
Generators in S160
(1 65 118 38 145)(2 66 119 39 146)(3 67 120 40 147)(4 68 121 41 148)(5 69 122 42 149)(6 70 123 43 150)(7 71 124 44 151)(8 72 125 45 152)(9 73 126 46 153)(10 74 127 47 154)(11 75 128 48 155)(12 76 113 33 156)(13 77 114 34 157)(14 78 115 35 158)(15 79 116 36 159)(16 80 117 37 160)(17 81 49 100 129)(18 82 50 101 130)(19 83 51 102 131)(20 84 52 103 132)(21 85 53 104 133)(22 86 54 105 134)(23 87 55 106 135)(24 88 56 107 136)(25 89 57 108 137)(26 90 58 109 138)(27 91 59 110 139)(28 92 60 111 140)(29 93 61 112 141)(30 94 62 97 142)(31 95 63 98 143)(32 96 64 99 144)
(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)
(1 111 9 103)(2 110 10 102)(3 109 11 101)(4 108 12 100)(5 107 13 99)(6 106 14 98)(7 105 15 97)(8 104 16 112)(17 121 25 113)(18 120 26 128)(19 119 27 127)(20 118 28 126)(21 117 29 125)(22 116 30 124)(23 115 31 123)(24 114 32 122)(33 81 41 89)(34 96 42 88)(35 95 43 87)(36 94 44 86)(37 93 45 85)(38 92 46 84)(39 91 47 83)(40 90 48 82)(49 148 57 156)(50 147 58 155)(51 146 59 154)(52 145 60 153)(53 160 61 152)(54 159 62 151)(55 158 63 150)(56 157 64 149)(65 140 73 132)(66 139 74 131)(67 138 75 130)(68 137 76 129)(69 136 77 144)(70 135 78 143)(71 134 79 142)(72 133 80 141)
(1 102)(2 111)(3 104)(4 97)(5 106)(6 99)(7 108)(8 101)(9 110)(10 103)(11 112)(12 105)(13 98)(14 107)(15 100)(16 109)(17 116)(18 125)(19 118)(20 127)(21 120)(22 113)(23 122)(24 115)(25 124)(26 117)(27 126)(28 119)(29 128)(30 121)(31 114)(32 123)(33 86)(34 95)(35 88)(36 81)(37 90)(38 83)(39 92)(40 85)(41 94)(42 87)(43 96)(44 89)(45 82)(46 91)(47 84)(48 93)(49 159)(50 152)(51 145)(52 154)(53 147)(54 156)(55 149)(56 158)(57 151)(58 160)(59 153)(60 146)(61 155)(62 148)(63 157)(64 150)(65 131)(66 140)(67 133)(68 142)(69 135)(70 144)(71 137)(72 130)(73 139)(74 132)(75 141)(76 134)(77 143)(78 136)(79 129)(80 138)
G:=sub<Sym(160)| (1,65,118,38,145)(2,66,119,39,146)(3,67,120,40,147)(4,68,121,41,148)(5,69,122,42,149)(6,70,123,43,150)(7,71,124,44,151)(8,72,125,45,152)(9,73,126,46,153)(10,74,127,47,154)(11,75,128,48,155)(12,76,113,33,156)(13,77,114,34,157)(14,78,115,35,158)(15,79,116,36,159)(16,80,117,37,160)(17,81,49,100,129)(18,82,50,101,130)(19,83,51,102,131)(20,84,52,103,132)(21,85,53,104,133)(22,86,54,105,134)(23,87,55,106,135)(24,88,56,107,136)(25,89,57,108,137)(26,90,58,109,138)(27,91,59,110,139)(28,92,60,111,140)(29,93,61,112,141)(30,94,62,97,142)(31,95,63,98,143)(32,96,64,99,144), (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), (1,111,9,103)(2,110,10,102)(3,109,11,101)(4,108,12,100)(5,107,13,99)(6,106,14,98)(7,105,15,97)(8,104,16,112)(17,121,25,113)(18,120,26,128)(19,119,27,127)(20,118,28,126)(21,117,29,125)(22,116,30,124)(23,115,31,123)(24,114,32,122)(33,81,41,89)(34,96,42,88)(35,95,43,87)(36,94,44,86)(37,93,45,85)(38,92,46,84)(39,91,47,83)(40,90,48,82)(49,148,57,156)(50,147,58,155)(51,146,59,154)(52,145,60,153)(53,160,61,152)(54,159,62,151)(55,158,63,150)(56,157,64,149)(65,140,73,132)(66,139,74,131)(67,138,75,130)(68,137,76,129)(69,136,77,144)(70,135,78,143)(71,134,79,142)(72,133,80,141), (1,102)(2,111)(3,104)(4,97)(5,106)(6,99)(7,108)(8,101)(9,110)(10,103)(11,112)(12,105)(13,98)(14,107)(15,100)(16,109)(17,116)(18,125)(19,118)(20,127)(21,120)(22,113)(23,122)(24,115)(25,124)(26,117)(27,126)(28,119)(29,128)(30,121)(31,114)(32,123)(33,86)(34,95)(35,88)(36,81)(37,90)(38,83)(39,92)(40,85)(41,94)(42,87)(43,96)(44,89)(45,82)(46,91)(47,84)(48,93)(49,159)(50,152)(51,145)(52,154)(53,147)(54,156)(55,149)(56,158)(57,151)(58,160)(59,153)(60,146)(61,155)(62,148)(63,157)(64,150)(65,131)(66,140)(67,133)(68,142)(69,135)(70,144)(71,137)(72,130)(73,139)(74,132)(75,141)(76,134)(77,143)(78,136)(79,129)(80,138)>;
G:=Group( (1,65,118,38,145)(2,66,119,39,146)(3,67,120,40,147)(4,68,121,41,148)(5,69,122,42,149)(6,70,123,43,150)(7,71,124,44,151)(8,72,125,45,152)(9,73,126,46,153)(10,74,127,47,154)(11,75,128,48,155)(12,76,113,33,156)(13,77,114,34,157)(14,78,115,35,158)(15,79,116,36,159)(16,80,117,37,160)(17,81,49,100,129)(18,82,50,101,130)(19,83,51,102,131)(20,84,52,103,132)(21,85,53,104,133)(22,86,54,105,134)(23,87,55,106,135)(24,88,56,107,136)(25,89,57,108,137)(26,90,58,109,138)(27,91,59,110,139)(28,92,60,111,140)(29,93,61,112,141)(30,94,62,97,142)(31,95,63,98,143)(32,96,64,99,144), (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), (1,111,9,103)(2,110,10,102)(3,109,11,101)(4,108,12,100)(5,107,13,99)(6,106,14,98)(7,105,15,97)(8,104,16,112)(17,121,25,113)(18,120,26,128)(19,119,27,127)(20,118,28,126)(21,117,29,125)(22,116,30,124)(23,115,31,123)(24,114,32,122)(33,81,41,89)(34,96,42,88)(35,95,43,87)(36,94,44,86)(37,93,45,85)(38,92,46,84)(39,91,47,83)(40,90,48,82)(49,148,57,156)(50,147,58,155)(51,146,59,154)(52,145,60,153)(53,160,61,152)(54,159,62,151)(55,158,63,150)(56,157,64,149)(65,140,73,132)(66,139,74,131)(67,138,75,130)(68,137,76,129)(69,136,77,144)(70,135,78,143)(71,134,79,142)(72,133,80,141), (1,102)(2,111)(3,104)(4,97)(5,106)(6,99)(7,108)(8,101)(9,110)(10,103)(11,112)(12,105)(13,98)(14,107)(15,100)(16,109)(17,116)(18,125)(19,118)(20,127)(21,120)(22,113)(23,122)(24,115)(25,124)(26,117)(27,126)(28,119)(29,128)(30,121)(31,114)(32,123)(33,86)(34,95)(35,88)(36,81)(37,90)(38,83)(39,92)(40,85)(41,94)(42,87)(43,96)(44,89)(45,82)(46,91)(47,84)(48,93)(49,159)(50,152)(51,145)(52,154)(53,147)(54,156)(55,149)(56,158)(57,151)(58,160)(59,153)(60,146)(61,155)(62,148)(63,157)(64,150)(65,131)(66,140)(67,133)(68,142)(69,135)(70,144)(71,137)(72,130)(73,139)(74,132)(75,141)(76,134)(77,143)(78,136)(79,129)(80,138) );
G=PermutationGroup([(1,65,118,38,145),(2,66,119,39,146),(3,67,120,40,147),(4,68,121,41,148),(5,69,122,42,149),(6,70,123,43,150),(7,71,124,44,151),(8,72,125,45,152),(9,73,126,46,153),(10,74,127,47,154),(11,75,128,48,155),(12,76,113,33,156),(13,77,114,34,157),(14,78,115,35,158),(15,79,116,36,159),(16,80,117,37,160),(17,81,49,100,129),(18,82,50,101,130),(19,83,51,102,131),(20,84,52,103,132),(21,85,53,104,133),(22,86,54,105,134),(23,87,55,106,135),(24,88,56,107,136),(25,89,57,108,137),(26,90,58,109,138),(27,91,59,110,139),(28,92,60,111,140),(29,93,61,112,141),(30,94,62,97,142),(31,95,63,98,143),(32,96,64,99,144)], [(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)], [(1,111,9,103),(2,110,10,102),(3,109,11,101),(4,108,12,100),(5,107,13,99),(6,106,14,98),(7,105,15,97),(8,104,16,112),(17,121,25,113),(18,120,26,128),(19,119,27,127),(20,118,28,126),(21,117,29,125),(22,116,30,124),(23,115,31,123),(24,114,32,122),(33,81,41,89),(34,96,42,88),(35,95,43,87),(36,94,44,86),(37,93,45,85),(38,92,46,84),(39,91,47,83),(40,90,48,82),(49,148,57,156),(50,147,58,155),(51,146,59,154),(52,145,60,153),(53,160,61,152),(54,159,62,151),(55,158,63,150),(56,157,64,149),(65,140,73,132),(66,139,74,131),(67,138,75,130),(68,137,76,129),(69,136,77,144),(70,135,78,143),(71,134,79,142),(72,133,80,141)], [(1,102),(2,111),(3,104),(4,97),(5,106),(6,99),(7,108),(8,101),(9,110),(10,103),(11,112),(12,105),(13,98),(14,107),(15,100),(16,109),(17,116),(18,125),(19,118),(20,127),(21,120),(22,113),(23,122),(24,115),(25,124),(26,117),(27,126),(28,119),(29,128),(30,121),(31,114),(32,123),(33,86),(34,95),(35,88),(36,81),(37,90),(38,83),(39,92),(40,85),(41,94),(42,87),(43,96),(44,89),(45,82),(46,91),(47,84),(48,93),(49,159),(50,152),(51,145),(52,154),(53,147),(54,156),(55,149),(56,158),(57,151),(58,160),(59,153),(60,146),(61,155),(62,148),(63,157),(64,150),(65,131),(66,140),(67,133),(68,142),(69,135),(70,144),(71,137),(72,130),(73,139),(74,132),(75,141),(76,134),(77,143),(78,136),(79,129),(80,138)])
Matrix representation ►G ⊆ GL6(𝔽241)
| 205 | 0 | 0 | 0 | 0 | 0 |
| 0 | 205 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 11 | 11 | 0 | 0 | 0 | 0 |
| 230 | 11 | 0 | 0 | 0 | 0 |
| 0 | 0 | 55 | 239 | 96 | 106 |
| 0 | 0 | 114 | 35 | 202 | 184 |
| 0 | 0 | 58 | 224 | 233 | 119 |
| 0 | 0 | 212 | 42 | 34 | 159 |
| 230 | 230 | 0 | 0 | 0 | 0 |
| 230 | 11 | 0 | 0 | 0 | 0 |
| 0 | 0 | 104 | 80 | 40 | 84 |
| 0 | 0 | 148 | 199 | 117 | 164 |
| 0 | 0 | 46 | 229 | 78 | 85 |
| 0 | 0 | 67 | 27 | 238 | 101 |
| 240 | 0 | 0 | 0 | 0 | 0 |
| 0 | 240 | 0 | 0 | 0 | 0 |
| 0 | 0 | 80 | 0 | 1 | 0 |
| 0 | 0 | 23 | 138 | 1 | 239 |
| 0 | 0 | 108 | 0 | 161 | 0 |
| 0 | 0 | 151 | 2 | 161 | 103 |
G:=sub<GL(6,GF(241))| [205,0,0,0,0,0,0,205,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[11,230,0,0,0,0,11,11,0,0,0,0,0,0,55,114,58,212,0,0,239,35,224,42,0,0,96,202,233,34,0,0,106,184,119,159],[230,230,0,0,0,0,230,11,0,0,0,0,0,0,104,148,46,67,0,0,80,199,229,27,0,0,40,117,78,238,0,0,84,164,85,101],[240,0,0,0,0,0,0,240,0,0,0,0,0,0,80,23,108,151,0,0,0,138,0,2,0,0,1,1,161,161,0,0,0,239,0,103] >;
80 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 5A | 5B | 5C | 5D | 8A | 8B | 8C | 10A | 10B | 10C | 10D | 10E | 10F | 10G | 10H | 10I | 10J | 10K | 10L | 16A | 16B | 16C | 16D | 20A | ··· | 20H | 20I | ··· | 20T | 40A | ··· | 40H | 40I | 40J | 40K | 40L | 80A | ··· | 80P |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 8 | 8 | 8 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 16 | 16 | 16 | 16 | 20 | ··· | 20 | 20 | ··· | 20 | 40 | ··· | 40 | 40 | 40 | 40 | 40 | 80 | ··· | 80 |
| size | 1 | 1 | 2 | 8 | 2 | 2 | 8 | 8 | 8 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 8 | 8 | 8 | 8 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 8 | ··· | 8 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
80 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | - | |||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C5 | C10 | C10 | C10 | C10 | C10 | D4 | D4 | D8 | D8 | C5×D4 | C5×D4 | C5×D8 | C5×D8 | Q32⋊C2 | C5×Q32⋊C2 |
| kernel | C5×Q32⋊C2 | C5×M5(2) | C5×SD32 | C5×Q32 | C10×Q16 | C5×C4○D8 | Q32⋊C2 | M5(2) | SD32 | Q32 | C2×Q16 | C4○D8 | C40 | C2×C20 | C20 | C2×C10 | C8 | C2×C4 | C4 | C22 | C5 | C1 |
| # reps | 1 | 1 | 2 | 2 | 1 | 1 | 4 | 4 | 8 | 8 | 4 | 4 | 1 | 1 | 2 | 2 | 4 | 4 | 8 | 8 | 2 | 8 |
In GAP, Magma, Sage, TeX
C_5\times Q_{32}\rtimes C_2 % in TeX
G:=Group("C5xQ32:C2"); // GroupNames label
G:=SmallGroup(320,1011);
// by ID
G=gap.SmallGroup(320,1011);
# by ID
G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-2,589,1128,3446,4204,2111,242,10085,5052,124]);
// Polycyclic
G:=Group<a,b,c,d|a^5=b^16=d^2=1,c^2=b^8,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^-1,d*b*d=b^9,c*d=d*c>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀