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), C4○D8.4C10, (C5×SD32)⋊6C2, D8.3(C2×C10), (C2×C10).28D8, C4.12(D4×C10), C10.89(C2×D8), C2.17(C10×D8), C22.6(C5×D8), (C10×Q16)⋊24C2, (C2×Q16)⋊10C10, C20.319(C2×D4), (C2×C20).347D4, (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
Generators and relations for C5×Q32⋊C2
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 >
Subgroups: 162 in 82 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, Q16, Q16, C2×Q8, C4○D4, C20, C20, C2×C10, C2×C10, M5(2), SD32, Q32, C2×Q16, C4○D8, C40, C2×C20, C2×C20, C5×D4, C5×Q8, Q32⋊C2, C80, C2×C40, C5×D8, C5×SD16, C5×Q16, C5×Q16, C5×Q16, Q8×C10, C5×C4○D4, C5×M5(2), C5×SD32, C5×Q32, C10×Q16, C5×C4○D8, C5×Q32⋊C2
Quotients: C1, C2, C22, C5, D4, C23, C10, D8, C2×D4, C2×C10, C2×D8, C5×D4, C22×C10, Q32⋊C2, C5×D8, D4×C10, C10×D8, C5×Q32⋊C2
►On 160 points
Generators in S160
(1 119 91 145 17)(2 120 92 146 18)(3 121 93 147 19)(4 122 94 148 20)(5 123 95 149 21)(6 124 96 150 22)(7 125 81 151 23)(8 126 82 152 24)(9 127 83 153 25)(10 128 84 154 26)(11 113 85 155 27)(12 114 86 156 28)(13 115 87 157 29)(14 116 88 158 30)(15 117 89 159 31)(16 118 90 160 32)(33 80 131 49 107)(34 65 132 50 108)(35 66 133 51 109)(36 67 134 52 110)(37 68 135 53 111)(38 69 136 54 112)(39 70 137 55 97)(40 71 138 56 98)(41 72 139 57 99)(42 73 140 58 100)(43 74 141 59 101)(44 75 142 60 102)(45 76 143 61 103)(46 77 144 62 104)(47 78 129 63 105)(48 79 130 64 106)
(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 43 9 35)(2 42 10 34)(3 41 11 33)(4 40 12 48)(5 39 13 47)(6 38 14 46)(7 37 15 45)(8 36 16 44)(17 101 25 109)(18 100 26 108)(19 99 27 107)(20 98 28 106)(21 97 29 105)(22 112 30 104)(23 111 31 103)(24 110 32 102)(49 147 57 155)(50 146 58 154)(51 145 59 153)(52 160 60 152)(53 159 61 151)(54 158 62 150)(55 157 63 149)(56 156 64 148)(65 120 73 128)(66 119 74 127)(67 118 75 126)(68 117 76 125)(69 116 77 124)(70 115 78 123)(71 114 79 122)(72 113 80 121)(81 135 89 143)(82 134 90 142)(83 133 91 141)(84 132 92 140)(85 131 93 139)(86 130 94 138)(87 129 95 137)(88 144 96 136)
(1 34)(2 43)(3 36)(4 45)(5 38)(6 47)(7 40)(8 33)(9 42)(10 35)(11 44)(12 37)(13 46)(14 39)(15 48)(16 41)(17 108)(18 101)(19 110)(20 103)(21 112)(22 105)(23 98)(24 107)(25 100)(26 109)(27 102)(28 111)(29 104)(30 97)(31 106)(32 99)(49 152)(50 145)(51 154)(52 147)(53 156)(54 149)(55 158)(56 151)(57 160)(58 153)(59 146)(60 155)(61 148)(62 157)(63 150)(64 159)(65 119)(66 128)(67 121)(68 114)(69 123)(70 116)(71 125)(72 118)(73 127)(74 120)(75 113)(76 122)(77 115)(78 124)(79 117)(80 126)(81 138)(82 131)(83 140)(84 133)(85 142)(86 135)(87 144)(88 137)(89 130)(90 139)(91 132)(92 141)(93 134)(94 143)(95 136)(96 129)
G:=sub<Sym(160)| (1,119,91,145,17)(2,120,92,146,18)(3,121,93,147,19)(4,122,94,148,20)(5,123,95,149,21)(6,124,96,150,22)(7,125,81,151,23)(8,126,82,152,24)(9,127,83,153,25)(10,128,84,154,26)(11,113,85,155,27)(12,114,86,156,28)(13,115,87,157,29)(14,116,88,158,30)(15,117,89,159,31)(16,118,90,160,32)(33,80,131,49,107)(34,65,132,50,108)(35,66,133,51,109)(36,67,134,52,110)(37,68,135,53,111)(38,69,136,54,112)(39,70,137,55,97)(40,71,138,56,98)(41,72,139,57,99)(42,73,140,58,100)(43,74,141,59,101)(44,75,142,60,102)(45,76,143,61,103)(46,77,144,62,104)(47,78,129,63,105)(48,79,130,64,106), (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), (1,43,9,35)(2,42,10,34)(3,41,11,33)(4,40,12,48)(5,39,13,47)(6,38,14,46)(7,37,15,45)(8,36,16,44)(17,101,25,109)(18,100,26,108)(19,99,27,107)(20,98,28,106)(21,97,29,105)(22,112,30,104)(23,111,31,103)(24,110,32,102)(49,147,57,155)(50,146,58,154)(51,145,59,153)(52,160,60,152)(53,159,61,151)(54,158,62,150)(55,157,63,149)(56,156,64,148)(65,120,73,128)(66,119,74,127)(67,118,75,126)(68,117,76,125)(69,116,77,124)(70,115,78,123)(71,114,79,122)(72,113,80,121)(81,135,89,143)(82,134,90,142)(83,133,91,141)(84,132,92,140)(85,131,93,139)(86,130,94,138)(87,129,95,137)(88,144,96,136), (1,34)(2,43)(3,36)(4,45)(5,38)(6,47)(7,40)(8,33)(9,42)(10,35)(11,44)(12,37)(13,46)(14,39)(15,48)(16,41)(17,108)(18,101)(19,110)(20,103)(21,112)(22,105)(23,98)(24,107)(25,100)(26,109)(27,102)(28,111)(29,104)(30,97)(31,106)(32,99)(49,152)(50,145)(51,154)(52,147)(53,156)(54,149)(55,158)(56,151)(57,160)(58,153)(59,146)(60,155)(61,148)(62,157)(63,150)(64,159)(65,119)(66,128)(67,121)(68,114)(69,123)(70,116)(71,125)(72,118)(73,127)(74,120)(75,113)(76,122)(77,115)(78,124)(79,117)(80,126)(81,138)(82,131)(83,140)(84,133)(85,142)(86,135)(87,144)(88,137)(89,130)(90,139)(91,132)(92,141)(93,134)(94,143)(95,136)(96,129)>;
G:=Group( (1,119,91,145,17)(2,120,92,146,18)(3,121,93,147,19)(4,122,94,148,20)(5,123,95,149,21)(6,124,96,150,22)(7,125,81,151,23)(8,126,82,152,24)(9,127,83,153,25)(10,128,84,154,26)(11,113,85,155,27)(12,114,86,156,28)(13,115,87,157,29)(14,116,88,158,30)(15,117,89,159,31)(16,118,90,160,32)(33,80,131,49,107)(34,65,132,50,108)(35,66,133,51,109)(36,67,134,52,110)(37,68,135,53,111)(38,69,136,54,112)(39,70,137,55,97)(40,71,138,56,98)(41,72,139,57,99)(42,73,140,58,100)(43,74,141,59,101)(44,75,142,60,102)(45,76,143,61,103)(46,77,144,62,104)(47,78,129,63,105)(48,79,130,64,106), (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), (1,43,9,35)(2,42,10,34)(3,41,11,33)(4,40,12,48)(5,39,13,47)(6,38,14,46)(7,37,15,45)(8,36,16,44)(17,101,25,109)(18,100,26,108)(19,99,27,107)(20,98,28,106)(21,97,29,105)(22,112,30,104)(23,111,31,103)(24,110,32,102)(49,147,57,155)(50,146,58,154)(51,145,59,153)(52,160,60,152)(53,159,61,151)(54,158,62,150)(55,157,63,149)(56,156,64,148)(65,120,73,128)(66,119,74,127)(67,118,75,126)(68,117,76,125)(69,116,77,124)(70,115,78,123)(71,114,79,122)(72,113,80,121)(81,135,89,143)(82,134,90,142)(83,133,91,141)(84,132,92,140)(85,131,93,139)(86,130,94,138)(87,129,95,137)(88,144,96,136), (1,34)(2,43)(3,36)(4,45)(5,38)(6,47)(7,40)(8,33)(9,42)(10,35)(11,44)(12,37)(13,46)(14,39)(15,48)(16,41)(17,108)(18,101)(19,110)(20,103)(21,112)(22,105)(23,98)(24,107)(25,100)(26,109)(27,102)(28,111)(29,104)(30,97)(31,106)(32,99)(49,152)(50,145)(51,154)(52,147)(53,156)(54,149)(55,158)(56,151)(57,160)(58,153)(59,146)(60,155)(61,148)(62,157)(63,150)(64,159)(65,119)(66,128)(67,121)(68,114)(69,123)(70,116)(71,125)(72,118)(73,127)(74,120)(75,113)(76,122)(77,115)(78,124)(79,117)(80,126)(81,138)(82,131)(83,140)(84,133)(85,142)(86,135)(87,144)(88,137)(89,130)(90,139)(91,132)(92,141)(93,134)(94,143)(95,136)(96,129) );
G=PermutationGroup([[(1,119,91,145,17),(2,120,92,146,18),(3,121,93,147,19),(4,122,94,148,20),(5,123,95,149,21),(6,124,96,150,22),(7,125,81,151,23),(8,126,82,152,24),(9,127,83,153,25),(10,128,84,154,26),(11,113,85,155,27),(12,114,86,156,28),(13,115,87,157,29),(14,116,88,158,30),(15,117,89,159,31),(16,118,90,160,32),(33,80,131,49,107),(34,65,132,50,108),(35,66,133,51,109),(36,67,134,52,110),(37,68,135,53,111),(38,69,136,54,112),(39,70,137,55,97),(40,71,138,56,98),(41,72,139,57,99),(42,73,140,58,100),(43,74,141,59,101),(44,75,142,60,102),(45,76,143,61,103),(46,77,144,62,104),(47,78,129,63,105),(48,79,130,64,106)], [(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)], [(1,43,9,35),(2,42,10,34),(3,41,11,33),(4,40,12,48),(5,39,13,47),(6,38,14,46),(7,37,15,45),(8,36,16,44),(17,101,25,109),(18,100,26,108),(19,99,27,107),(20,98,28,106),(21,97,29,105),(22,112,30,104),(23,111,31,103),(24,110,32,102),(49,147,57,155),(50,146,58,154),(51,145,59,153),(52,160,60,152),(53,159,61,151),(54,158,62,150),(55,157,63,149),(56,156,64,148),(65,120,73,128),(66,119,74,127),(67,118,75,126),(68,117,76,125),(69,116,77,124),(70,115,78,123),(71,114,79,122),(72,113,80,121),(81,135,89,143),(82,134,90,142),(83,133,91,141),(84,132,92,140),(85,131,93,139),(86,130,94,138),(87,129,95,137),(88,144,96,136)], [(1,34),(2,43),(3,36),(4,45),(5,38),(6,47),(7,40),(8,33),(9,42),(10,35),(11,44),(12,37),(13,46),(14,39),(15,48),(16,41),(17,108),(18,101),(19,110),(20,103),(21,112),(22,105),(23,98),(24,107),(25,100),(26,109),(27,102),(28,111),(29,104),(30,97),(31,106),(32,99),(49,152),(50,145),(51,154),(52,147),(53,156),(54,149),(55,158),(56,151),(57,160),(58,153),(59,146),(60,155),(61,148),(62,157),(63,150),(64,159),(65,119),(66,128),(67,121),(68,114),(69,123),(70,116),(71,125),(72,118),(73,127),(74,120),(75,113),(76,122),(77,115),(78,124),(79,117),(80,126),(81,138),(82,131),(83,140),(84,133),(85,142),(86,135),(87,144),(88,137),(89,130),(90,139),(91,132),(92,141),(93,134),(94,143),(95,136),(96,129)]])
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 |
Matrix representation of C5×Q32⋊C2 ►in 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] >;
C5×Q32⋊C2 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