direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C5×C22⋊Q16, (C2×C10)⋊7Q16, Q8.6(C5×D4), (C2×Q16)⋊1C10, C4.25(D4×C10), (C5×Q8).40D4, C2.4(C10×Q16), C22⋊2(C5×Q16), Q8⋊C4⋊5C10, (C10×Q16)⋊15C2, (C2×C20).320D4, C20.386(C2×D4), C22⋊C8.3C10, C10.51(C2×Q16), C23.44(C5×D4), C22⋊Q8.2C10, C10.98C22≀C2, C22.81(D4×C10), (C22×Q8).6C10, (C2×C40).254C22, (C2×C20).916C23, (C22×C10).166D4, (Q8×C10).260C22, C10.132(C8.C22), (C22×C20).423C22, C4⋊C4.3(C2×C10), (C2×C8).2(C2×C10), (C2×C4).29(C5×D4), (Q8×C2×C10).16C2, C2.7(C5×C8.C22), (C5×Q8⋊C4)⋊28C2, C2.12(C5×C22≀C2), (C2×C10).637(C2×D4), (C5×C22⋊C8).12C2, (C2×Q8).45(C2×C10), (C5×C22⋊Q8).12C2, (C5×C4⋊C4).225C22, (C22×C4).41(C2×C10), (C2×C4).91(C22×C10), SmallGroup(320,952)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5×C22⋊Q16
G = < a,b,c,d,e | a5=b2=c2=d8=1, e2=d4, ab=ba, ac=ca, ad=da, ae=ea, dbd-1=bc=cb, be=eb, cd=dc, ce=ec, ede-1=d-1 >
Subgroups: 242 in 148 conjugacy classes, 62 normal (30 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C5, C8, C2×C4, C2×C4, Q8, Q8, C23, C10, C10, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, Q16, C22×C4, C22×C4, C2×Q8, C2×Q8, C2×Q8, C20, C20, C2×C10, C2×C10, C2×C10, C22⋊C8, Q8⋊C4, C22⋊Q8, C2×Q16, C22×Q8, C40, C2×C20, C2×C20, C5×Q8, C5×Q8, C22×C10, C22⋊Q16, C5×C22⋊C4, C5×C4⋊C4, C5×C4⋊C4, C2×C40, C5×Q16, C22×C20, C22×C20, Q8×C10, Q8×C10, Q8×C10, C5×C22⋊C8, C5×Q8⋊C4, C5×C22⋊Q8, C10×Q16, Q8×C2×C10, C5×C22⋊Q16
Quotients: C1, C2, C22, C5, D4, C23, C10, Q16, C2×D4, C2×C10, C22≀C2, C2×Q16, C8.C22, C5×D4, C22×C10, C22⋊Q16, C5×Q16, D4×C10, C5×C22≀C2, C10×Q16, C5×C8.C22, C5×C22⋊Q16
►On 160 points
Generators in S160
(1 44 81 54 95)(2 45 82 55 96)(3 46 83 56 89)(4 47 84 49 90)(5 48 85 50 91)(6 41 86 51 92)(7 42 87 52 93)(8 43 88 53 94)(9 113 137 17 129)(10 114 138 18 130)(11 115 139 19 131)(12 116 140 20 132)(13 117 141 21 133)(14 118 142 22 134)(15 119 143 23 135)(16 120 144 24 136)(25 126 153 33 145)(26 127 154 34 146)(27 128 155 35 147)(28 121 156 36 148)(29 122 157 37 149)(30 123 158 38 150)(31 124 159 39 151)(32 125 160 40 152)(57 78 111 70 103)(58 79 112 71 104)(59 80 105 72 97)(60 73 106 65 98)(61 74 107 66 99)(62 75 108 67 100)(63 76 109 68 101)(64 77 110 69 102)
(2 28)(4 30)(6 32)(8 26)(10 67)(12 69)(14 71)(16 65)(18 75)(20 77)(22 79)(24 73)(34 53)(36 55)(38 49)(40 51)(41 125)(43 127)(45 121)(47 123)(58 142)(60 144)(62 138)(64 140)(82 156)(84 158)(86 160)(88 154)(90 150)(92 152)(94 146)(96 148)(98 120)(100 114)(102 116)(104 118)(106 136)(108 130)(110 132)(112 134)
(1 27)(2 28)(3 29)(4 30)(5 31)(6 32)(7 25)(8 26)(9 66)(10 67)(11 68)(12 69)(13 70)(14 71)(15 72)(16 65)(17 74)(18 75)(19 76)(20 77)(21 78)(22 79)(23 80)(24 73)(33 52)(34 53)(35 54)(36 55)(37 56)(38 49)(39 50)(40 51)(41 125)(42 126)(43 127)(44 128)(45 121)(46 122)(47 123)(48 124)(57 141)(58 142)(59 143)(60 144)(61 137)(62 138)(63 139)(64 140)(81 155)(82 156)(83 157)(84 158)(85 159)(86 160)(87 153)(88 154)(89 149)(90 150)(91 151)(92 152)(93 145)(94 146)(95 147)(96 148)(97 119)(98 120)(99 113)(100 114)(101 115)(102 116)(103 117)(104 118)(105 135)(106 136)(107 129)(108 130)(109 131)(110 132)(111 133)(112 134)
(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 97 5 101)(2 104 6 100)(3 103 7 99)(4 102 8 98)(9 149 13 145)(10 148 14 152)(11 147 15 151)(12 146 16 150)(17 157 21 153)(18 156 22 160)(19 155 23 159)(20 154 24 158)(25 113 29 117)(26 120 30 116)(27 119 31 115)(28 118 32 114)(33 129 37 133)(34 136 38 132)(35 135 39 131)(36 134 40 130)(41 62 45 58)(42 61 46 57)(43 60 47 64)(44 59 48 63)(49 110 53 106)(50 109 54 105)(51 108 55 112)(52 107 56 111)(65 90 69 94)(66 89 70 93)(67 96 71 92)(68 95 72 91)(73 84 77 88)(74 83 78 87)(75 82 79 86)(76 81 80 85)(121 142 125 138)(122 141 126 137)(123 140 127 144)(124 139 128 143)
G:=sub<Sym(160)| (1,44,81,54,95)(2,45,82,55,96)(3,46,83,56,89)(4,47,84,49,90)(5,48,85,50,91)(6,41,86,51,92)(7,42,87,52,93)(8,43,88,53,94)(9,113,137,17,129)(10,114,138,18,130)(11,115,139,19,131)(12,116,140,20,132)(13,117,141,21,133)(14,118,142,22,134)(15,119,143,23,135)(16,120,144,24,136)(25,126,153,33,145)(26,127,154,34,146)(27,128,155,35,147)(28,121,156,36,148)(29,122,157,37,149)(30,123,158,38,150)(31,124,159,39,151)(32,125,160,40,152)(57,78,111,70,103)(58,79,112,71,104)(59,80,105,72,97)(60,73,106,65,98)(61,74,107,66,99)(62,75,108,67,100)(63,76,109,68,101)(64,77,110,69,102), (2,28)(4,30)(6,32)(8,26)(10,67)(12,69)(14,71)(16,65)(18,75)(20,77)(22,79)(24,73)(34,53)(36,55)(38,49)(40,51)(41,125)(43,127)(45,121)(47,123)(58,142)(60,144)(62,138)(64,140)(82,156)(84,158)(86,160)(88,154)(90,150)(92,152)(94,146)(96,148)(98,120)(100,114)(102,116)(104,118)(106,136)(108,130)(110,132)(112,134), (1,27)(2,28)(3,29)(4,30)(5,31)(6,32)(7,25)(8,26)(9,66)(10,67)(11,68)(12,69)(13,70)(14,71)(15,72)(16,65)(17,74)(18,75)(19,76)(20,77)(21,78)(22,79)(23,80)(24,73)(33,52)(34,53)(35,54)(36,55)(37,56)(38,49)(39,50)(40,51)(41,125)(42,126)(43,127)(44,128)(45,121)(46,122)(47,123)(48,124)(57,141)(58,142)(59,143)(60,144)(61,137)(62,138)(63,139)(64,140)(81,155)(82,156)(83,157)(84,158)(85,159)(86,160)(87,153)(88,154)(89,149)(90,150)(91,151)(92,152)(93,145)(94,146)(95,147)(96,148)(97,119)(98,120)(99,113)(100,114)(101,115)(102,116)(103,117)(104,118)(105,135)(106,136)(107,129)(108,130)(109,131)(110,132)(111,133)(112,134), (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), (1,97,5,101)(2,104,6,100)(3,103,7,99)(4,102,8,98)(9,149,13,145)(10,148,14,152)(11,147,15,151)(12,146,16,150)(17,157,21,153)(18,156,22,160)(19,155,23,159)(20,154,24,158)(25,113,29,117)(26,120,30,116)(27,119,31,115)(28,118,32,114)(33,129,37,133)(34,136,38,132)(35,135,39,131)(36,134,40,130)(41,62,45,58)(42,61,46,57)(43,60,47,64)(44,59,48,63)(49,110,53,106)(50,109,54,105)(51,108,55,112)(52,107,56,111)(65,90,69,94)(66,89,70,93)(67,96,71,92)(68,95,72,91)(73,84,77,88)(74,83,78,87)(75,82,79,86)(76,81,80,85)(121,142,125,138)(122,141,126,137)(123,140,127,144)(124,139,128,143)>;
G:=Group( (1,44,81,54,95)(2,45,82,55,96)(3,46,83,56,89)(4,47,84,49,90)(5,48,85,50,91)(6,41,86,51,92)(7,42,87,52,93)(8,43,88,53,94)(9,113,137,17,129)(10,114,138,18,130)(11,115,139,19,131)(12,116,140,20,132)(13,117,141,21,133)(14,118,142,22,134)(15,119,143,23,135)(16,120,144,24,136)(25,126,153,33,145)(26,127,154,34,146)(27,128,155,35,147)(28,121,156,36,148)(29,122,157,37,149)(30,123,158,38,150)(31,124,159,39,151)(32,125,160,40,152)(57,78,111,70,103)(58,79,112,71,104)(59,80,105,72,97)(60,73,106,65,98)(61,74,107,66,99)(62,75,108,67,100)(63,76,109,68,101)(64,77,110,69,102), (2,28)(4,30)(6,32)(8,26)(10,67)(12,69)(14,71)(16,65)(18,75)(20,77)(22,79)(24,73)(34,53)(36,55)(38,49)(40,51)(41,125)(43,127)(45,121)(47,123)(58,142)(60,144)(62,138)(64,140)(82,156)(84,158)(86,160)(88,154)(90,150)(92,152)(94,146)(96,148)(98,120)(100,114)(102,116)(104,118)(106,136)(108,130)(110,132)(112,134), (1,27)(2,28)(3,29)(4,30)(5,31)(6,32)(7,25)(8,26)(9,66)(10,67)(11,68)(12,69)(13,70)(14,71)(15,72)(16,65)(17,74)(18,75)(19,76)(20,77)(21,78)(22,79)(23,80)(24,73)(33,52)(34,53)(35,54)(36,55)(37,56)(38,49)(39,50)(40,51)(41,125)(42,126)(43,127)(44,128)(45,121)(46,122)(47,123)(48,124)(57,141)(58,142)(59,143)(60,144)(61,137)(62,138)(63,139)(64,140)(81,155)(82,156)(83,157)(84,158)(85,159)(86,160)(87,153)(88,154)(89,149)(90,150)(91,151)(92,152)(93,145)(94,146)(95,147)(96,148)(97,119)(98,120)(99,113)(100,114)(101,115)(102,116)(103,117)(104,118)(105,135)(106,136)(107,129)(108,130)(109,131)(110,132)(111,133)(112,134), (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), (1,97,5,101)(2,104,6,100)(3,103,7,99)(4,102,8,98)(9,149,13,145)(10,148,14,152)(11,147,15,151)(12,146,16,150)(17,157,21,153)(18,156,22,160)(19,155,23,159)(20,154,24,158)(25,113,29,117)(26,120,30,116)(27,119,31,115)(28,118,32,114)(33,129,37,133)(34,136,38,132)(35,135,39,131)(36,134,40,130)(41,62,45,58)(42,61,46,57)(43,60,47,64)(44,59,48,63)(49,110,53,106)(50,109,54,105)(51,108,55,112)(52,107,56,111)(65,90,69,94)(66,89,70,93)(67,96,71,92)(68,95,72,91)(73,84,77,88)(74,83,78,87)(75,82,79,86)(76,81,80,85)(121,142,125,138)(122,141,126,137)(123,140,127,144)(124,139,128,143) );
G=PermutationGroup([[(1,44,81,54,95),(2,45,82,55,96),(3,46,83,56,89),(4,47,84,49,90),(5,48,85,50,91),(6,41,86,51,92),(7,42,87,52,93),(8,43,88,53,94),(9,113,137,17,129),(10,114,138,18,130),(11,115,139,19,131),(12,116,140,20,132),(13,117,141,21,133),(14,118,142,22,134),(15,119,143,23,135),(16,120,144,24,136),(25,126,153,33,145),(26,127,154,34,146),(27,128,155,35,147),(28,121,156,36,148),(29,122,157,37,149),(30,123,158,38,150),(31,124,159,39,151),(32,125,160,40,152),(57,78,111,70,103),(58,79,112,71,104),(59,80,105,72,97),(60,73,106,65,98),(61,74,107,66,99),(62,75,108,67,100),(63,76,109,68,101),(64,77,110,69,102)], [(2,28),(4,30),(6,32),(8,26),(10,67),(12,69),(14,71),(16,65),(18,75),(20,77),(22,79),(24,73),(34,53),(36,55),(38,49),(40,51),(41,125),(43,127),(45,121),(47,123),(58,142),(60,144),(62,138),(64,140),(82,156),(84,158),(86,160),(88,154),(90,150),(92,152),(94,146),(96,148),(98,120),(100,114),(102,116),(104,118),(106,136),(108,130),(110,132),(112,134)], [(1,27),(2,28),(3,29),(4,30),(5,31),(6,32),(7,25),(8,26),(9,66),(10,67),(11,68),(12,69),(13,70),(14,71),(15,72),(16,65),(17,74),(18,75),(19,76),(20,77),(21,78),(22,79),(23,80),(24,73),(33,52),(34,53),(35,54),(36,55),(37,56),(38,49),(39,50),(40,51),(41,125),(42,126),(43,127),(44,128),(45,121),(46,122),(47,123),(48,124),(57,141),(58,142),(59,143),(60,144),(61,137),(62,138),(63,139),(64,140),(81,155),(82,156),(83,157),(84,158),(85,159),(86,160),(87,153),(88,154),(89,149),(90,150),(91,151),(92,152),(93,145),(94,146),(95,147),(96,148),(97,119),(98,120),(99,113),(100,114),(101,115),(102,116),(103,117),(104,118),(105,135),(106,136),(107,129),(108,130),(109,131),(110,132),(111,133),(112,134)], [(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)], [(1,97,5,101),(2,104,6,100),(3,103,7,99),(4,102,8,98),(9,149,13,145),(10,148,14,152),(11,147,15,151),(12,146,16,150),(17,157,21,153),(18,156,22,160),(19,155,23,159),(20,154,24,158),(25,113,29,117),(26,120,30,116),(27,119,31,115),(28,118,32,114),(33,129,37,133),(34,136,38,132),(35,135,39,131),(36,134,40,130),(41,62,45,58),(42,61,46,57),(43,60,47,64),(44,59,48,63),(49,110,53,106),(50,109,54,105),(51,108,55,112),(52,107,56,111),(65,90,69,94),(66,89,70,93),(67,96,71,92),(68,95,72,91),(73,84,77,88),(74,83,78,87),(75,82,79,86),(76,81,80,85),(121,142,125,138),(122,141,126,137),(123,140,127,144),(124,139,128,143)]])
95 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | ··· | 4G | 4H | 4I | 5A | 5B | 5C | 5D | 8A | 8B | 8C | 8D | 10A | ··· | 10L | 10M | ··· | 10T | 20A | ··· | 20H | 20I | ··· | 20AB | 20AC | ··· | 20AJ | 40A | ··· | 40P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 | 20 | ··· | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 8 | 8 | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 4 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | ··· | 8 | 4 | ··· | 4 |
95 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 | D4 | Q16 | C5×D4 | C5×D4 | C5×D4 | C5×Q16 | C8.C22 | C5×C8.C22 |
| kernel | C5×C22⋊Q16 | C5×C22⋊C8 | C5×Q8⋊C4 | C5×C22⋊Q8 | C10×Q16 | Q8×C2×C10 | C22⋊Q16 | C22⋊C8 | Q8⋊C4 | C22⋊Q8 | C2×Q16 | C22×Q8 | C2×C20 | C5×Q8 | C22×C10 | C2×C10 | C2×C4 | Q8 | C23 | C22 | C10 | C2 |
| # reps | 1 | 1 | 2 | 1 | 2 | 1 | 4 | 4 | 8 | 4 | 8 | 4 | 1 | 4 | 1 | 4 | 4 | 16 | 4 | 16 | 1 | 4 |
Matrix representation of C5×C22⋊Q16 ►in GL4(𝔽41) generated by
| 18 | 0 | 0 | 0 |
| 0 | 18 | 0 | 0 |
| 0 | 0 | 18 | 0 |
| 0 | 0 | 0 | 18 |
| 1 | 33 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 40 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 33 | 12 | 0 | 0 |
| 39 | 8 | 0 | 0 |
| 0 | 0 | 0 | 24 |
| 0 | 0 | 29 | 24 |
| 40 | 8 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 7 | 13 |
| 0 | 0 | 34 | 34 |
G:=sub<GL(4,GF(41))| [18,0,0,0,0,18,0,0,0,0,18,0,0,0,0,18],[1,0,0,0,33,40,0,0,0,0,1,0,0,0,0,1],[40,0,0,0,0,40,0,0,0,0,1,0,0,0,0,1],[33,39,0,0,12,8,0,0,0,0,0,29,0,0,24,24],[40,0,0,0,8,1,0,0,0,0,7,34,0,0,13,34] >;
C5×C22⋊Q16 in GAP, Magma, Sage, TeX
C_5\times C_2^2\rtimes Q_{16} % in TeX
G:=Group("C5xC2^2:Q16"); // GroupNames label
G:=SmallGroup(320,952);
// by ID
G=gap.SmallGroup(320,952);
# by ID
G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-2,1120,589,1128,1766,7004,3511,172]);
// Polycyclic
G:=Group<a,b,c,d,e|a^5=b^2=c^2=d^8=1,e^2=d^4,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,d*b*d^-1=b*c=c*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=d^-1>;
// generators/relations