direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C5×C22.M4(2), (C2×C4)⋊C40, (C2×C20)⋊7C8, (C2×C20).442D4, C22⋊C8.1C10, C22.3(C2×C40), (C22×C4).2C20, (C22×C20).5C4, C23.22(C2×C20), C10.36(C22⋊C8), C10.49(C23⋊C4), (C2×C10).33M4(2), (C22×C20).2C22, C22.3(C5×M4(2)), C10.15(C4.10D4), (C2×C4⋊C4).1C10, (C2×C4).92(C5×D4), C2.4(C5×C22⋊C8), C2.2(C5×C23⋊C4), (C10×C4⋊C4).28C2, (C2×C10).49(C2×C8), (C5×C22⋊C8).3C2, (C22×C4).2(C2×C10), C2.1(C5×C4.10D4), C22.24(C5×C22⋊C4), (C22×C10).175(C2×C4), (C2×C10).183(C22⋊C4), SmallGroup(320,129)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5×C22.M4(2)
G = < a,b,c,d,e | a5=b2=c2=d8=1, e2=c, ab=ba, ac=ca, ad=da, ae=ea, dbd-1=bc=cb, be=eb, cd=dc, ce=ec, ede-1=bcd5 >
Subgroups: 138 in 78 conjugacy classes, 38 normal (26 characteristic)
C1, C2 [×3], C2 [×2], C4 [×5], C22 [×3], C22 [×2], C5, C8 [×2], C2×C4 [×4], C2×C4 [×5], C23, C10 [×3], C10 [×2], C4⋊C4 [×2], C2×C8 [×2], C22×C4 [×3], C20 [×5], C2×C10 [×3], C2×C10 [×2], C22⋊C8 [×2], C2×C4⋊C4, C40 [×2], C2×C20 [×4], C2×C20 [×5], C22×C10, C22.M4(2), C5×C4⋊C4 [×2], C2×C40 [×2], C22×C20 [×3], C5×C22⋊C8 [×2], C10×C4⋊C4, C5×C22.M4(2)
Quotients: C1, C2 [×3], C4 [×2], C22, C5, C8 [×2], C2×C4, D4 [×2], C10 [×3], C22⋊C4, C2×C8, M4(2), C20 [×2], C2×C10, C22⋊C8, C23⋊C4, C4.10D4, C40 [×2], C2×C20, C5×D4 [×2], C22.M4(2), C5×C22⋊C4, C2×C40, C5×M4(2), C5×C22⋊C8, C5×C23⋊C4, C5×C4.10D4, C5×C22.M4(2)
►On 160 points
Generators in S160
(1 122 155 35 147)(2 123 156 36 148)(3 124 157 37 149)(4 125 158 38 150)(5 126 159 39 151)(6 127 160 40 152)(7 128 153 33 145)(8 121 154 34 146)(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 41 91 49 87)(26 42 92 50 88)(27 43 93 51 81)(28 44 94 52 82)(29 45 95 53 83)(30 46 96 54 84)(31 47 89 55 85)(32 48 90 56 86)(57 73 107 65 99)(58 74 108 66 100)(59 75 109 67 101)(60 76 110 68 102)(61 77 111 69 103)(62 78 112 70 104)(63 79 105 71 97)(64 80 106 72 98)
(2 54)(4 56)(6 50)(8 52)(10 97)(12 99)(14 101)(16 103)(18 105)(20 107)(22 109)(24 111)(26 160)(28 154)(30 156)(32 158)(34 44)(36 46)(38 48)(40 42)(57 116)(59 118)(61 120)(63 114)(65 132)(67 134)(69 136)(71 130)(73 140)(75 142)(77 144)(79 138)(82 121)(84 123)(86 125)(88 127)(90 150)(92 152)(94 146)(96 148)
(1 53)(2 54)(3 55)(4 56)(5 49)(6 50)(7 51)(8 52)(9 104)(10 97)(11 98)(12 99)(13 100)(14 101)(15 102)(16 103)(17 112)(18 105)(19 106)(20 107)(21 108)(22 109)(23 110)(24 111)(25 159)(26 160)(27 153)(28 154)(29 155)(30 156)(31 157)(32 158)(33 43)(34 44)(35 45)(36 46)(37 47)(38 48)(39 41)(40 42)(57 116)(58 117)(59 118)(60 119)(61 120)(62 113)(63 114)(64 115)(65 132)(66 133)(67 134)(68 135)(69 136)(70 129)(71 130)(72 131)(73 140)(74 141)(75 142)(76 143)(77 144)(78 137)(79 138)(80 139)(81 128)(82 121)(83 122)(84 123)(85 124)(86 125)(87 126)(88 127)(89 149)(90 150)(91 151)(92 152)(93 145)(94 146)(95 147)(96 148)
(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 64 53 115)(2 120 54 61)(3 117 55 58)(4 63 56 114)(5 60 49 119)(6 116 50 57)(7 113 51 62)(8 59 52 118)(9 93 104 145)(10 150 97 90)(11 147 98 95)(12 92 99 152)(13 89 100 149)(14 146 101 94)(15 151 102 91)(16 96 103 148)(17 27 112 153)(18 158 105 32)(19 155 106 29)(20 26 107 160)(21 31 108 157)(22 154 109 28)(23 159 110 25)(24 30 111 156)(33 129 43 70)(34 67 44 134)(35 72 45 131)(36 136 46 69)(37 133 47 66)(38 71 48 130)(39 68 41 135)(40 132 42 65)(73 127 140 88)(74 124 141 85)(75 82 142 121)(76 87 143 126)(77 123 144 84)(78 128 137 81)(79 86 138 125)(80 83 139 122)
G:=sub<Sym(160)| (1,122,155,35,147)(2,123,156,36,148)(3,124,157,37,149)(4,125,158,38,150)(5,126,159,39,151)(6,127,160,40,152)(7,128,153,33,145)(8,121,154,34,146)(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)(25,41,91,49,87)(26,42,92,50,88)(27,43,93,51,81)(28,44,94,52,82)(29,45,95,53,83)(30,46,96,54,84)(31,47,89,55,85)(32,48,90,56,86)(57,73,107,65,99)(58,74,108,66,100)(59,75,109,67,101)(60,76,110,68,102)(61,77,111,69,103)(62,78,112,70,104)(63,79,105,71,97)(64,80,106,72,98), (2,54)(4,56)(6,50)(8,52)(10,97)(12,99)(14,101)(16,103)(18,105)(20,107)(22,109)(24,111)(26,160)(28,154)(30,156)(32,158)(34,44)(36,46)(38,48)(40,42)(57,116)(59,118)(61,120)(63,114)(65,132)(67,134)(69,136)(71,130)(73,140)(75,142)(77,144)(79,138)(82,121)(84,123)(86,125)(88,127)(90,150)(92,152)(94,146)(96,148), (1,53)(2,54)(3,55)(4,56)(5,49)(6,50)(7,51)(8,52)(9,104)(10,97)(11,98)(12,99)(13,100)(14,101)(15,102)(16,103)(17,112)(18,105)(19,106)(20,107)(21,108)(22,109)(23,110)(24,111)(25,159)(26,160)(27,153)(28,154)(29,155)(30,156)(31,157)(32,158)(33,43)(34,44)(35,45)(36,46)(37,47)(38,48)(39,41)(40,42)(57,116)(58,117)(59,118)(60,119)(61,120)(62,113)(63,114)(64,115)(65,132)(66,133)(67,134)(68,135)(69,136)(70,129)(71,130)(72,131)(73,140)(74,141)(75,142)(76,143)(77,144)(78,137)(79,138)(80,139)(81,128)(82,121)(83,122)(84,123)(85,124)(86,125)(87,126)(88,127)(89,149)(90,150)(91,151)(92,152)(93,145)(94,146)(95,147)(96,148), (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), (1,64,53,115)(2,120,54,61)(3,117,55,58)(4,63,56,114)(5,60,49,119)(6,116,50,57)(7,113,51,62)(8,59,52,118)(9,93,104,145)(10,150,97,90)(11,147,98,95)(12,92,99,152)(13,89,100,149)(14,146,101,94)(15,151,102,91)(16,96,103,148)(17,27,112,153)(18,158,105,32)(19,155,106,29)(20,26,107,160)(21,31,108,157)(22,154,109,28)(23,159,110,25)(24,30,111,156)(33,129,43,70)(34,67,44,134)(35,72,45,131)(36,136,46,69)(37,133,47,66)(38,71,48,130)(39,68,41,135)(40,132,42,65)(73,127,140,88)(74,124,141,85)(75,82,142,121)(76,87,143,126)(77,123,144,84)(78,128,137,81)(79,86,138,125)(80,83,139,122)>;
G:=Group( (1,122,155,35,147)(2,123,156,36,148)(3,124,157,37,149)(4,125,158,38,150)(5,126,159,39,151)(6,127,160,40,152)(7,128,153,33,145)(8,121,154,34,146)(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)(25,41,91,49,87)(26,42,92,50,88)(27,43,93,51,81)(28,44,94,52,82)(29,45,95,53,83)(30,46,96,54,84)(31,47,89,55,85)(32,48,90,56,86)(57,73,107,65,99)(58,74,108,66,100)(59,75,109,67,101)(60,76,110,68,102)(61,77,111,69,103)(62,78,112,70,104)(63,79,105,71,97)(64,80,106,72,98), (2,54)(4,56)(6,50)(8,52)(10,97)(12,99)(14,101)(16,103)(18,105)(20,107)(22,109)(24,111)(26,160)(28,154)(30,156)(32,158)(34,44)(36,46)(38,48)(40,42)(57,116)(59,118)(61,120)(63,114)(65,132)(67,134)(69,136)(71,130)(73,140)(75,142)(77,144)(79,138)(82,121)(84,123)(86,125)(88,127)(90,150)(92,152)(94,146)(96,148), (1,53)(2,54)(3,55)(4,56)(5,49)(6,50)(7,51)(8,52)(9,104)(10,97)(11,98)(12,99)(13,100)(14,101)(15,102)(16,103)(17,112)(18,105)(19,106)(20,107)(21,108)(22,109)(23,110)(24,111)(25,159)(26,160)(27,153)(28,154)(29,155)(30,156)(31,157)(32,158)(33,43)(34,44)(35,45)(36,46)(37,47)(38,48)(39,41)(40,42)(57,116)(58,117)(59,118)(60,119)(61,120)(62,113)(63,114)(64,115)(65,132)(66,133)(67,134)(68,135)(69,136)(70,129)(71,130)(72,131)(73,140)(74,141)(75,142)(76,143)(77,144)(78,137)(79,138)(80,139)(81,128)(82,121)(83,122)(84,123)(85,124)(86,125)(87,126)(88,127)(89,149)(90,150)(91,151)(92,152)(93,145)(94,146)(95,147)(96,148), (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), (1,64,53,115)(2,120,54,61)(3,117,55,58)(4,63,56,114)(5,60,49,119)(6,116,50,57)(7,113,51,62)(8,59,52,118)(9,93,104,145)(10,150,97,90)(11,147,98,95)(12,92,99,152)(13,89,100,149)(14,146,101,94)(15,151,102,91)(16,96,103,148)(17,27,112,153)(18,158,105,32)(19,155,106,29)(20,26,107,160)(21,31,108,157)(22,154,109,28)(23,159,110,25)(24,30,111,156)(33,129,43,70)(34,67,44,134)(35,72,45,131)(36,136,46,69)(37,133,47,66)(38,71,48,130)(39,68,41,135)(40,132,42,65)(73,127,140,88)(74,124,141,85)(75,82,142,121)(76,87,143,126)(77,123,144,84)(78,128,137,81)(79,86,138,125)(80,83,139,122) );
G=PermutationGroup([(1,122,155,35,147),(2,123,156,36,148),(3,124,157,37,149),(4,125,158,38,150),(5,126,159,39,151),(6,127,160,40,152),(7,128,153,33,145),(8,121,154,34,146),(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),(25,41,91,49,87),(26,42,92,50,88),(27,43,93,51,81),(28,44,94,52,82),(29,45,95,53,83),(30,46,96,54,84),(31,47,89,55,85),(32,48,90,56,86),(57,73,107,65,99),(58,74,108,66,100),(59,75,109,67,101),(60,76,110,68,102),(61,77,111,69,103),(62,78,112,70,104),(63,79,105,71,97),(64,80,106,72,98)], [(2,54),(4,56),(6,50),(8,52),(10,97),(12,99),(14,101),(16,103),(18,105),(20,107),(22,109),(24,111),(26,160),(28,154),(30,156),(32,158),(34,44),(36,46),(38,48),(40,42),(57,116),(59,118),(61,120),(63,114),(65,132),(67,134),(69,136),(71,130),(73,140),(75,142),(77,144),(79,138),(82,121),(84,123),(86,125),(88,127),(90,150),(92,152),(94,146),(96,148)], [(1,53),(2,54),(3,55),(4,56),(5,49),(6,50),(7,51),(8,52),(9,104),(10,97),(11,98),(12,99),(13,100),(14,101),(15,102),(16,103),(17,112),(18,105),(19,106),(20,107),(21,108),(22,109),(23,110),(24,111),(25,159),(26,160),(27,153),(28,154),(29,155),(30,156),(31,157),(32,158),(33,43),(34,44),(35,45),(36,46),(37,47),(38,48),(39,41),(40,42),(57,116),(58,117),(59,118),(60,119),(61,120),(62,113),(63,114),(64,115),(65,132),(66,133),(67,134),(68,135),(69,136),(70,129),(71,130),(72,131),(73,140),(74,141),(75,142),(76,143),(77,144),(78,137),(79,138),(80,139),(81,128),(82,121),(83,122),(84,123),(85,124),(86,125),(87,126),(88,127),(89,149),(90,150),(91,151),(92,152),(93,145),(94,146),(95,147),(96,148)], [(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)], [(1,64,53,115),(2,120,54,61),(3,117,55,58),(4,63,56,114),(5,60,49,119),(6,116,50,57),(7,113,51,62),(8,59,52,118),(9,93,104,145),(10,150,97,90),(11,147,98,95),(12,92,99,152),(13,89,100,149),(14,146,101,94),(15,151,102,91),(16,96,103,148),(17,27,112,153),(18,158,105,32),(19,155,106,29),(20,26,107,160),(21,31,108,157),(22,154,109,28),(23,159,110,25),(24,30,111,156),(33,129,43,70),(34,67,44,134),(35,72,45,131),(36,136,46,69),(37,133,47,66),(38,71,48,130),(39,68,41,135),(40,132,42,65),(73,127,140,88),(74,124,141,85),(75,82,142,121),(76,87,143,126),(77,123,144,84),(78,128,137,81),(79,86,138,125),(80,83,139,122)])
110 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 5A | 5B | 5C | 5D | 8A | ··· | 8H | 10A | ··· | 10L | 10M | ··· | 10T | 20A | ··· | 20P | 20Q | ··· | 20AF | 40A | ··· | 40AF |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 8 | ··· | 8 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 1 | 1 | 1 | 1 | 4 | ··· | 4 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
110 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | - | ||||||||||||
| image | C1 | C2 | C2 | C4 | C5 | C8 | C10 | C10 | C20 | C40 | D4 | M4(2) | C5×D4 | C5×M4(2) | C23⋊C4 | C4.10D4 | C5×C23⋊C4 | C5×C4.10D4 |
| kernel | C5×C22.M4(2) | C5×C22⋊C8 | C10×C4⋊C4 | C22×C20 | C22.M4(2) | C2×C20 | C22⋊C8 | C2×C4⋊C4 | C22×C4 | C2×C4 | C2×C20 | C2×C10 | C2×C4 | C22 | C10 | C10 | C2 | C2 |
| # reps | 1 | 2 | 1 | 4 | 4 | 8 | 8 | 4 | 16 | 32 | 2 | 2 | 8 | 8 | 1 | 1 | 4 | 4 |
Matrix representation of C5×C22.M4(2) ►in GL6(𝔽41)
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 16 | 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 |
| 1 | 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 | 38 | 39 | 40 | 0 |
| 0 | 0 | 34 | 0 | 0 | 40 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 32 | 7 | 0 | 0 | 0 | 0 |
| 34 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 14 | 23 | 23 | 0 |
| 0 | 0 | 30 | 40 | 40 | 9 |
| 0 | 0 | 11 | 19 | 28 | 32 |
| 0 | 0 | 17 | 22 | 22 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 9 | 0 | 0 | 0 |
| 0 | 0 | 1 | 32 | 0 | 0 |
| 0 | 0 | 40 | 18 | 9 | 0 |
| 0 | 0 | 19 | 0 | 0 | 32 |
G:=sub<GL(6,GF(41))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,38,34,0,0,0,1,39,0,0,0,0,0,40,0,0,0,0,0,0,40],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[32,34,0,0,0,0,7,9,0,0,0,0,0,0,14,30,11,17,0,0,23,40,19,22,0,0,23,40,28,22,0,0,0,9,32,0],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,9,1,40,19,0,0,0,32,18,0,0,0,0,0,9,0,0,0,0,0,0,32] >;
C5×C22.M4(2) in GAP, Magma, Sage, TeX
C_5\times C_2^2.M_4(2)
% in TeX
G:=Group("C5xC2^2.M4(2)"); // GroupNames label
G:=SmallGroup(320,129);
// by ID
G=gap.SmallGroup(320,129);
# by ID
G:=PCGroup([7,-2,-2,-5,-2,-2,-2,-2,280,309,568,2803,2111,172]);
// Polycyclic
G:=Group<a,b,c,d,e|a^5=b^2=c^2=d^8=1,e^2=c,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=b*c*d^5>;
// generators/relations