direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C5×C42.C22, C8⋊C4⋊6C10, C10.36C4≀C2, (C2×D4).1C20, (C2×Q8).1C20, (D4×C10).17C4, (C2×C20).445D4, C42.1(C2×C10), (Q8×C10).14C4, C4.4D4.1C10, (C4×C20).241C22, C10.19(C4.D4), C2.6(C5×C4≀C2), (C2×C4).9(C2×C20), (C5×C8⋊C4)⋊16C2, (C2×C4).97(C5×D4), C2.3(C5×C4.D4), (C2×C20).349(C2×C4), (C5×C4.4D4).10C2, C22.37(C5×C22⋊C4), (C2×C10).188(C22⋊C4), SmallGroup(320,134)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5×C42.C22
G = < a,b,c,d,e | a5=b4=c4=e2=1, d2=c, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, dbd-1=bc2, ebe=b-1, cd=dc, ece=b2c-1, ede=b-1c2d >
Subgroups: 146 in 70 conjugacy classes, 30 normal (18 characteristic)
C1, C2, C2, C2, C4, C22, C22, C5, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C10, C10, C10, C42, C22⋊C4, C2×C8, C2×D4, C2×Q8, C20, C2×C10, C2×C10, C8⋊C4, C4.4D4, C40, C2×C20, C2×C20, C2×C20, C5×D4, C5×Q8, C22×C10, C42.C22, C4×C20, C5×C22⋊C4, C2×C40, D4×C10, Q8×C10, C5×C8⋊C4, C5×C4.4D4, C5×C42.C22
Quotients: C1, C2, C4, C22, C5, C2×C4, D4, C10, C22⋊C4, C20, C2×C10, C4.D4, C4≀C2, C2×C20, C5×D4, C42.C22, C5×C22⋊C4, C5×C4.D4, C5×C4≀C2, C5×C42.C22
►On 160 points
Generators in S160
(1 47 39 31 23)(2 48 40 32 24)(3 41 33 25 17)(4 42 34 26 18)(5 43 35 27 19)(6 44 36 28 20)(7 45 37 29 21)(8 46 38 30 22)(9 113 105 97 89)(10 114 106 98 90)(11 115 107 99 91)(12 116 108 100 92)(13 117 109 101 93)(14 118 110 102 94)(15 119 111 103 95)(16 120 112 104 96)(49 88 73 65 57)(50 81 74 66 58)(51 82 75 67 59)(52 83 76 68 60)(53 84 77 69 61)(54 85 78 70 62)(55 86 79 71 63)(56 87 80 72 64)(121 153 145 137 129)(122 154 146 138 130)(123 155 147 139 131)(124 156 148 140 132)(125 157 149 141 133)(126 158 150 142 134)(127 159 151 143 135)(128 160 152 144 136)
(1 55 127 95)(2 52 128 92)(3 49 121 89)(4 54 122 94)(5 51 123 91)(6 56 124 96)(7 53 125 93)(8 50 126 90)(9 41 88 153)(10 46 81 158)(11 43 82 155)(12 48 83 160)(13 45 84 157)(14 42 85 154)(15 47 86 159)(16 44 87 156)(17 57 129 97)(18 62 130 102)(19 59 131 99)(20 64 132 104)(21 61 133 101)(22 58 134 98)(23 63 135 103)(24 60 136 100)(25 65 137 105)(26 70 138 110)(27 67 139 107)(28 72 140 112)(29 69 141 109)(30 66 142 106)(31 71 143 111)(32 68 144 108)(33 73 145 113)(34 78 146 118)(35 75 147 115)(36 80 148 120)(37 77 149 117)(38 74 150 114)(39 79 151 119)(40 76 152 116)
(1 3 5 7)(2 4 6 8)(9 11 13 15)(10 12 14 16)(17 19 21 23)(18 20 22 24)(25 27 29 31)(26 28 30 32)(33 35 37 39)(34 36 38 40)(41 43 45 47)(42 44 46 48)(49 51 53 55)(50 52 54 56)(57 59 61 63)(58 60 62 64)(65 67 69 71)(66 68 70 72)(73 75 77 79)(74 76 78 80)(81 83 85 87)(82 84 86 88)(89 91 93 95)(90 92 94 96)(97 99 101 103)(98 100 102 104)(105 107 109 111)(106 108 110 112)(113 115 117 119)(114 116 118 120)(121 123 125 127)(122 124 126 128)(129 131 133 135)(130 132 134 136)(137 139 141 143)(138 140 142 144)(145 147 149 151)(146 148 150 152)(153 155 157 159)(154 156 158 160)
(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 92)(3 125)(4 50)(6 96)(7 121)(8 54)(9 13)(10 154)(11 82)(12 48)(14 158)(15 86)(16 44)(17 133)(18 58)(20 104)(21 129)(22 62)(24 100)(25 141)(26 66)(28 112)(29 137)(30 70)(32 108)(33 149)(34 74)(36 120)(37 145)(38 78)(40 116)(41 157)(42 81)(45 153)(46 85)(49 53)(51 91)(52 128)(55 95)(56 124)(57 61)(59 99)(60 136)(63 103)(64 132)(65 69)(67 107)(68 144)(71 111)(72 140)(73 77)(75 115)(76 152)(79 119)(80 148)(83 160)(84 88)(87 156)(89 93)(90 122)(94 126)(97 101)(98 130)(102 134)(105 109)(106 138)(110 142)(113 117)(114 146)(118 150)
G:=sub<Sym(160)| (1,47,39,31,23)(2,48,40,32,24)(3,41,33,25,17)(4,42,34,26,18)(5,43,35,27,19)(6,44,36,28,20)(7,45,37,29,21)(8,46,38,30,22)(9,113,105,97,89)(10,114,106,98,90)(11,115,107,99,91)(12,116,108,100,92)(13,117,109,101,93)(14,118,110,102,94)(15,119,111,103,95)(16,120,112,104,96)(49,88,73,65,57)(50,81,74,66,58)(51,82,75,67,59)(52,83,76,68,60)(53,84,77,69,61)(54,85,78,70,62)(55,86,79,71,63)(56,87,80,72,64)(121,153,145,137,129)(122,154,146,138,130)(123,155,147,139,131)(124,156,148,140,132)(125,157,149,141,133)(126,158,150,142,134)(127,159,151,143,135)(128,160,152,144,136), (1,55,127,95)(2,52,128,92)(3,49,121,89)(4,54,122,94)(5,51,123,91)(6,56,124,96)(7,53,125,93)(8,50,126,90)(9,41,88,153)(10,46,81,158)(11,43,82,155)(12,48,83,160)(13,45,84,157)(14,42,85,154)(15,47,86,159)(16,44,87,156)(17,57,129,97)(18,62,130,102)(19,59,131,99)(20,64,132,104)(21,61,133,101)(22,58,134,98)(23,63,135,103)(24,60,136,100)(25,65,137,105)(26,70,138,110)(27,67,139,107)(28,72,140,112)(29,69,141,109)(30,66,142,106)(31,71,143,111)(32,68,144,108)(33,73,145,113)(34,78,146,118)(35,75,147,115)(36,80,148,120)(37,77,149,117)(38,74,150,114)(39,79,151,119)(40,76,152,116), (1,3,5,7)(2,4,6,8)(9,11,13,15)(10,12,14,16)(17,19,21,23)(18,20,22,24)(25,27,29,31)(26,28,30,32)(33,35,37,39)(34,36,38,40)(41,43,45,47)(42,44,46,48)(49,51,53,55)(50,52,54,56)(57,59,61,63)(58,60,62,64)(65,67,69,71)(66,68,70,72)(73,75,77,79)(74,76,78,80)(81,83,85,87)(82,84,86,88)(89,91,93,95)(90,92,94,96)(97,99,101,103)(98,100,102,104)(105,107,109,111)(106,108,110,112)(113,115,117,119)(114,116,118,120)(121,123,125,127)(122,124,126,128)(129,131,133,135)(130,132,134,136)(137,139,141,143)(138,140,142,144)(145,147,149,151)(146,148,150,152)(153,155,157,159)(154,156,158,160), (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), (2,92)(3,125)(4,50)(6,96)(7,121)(8,54)(9,13)(10,154)(11,82)(12,48)(14,158)(15,86)(16,44)(17,133)(18,58)(20,104)(21,129)(22,62)(24,100)(25,141)(26,66)(28,112)(29,137)(30,70)(32,108)(33,149)(34,74)(36,120)(37,145)(38,78)(40,116)(41,157)(42,81)(45,153)(46,85)(49,53)(51,91)(52,128)(55,95)(56,124)(57,61)(59,99)(60,136)(63,103)(64,132)(65,69)(67,107)(68,144)(71,111)(72,140)(73,77)(75,115)(76,152)(79,119)(80,148)(83,160)(84,88)(87,156)(89,93)(90,122)(94,126)(97,101)(98,130)(102,134)(105,109)(106,138)(110,142)(113,117)(114,146)(118,150)>;
G:=Group( (1,47,39,31,23)(2,48,40,32,24)(3,41,33,25,17)(4,42,34,26,18)(5,43,35,27,19)(6,44,36,28,20)(7,45,37,29,21)(8,46,38,30,22)(9,113,105,97,89)(10,114,106,98,90)(11,115,107,99,91)(12,116,108,100,92)(13,117,109,101,93)(14,118,110,102,94)(15,119,111,103,95)(16,120,112,104,96)(49,88,73,65,57)(50,81,74,66,58)(51,82,75,67,59)(52,83,76,68,60)(53,84,77,69,61)(54,85,78,70,62)(55,86,79,71,63)(56,87,80,72,64)(121,153,145,137,129)(122,154,146,138,130)(123,155,147,139,131)(124,156,148,140,132)(125,157,149,141,133)(126,158,150,142,134)(127,159,151,143,135)(128,160,152,144,136), (1,55,127,95)(2,52,128,92)(3,49,121,89)(4,54,122,94)(5,51,123,91)(6,56,124,96)(7,53,125,93)(8,50,126,90)(9,41,88,153)(10,46,81,158)(11,43,82,155)(12,48,83,160)(13,45,84,157)(14,42,85,154)(15,47,86,159)(16,44,87,156)(17,57,129,97)(18,62,130,102)(19,59,131,99)(20,64,132,104)(21,61,133,101)(22,58,134,98)(23,63,135,103)(24,60,136,100)(25,65,137,105)(26,70,138,110)(27,67,139,107)(28,72,140,112)(29,69,141,109)(30,66,142,106)(31,71,143,111)(32,68,144,108)(33,73,145,113)(34,78,146,118)(35,75,147,115)(36,80,148,120)(37,77,149,117)(38,74,150,114)(39,79,151,119)(40,76,152,116), (1,3,5,7)(2,4,6,8)(9,11,13,15)(10,12,14,16)(17,19,21,23)(18,20,22,24)(25,27,29,31)(26,28,30,32)(33,35,37,39)(34,36,38,40)(41,43,45,47)(42,44,46,48)(49,51,53,55)(50,52,54,56)(57,59,61,63)(58,60,62,64)(65,67,69,71)(66,68,70,72)(73,75,77,79)(74,76,78,80)(81,83,85,87)(82,84,86,88)(89,91,93,95)(90,92,94,96)(97,99,101,103)(98,100,102,104)(105,107,109,111)(106,108,110,112)(113,115,117,119)(114,116,118,120)(121,123,125,127)(122,124,126,128)(129,131,133,135)(130,132,134,136)(137,139,141,143)(138,140,142,144)(145,147,149,151)(146,148,150,152)(153,155,157,159)(154,156,158,160), (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), (2,92)(3,125)(4,50)(6,96)(7,121)(8,54)(9,13)(10,154)(11,82)(12,48)(14,158)(15,86)(16,44)(17,133)(18,58)(20,104)(21,129)(22,62)(24,100)(25,141)(26,66)(28,112)(29,137)(30,70)(32,108)(33,149)(34,74)(36,120)(37,145)(38,78)(40,116)(41,157)(42,81)(45,153)(46,85)(49,53)(51,91)(52,128)(55,95)(56,124)(57,61)(59,99)(60,136)(63,103)(64,132)(65,69)(67,107)(68,144)(71,111)(72,140)(73,77)(75,115)(76,152)(79,119)(80,148)(83,160)(84,88)(87,156)(89,93)(90,122)(94,126)(97,101)(98,130)(102,134)(105,109)(106,138)(110,142)(113,117)(114,146)(118,150) );
G=PermutationGroup([[(1,47,39,31,23),(2,48,40,32,24),(3,41,33,25,17),(4,42,34,26,18),(5,43,35,27,19),(6,44,36,28,20),(7,45,37,29,21),(8,46,38,30,22),(9,113,105,97,89),(10,114,106,98,90),(11,115,107,99,91),(12,116,108,100,92),(13,117,109,101,93),(14,118,110,102,94),(15,119,111,103,95),(16,120,112,104,96),(49,88,73,65,57),(50,81,74,66,58),(51,82,75,67,59),(52,83,76,68,60),(53,84,77,69,61),(54,85,78,70,62),(55,86,79,71,63),(56,87,80,72,64),(121,153,145,137,129),(122,154,146,138,130),(123,155,147,139,131),(124,156,148,140,132),(125,157,149,141,133),(126,158,150,142,134),(127,159,151,143,135),(128,160,152,144,136)], [(1,55,127,95),(2,52,128,92),(3,49,121,89),(4,54,122,94),(5,51,123,91),(6,56,124,96),(7,53,125,93),(8,50,126,90),(9,41,88,153),(10,46,81,158),(11,43,82,155),(12,48,83,160),(13,45,84,157),(14,42,85,154),(15,47,86,159),(16,44,87,156),(17,57,129,97),(18,62,130,102),(19,59,131,99),(20,64,132,104),(21,61,133,101),(22,58,134,98),(23,63,135,103),(24,60,136,100),(25,65,137,105),(26,70,138,110),(27,67,139,107),(28,72,140,112),(29,69,141,109),(30,66,142,106),(31,71,143,111),(32,68,144,108),(33,73,145,113),(34,78,146,118),(35,75,147,115),(36,80,148,120),(37,77,149,117),(38,74,150,114),(39,79,151,119),(40,76,152,116)], [(1,3,5,7),(2,4,6,8),(9,11,13,15),(10,12,14,16),(17,19,21,23),(18,20,22,24),(25,27,29,31),(26,28,30,32),(33,35,37,39),(34,36,38,40),(41,43,45,47),(42,44,46,48),(49,51,53,55),(50,52,54,56),(57,59,61,63),(58,60,62,64),(65,67,69,71),(66,68,70,72),(73,75,77,79),(74,76,78,80),(81,83,85,87),(82,84,86,88),(89,91,93,95),(90,92,94,96),(97,99,101,103),(98,100,102,104),(105,107,109,111),(106,108,110,112),(113,115,117,119),(114,116,118,120),(121,123,125,127),(122,124,126,128),(129,131,133,135),(130,132,134,136),(137,139,141,143),(138,140,142,144),(145,147,149,151),(146,148,150,152),(153,155,157,159),(154,156,158,160)], [(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)], [(2,92),(3,125),(4,50),(6,96),(7,121),(8,54),(9,13),(10,154),(11,82),(12,48),(14,158),(15,86),(16,44),(17,133),(18,58),(20,104),(21,129),(22,62),(24,100),(25,141),(26,66),(28,112),(29,137),(30,70),(32,108),(33,149),(34,74),(36,120),(37,145),(38,78),(40,116),(41,157),(42,81),(45,153),(46,85),(49,53),(51,91),(52,128),(55,95),(56,124),(57,61),(59,99),(60,136),(63,103),(64,132),(65,69),(67,107),(68,144),(71,111),(72,140),(73,77),(75,115),(76,152),(79,119),(80,148),(83,160),(84,88),(87,156),(89,93),(90,122),(94,126),(97,101),(98,130),(102,134),(105,109),(106,138),(110,142),(113,117),(114,146),(118,150)]])
95 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | 4D | 4E | 4F | 5A | 5B | 5C | 5D | 8A | ··· | 8H | 10A | ··· | 10L | 10M | 10N | 10O | 10P | 20A | ··· | 20P | 20Q | 20R | 20S | 20T | 20U | 20V | 20W | 20X | 40A | ··· | 40AF |
| order | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 8 | ··· | 8 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 20 | ··· | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 8 | 2 | 2 | 2 | 2 | 4 | 8 | 1 | 1 | 1 | 1 | 4 | ··· | 4 | 1 | ··· | 1 | 8 | 8 | 8 | 8 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 4 | ··· | 4 |
95 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | |||||||||||
| image | C1 | C2 | C2 | C4 | C4 | C5 | C10 | C10 | C20 | C20 | D4 | C4≀C2 | C5×D4 | C5×C4≀C2 | C4.D4 | C5×C4.D4 |
| kernel | C5×C42.C22 | C5×C8⋊C4 | C5×C4.4D4 | D4×C10 | Q8×C10 | C42.C22 | C8⋊C4 | C4.4D4 | C2×D4 | C2×Q8 | C2×C20 | C10 | C2×C4 | C2 | C10 | C2 |
| # reps | 1 | 2 | 1 | 2 | 2 | 4 | 8 | 4 | 8 | 8 | 2 | 8 | 8 | 32 | 1 | 4 |
Matrix representation of C5×C42.C22 ►in GL4(𝔽41) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 37 | 0 |
| 0 | 0 | 0 | 37 |
| 0 | 1 | 0 | 0 |
| 40 | 0 | 0 | 0 |
| 0 | 0 | 1 | 9 |
| 0 | 0 | 18 | 40 |
| 9 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 |
| 0 | 0 | 32 | 1 |
| 0 | 0 | 2 | 9 |
| 36 | 5 | 0 | 0 |
| 5 | 5 | 0 | 0 |
| 0 | 0 | 8 | 36 |
| 0 | 0 | 31 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 1 | 9 |
| 0 | 0 | 0 | 40 |
G:=sub<GL(4,GF(41))| [1,0,0,0,0,1,0,0,0,0,37,0,0,0,0,37],[0,40,0,0,1,0,0,0,0,0,1,18,0,0,9,40],[9,0,0,0,0,9,0,0,0,0,32,2,0,0,1,9],[36,5,0,0,5,5,0,0,0,0,8,31,0,0,36,0],[1,0,0,0,0,40,0,0,0,0,1,0,0,0,9,40] >;
C5×C42.C22 in GAP, Magma, Sage, TeX
C_5\times C_4^2.C_2^2
% in TeX
G:=Group("C5xC4^2.C2^2"); // GroupNames label
G:=SmallGroup(320,134);
// by ID
G=gap.SmallGroup(320,134);
# by ID
G:=PCGroup([7,-2,-2,-5,-2,-2,-2,-2,280,309,2803,2530,248,4911,102]);
// Polycyclic
G:=Group<a,b,c,d,e|a^5=b^4=c^4=e^2=1,d^2=c,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,d*b*d^-1=b*c^2,e*b*e=b^-1,c*d=d*c,e*c*e=b^2*c^-1,e*d*e=b^-1*c^2*d>;
// generators/relations