direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C22×C4.Dic5, C20.74C24, C24.5Dic5, (C23×C4).9D5, C5⋊2C8⋊13C23, C10⋊6(C2×M4(2)), C4.73(C23×D5), C5⋊6(C22×M4(2)), (C2×C10)⋊15M4(2), (C22×C20).56C4, (C23×C20).16C2, C10.62(C23×C4), (C23×C10).17C4, C2.3(C23×Dic5), C20.239(C22×C4), (C2×C20).799C23, (C22×C4).472D10, C23.35(C2×Dic5), (C22×C4).18Dic5, C4.38(C22×Dic5), (C22×C20).546C22, C22.28(C22×Dic5), (C2×C20).473(C2×C4), (C2×C5⋊2C8)⋊48C22, (C22×C5⋊2C8)⋊23C2, (C2×C4).86(C2×Dic5), (C2×C4).827(C22×D5), (C22×C10).208(C2×C4), (C2×C10).306(C22×C4), SmallGroup(320,1453)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C22×C4.Dic5
G = < a,b,c,d,e | a2=b2=c4=1, d10=c2, e2=d5, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece-1=c-1, ede-1=d9 >
Subgroups: 542 in 298 conjugacy classes, 207 normal (17 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C23, C23, C23, C10, C10, C10, C2×C8, M4(2), C22×C4, C22×C4, C24, C20, C20, C2×C10, C2×C10, C22×C8, C2×M4(2), C23×C4, C5⋊2C8, C2×C20, C22×C10, C22×C10, C22×C10, C22×M4(2), C2×C5⋊2C8, C4.Dic5, C22×C20, C22×C20, C23×C10, C22×C5⋊2C8, C2×C4.Dic5, C23×C20, C22×C4.Dic5
Quotients: C1, C2, C4, C22, C2×C4, C23, D5, M4(2), C22×C4, C24, Dic5, D10, C2×M4(2), C23×C4, C2×Dic5, C22×D5, C22×M4(2), C4.Dic5, C22×Dic5, C23×D5, C2×C4.Dic5, C23×Dic5, C22×C4.Dic5
►On 160 points
Generators in S160
(1 56)(2 57)(3 58)(4 59)(5 60)(6 41)(7 42)(8 43)(9 44)(10 45)(11 46)(12 47)(13 48)(14 49)(15 50)(16 51)(17 52)(18 53)(19 54)(20 55)(21 76)(22 77)(23 78)(24 79)(25 80)(26 61)(27 62)(28 63)(29 64)(30 65)(31 66)(32 67)(33 68)(34 69)(35 70)(36 71)(37 72)(38 73)(39 74)(40 75)(81 121)(82 122)(83 123)(84 124)(85 125)(86 126)(87 127)(88 128)(89 129)(90 130)(91 131)(92 132)(93 133)(94 134)(95 135)(96 136)(97 137)(98 138)(99 139)(100 140)(101 148)(102 149)(103 150)(104 151)(105 152)(106 153)(107 154)(108 155)(109 156)(110 157)(111 158)(112 159)(113 160)(114 141)(115 142)(116 143)(117 144)(118 145)(119 146)(120 147)
(1 31)(2 32)(3 33)(4 34)(5 35)(6 36)(7 37)(8 38)(9 39)(10 40)(11 21)(12 22)(13 23)(14 24)(15 25)(16 26)(17 27)(18 28)(19 29)(20 30)(41 71)(42 72)(43 73)(44 74)(45 75)(46 76)(47 77)(48 78)(49 79)(50 80)(51 61)(52 62)(53 63)(54 64)(55 65)(56 66)(57 67)(58 68)(59 69)(60 70)(81 115)(82 116)(83 117)(84 118)(85 119)(86 120)(87 101)(88 102)(89 103)(90 104)(91 105)(92 106)(93 107)(94 108)(95 109)(96 110)(97 111)(98 112)(99 113)(100 114)(121 142)(122 143)(123 144)(124 145)(125 146)(126 147)(127 148)(128 149)(129 150)(130 151)(131 152)(132 153)(133 154)(134 155)(135 156)(136 157)(137 158)(138 159)(139 160)(140 141)
(1 36 11 26)(2 37 12 27)(3 38 13 28)(4 39 14 29)(5 40 15 30)(6 21 16 31)(7 22 17 32)(8 23 18 33)(9 24 19 34)(10 25 20 35)(41 76 51 66)(42 77 52 67)(43 78 53 68)(44 79 54 69)(45 80 55 70)(46 61 56 71)(47 62 57 72)(48 63 58 73)(49 64 59 74)(50 65 60 75)(81 110 91 120)(82 111 92 101)(83 112 93 102)(84 113 94 103)(85 114 95 104)(86 115 96 105)(87 116 97 106)(88 117 98 107)(89 118 99 108)(90 119 100 109)(121 157 131 147)(122 158 132 148)(123 159 133 149)(124 160 134 150)(125 141 135 151)(126 142 136 152)(127 143 137 153)(128 144 138 154)(129 145 139 155)(130 146 140 156)
(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 95 6 100 11 85 16 90)(2 84 7 89 12 94 17 99)(3 93 8 98 13 83 18 88)(4 82 9 87 14 92 19 97)(5 91 10 96 15 81 20 86)(21 119 26 104 31 109 36 114)(22 108 27 113 32 118 37 103)(23 117 28 102 33 107 38 112)(24 106 29 111 34 116 39 101)(25 115 30 120 35 105 40 110)(41 140 46 125 51 130 56 135)(42 129 47 134 52 139 57 124)(43 138 48 123 53 128 58 133)(44 127 49 132 54 137 59 122)(45 136 50 121 55 126 60 131)(61 151 66 156 71 141 76 146)(62 160 67 145 72 150 77 155)(63 149 68 154 73 159 78 144)(64 158 69 143 74 148 79 153)(65 147 70 152 75 157 80 142)
G:=sub<Sym(160)| (1,56)(2,57)(3,58)(4,59)(5,60)(6,41)(7,42)(8,43)(9,44)(10,45)(11,46)(12,47)(13,48)(14,49)(15,50)(16,51)(17,52)(18,53)(19,54)(20,55)(21,76)(22,77)(23,78)(24,79)(25,80)(26,61)(27,62)(28,63)(29,64)(30,65)(31,66)(32,67)(33,68)(34,69)(35,70)(36,71)(37,72)(38,73)(39,74)(40,75)(81,121)(82,122)(83,123)(84,124)(85,125)(86,126)(87,127)(88,128)(89,129)(90,130)(91,131)(92,132)(93,133)(94,134)(95,135)(96,136)(97,137)(98,138)(99,139)(100,140)(101,148)(102,149)(103,150)(104,151)(105,152)(106,153)(107,154)(108,155)(109,156)(110,157)(111,158)(112,159)(113,160)(114,141)(115,142)(116,143)(117,144)(118,145)(119,146)(120,147), (1,31)(2,32)(3,33)(4,34)(5,35)(6,36)(7,37)(8,38)(9,39)(10,40)(11,21)(12,22)(13,23)(14,24)(15,25)(16,26)(17,27)(18,28)(19,29)(20,30)(41,71)(42,72)(43,73)(44,74)(45,75)(46,76)(47,77)(48,78)(49,79)(50,80)(51,61)(52,62)(53,63)(54,64)(55,65)(56,66)(57,67)(58,68)(59,69)(60,70)(81,115)(82,116)(83,117)(84,118)(85,119)(86,120)(87,101)(88,102)(89,103)(90,104)(91,105)(92,106)(93,107)(94,108)(95,109)(96,110)(97,111)(98,112)(99,113)(100,114)(121,142)(122,143)(123,144)(124,145)(125,146)(126,147)(127,148)(128,149)(129,150)(130,151)(131,152)(132,153)(133,154)(134,155)(135,156)(136,157)(137,158)(138,159)(139,160)(140,141), (1,36,11,26)(2,37,12,27)(3,38,13,28)(4,39,14,29)(5,40,15,30)(6,21,16,31)(7,22,17,32)(8,23,18,33)(9,24,19,34)(10,25,20,35)(41,76,51,66)(42,77,52,67)(43,78,53,68)(44,79,54,69)(45,80,55,70)(46,61,56,71)(47,62,57,72)(48,63,58,73)(49,64,59,74)(50,65,60,75)(81,110,91,120)(82,111,92,101)(83,112,93,102)(84,113,94,103)(85,114,95,104)(86,115,96,105)(87,116,97,106)(88,117,98,107)(89,118,99,108)(90,119,100,109)(121,157,131,147)(122,158,132,148)(123,159,133,149)(124,160,134,150)(125,141,135,151)(126,142,136,152)(127,143,137,153)(128,144,138,154)(129,145,139,155)(130,146,140,156), (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), (1,95,6,100,11,85,16,90)(2,84,7,89,12,94,17,99)(3,93,8,98,13,83,18,88)(4,82,9,87,14,92,19,97)(5,91,10,96,15,81,20,86)(21,119,26,104,31,109,36,114)(22,108,27,113,32,118,37,103)(23,117,28,102,33,107,38,112)(24,106,29,111,34,116,39,101)(25,115,30,120,35,105,40,110)(41,140,46,125,51,130,56,135)(42,129,47,134,52,139,57,124)(43,138,48,123,53,128,58,133)(44,127,49,132,54,137,59,122)(45,136,50,121,55,126,60,131)(61,151,66,156,71,141,76,146)(62,160,67,145,72,150,77,155)(63,149,68,154,73,159,78,144)(64,158,69,143,74,148,79,153)(65,147,70,152,75,157,80,142)>;
G:=Group( (1,56)(2,57)(3,58)(4,59)(5,60)(6,41)(7,42)(8,43)(9,44)(10,45)(11,46)(12,47)(13,48)(14,49)(15,50)(16,51)(17,52)(18,53)(19,54)(20,55)(21,76)(22,77)(23,78)(24,79)(25,80)(26,61)(27,62)(28,63)(29,64)(30,65)(31,66)(32,67)(33,68)(34,69)(35,70)(36,71)(37,72)(38,73)(39,74)(40,75)(81,121)(82,122)(83,123)(84,124)(85,125)(86,126)(87,127)(88,128)(89,129)(90,130)(91,131)(92,132)(93,133)(94,134)(95,135)(96,136)(97,137)(98,138)(99,139)(100,140)(101,148)(102,149)(103,150)(104,151)(105,152)(106,153)(107,154)(108,155)(109,156)(110,157)(111,158)(112,159)(113,160)(114,141)(115,142)(116,143)(117,144)(118,145)(119,146)(120,147), (1,31)(2,32)(3,33)(4,34)(5,35)(6,36)(7,37)(8,38)(9,39)(10,40)(11,21)(12,22)(13,23)(14,24)(15,25)(16,26)(17,27)(18,28)(19,29)(20,30)(41,71)(42,72)(43,73)(44,74)(45,75)(46,76)(47,77)(48,78)(49,79)(50,80)(51,61)(52,62)(53,63)(54,64)(55,65)(56,66)(57,67)(58,68)(59,69)(60,70)(81,115)(82,116)(83,117)(84,118)(85,119)(86,120)(87,101)(88,102)(89,103)(90,104)(91,105)(92,106)(93,107)(94,108)(95,109)(96,110)(97,111)(98,112)(99,113)(100,114)(121,142)(122,143)(123,144)(124,145)(125,146)(126,147)(127,148)(128,149)(129,150)(130,151)(131,152)(132,153)(133,154)(134,155)(135,156)(136,157)(137,158)(138,159)(139,160)(140,141), (1,36,11,26)(2,37,12,27)(3,38,13,28)(4,39,14,29)(5,40,15,30)(6,21,16,31)(7,22,17,32)(8,23,18,33)(9,24,19,34)(10,25,20,35)(41,76,51,66)(42,77,52,67)(43,78,53,68)(44,79,54,69)(45,80,55,70)(46,61,56,71)(47,62,57,72)(48,63,58,73)(49,64,59,74)(50,65,60,75)(81,110,91,120)(82,111,92,101)(83,112,93,102)(84,113,94,103)(85,114,95,104)(86,115,96,105)(87,116,97,106)(88,117,98,107)(89,118,99,108)(90,119,100,109)(121,157,131,147)(122,158,132,148)(123,159,133,149)(124,160,134,150)(125,141,135,151)(126,142,136,152)(127,143,137,153)(128,144,138,154)(129,145,139,155)(130,146,140,156), (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), (1,95,6,100,11,85,16,90)(2,84,7,89,12,94,17,99)(3,93,8,98,13,83,18,88)(4,82,9,87,14,92,19,97)(5,91,10,96,15,81,20,86)(21,119,26,104,31,109,36,114)(22,108,27,113,32,118,37,103)(23,117,28,102,33,107,38,112)(24,106,29,111,34,116,39,101)(25,115,30,120,35,105,40,110)(41,140,46,125,51,130,56,135)(42,129,47,134,52,139,57,124)(43,138,48,123,53,128,58,133)(44,127,49,132,54,137,59,122)(45,136,50,121,55,126,60,131)(61,151,66,156,71,141,76,146)(62,160,67,145,72,150,77,155)(63,149,68,154,73,159,78,144)(64,158,69,143,74,148,79,153)(65,147,70,152,75,157,80,142) );
G=PermutationGroup([[(1,56),(2,57),(3,58),(4,59),(5,60),(6,41),(7,42),(8,43),(9,44),(10,45),(11,46),(12,47),(13,48),(14,49),(15,50),(16,51),(17,52),(18,53),(19,54),(20,55),(21,76),(22,77),(23,78),(24,79),(25,80),(26,61),(27,62),(28,63),(29,64),(30,65),(31,66),(32,67),(33,68),(34,69),(35,70),(36,71),(37,72),(38,73),(39,74),(40,75),(81,121),(82,122),(83,123),(84,124),(85,125),(86,126),(87,127),(88,128),(89,129),(90,130),(91,131),(92,132),(93,133),(94,134),(95,135),(96,136),(97,137),(98,138),(99,139),(100,140),(101,148),(102,149),(103,150),(104,151),(105,152),(106,153),(107,154),(108,155),(109,156),(110,157),(111,158),(112,159),(113,160),(114,141),(115,142),(116,143),(117,144),(118,145),(119,146),(120,147)], [(1,31),(2,32),(3,33),(4,34),(5,35),(6,36),(7,37),(8,38),(9,39),(10,40),(11,21),(12,22),(13,23),(14,24),(15,25),(16,26),(17,27),(18,28),(19,29),(20,30),(41,71),(42,72),(43,73),(44,74),(45,75),(46,76),(47,77),(48,78),(49,79),(50,80),(51,61),(52,62),(53,63),(54,64),(55,65),(56,66),(57,67),(58,68),(59,69),(60,70),(81,115),(82,116),(83,117),(84,118),(85,119),(86,120),(87,101),(88,102),(89,103),(90,104),(91,105),(92,106),(93,107),(94,108),(95,109),(96,110),(97,111),(98,112),(99,113),(100,114),(121,142),(122,143),(123,144),(124,145),(125,146),(126,147),(127,148),(128,149),(129,150),(130,151),(131,152),(132,153),(133,154),(134,155),(135,156),(136,157),(137,158),(138,159),(139,160),(140,141)], [(1,36,11,26),(2,37,12,27),(3,38,13,28),(4,39,14,29),(5,40,15,30),(6,21,16,31),(7,22,17,32),(8,23,18,33),(9,24,19,34),(10,25,20,35),(41,76,51,66),(42,77,52,67),(43,78,53,68),(44,79,54,69),(45,80,55,70),(46,61,56,71),(47,62,57,72),(48,63,58,73),(49,64,59,74),(50,65,60,75),(81,110,91,120),(82,111,92,101),(83,112,93,102),(84,113,94,103),(85,114,95,104),(86,115,96,105),(87,116,97,106),(88,117,98,107),(89,118,99,108),(90,119,100,109),(121,157,131,147),(122,158,132,148),(123,159,133,149),(124,160,134,150),(125,141,135,151),(126,142,136,152),(127,143,137,153),(128,144,138,154),(129,145,139,155),(130,146,140,156)], [(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)], [(1,95,6,100,11,85,16,90),(2,84,7,89,12,94,17,99),(3,93,8,98,13,83,18,88),(4,82,9,87,14,92,19,97),(5,91,10,96,15,81,20,86),(21,119,26,104,31,109,36,114),(22,108,27,113,32,118,37,103),(23,117,28,102,33,107,38,112),(24,106,29,111,34,116,39,101),(25,115,30,120,35,105,40,110),(41,140,46,125,51,130,56,135),(42,129,47,134,52,139,57,124),(43,138,48,123,53,128,58,133),(44,127,49,132,54,137,59,122),(45,136,50,121,55,126,60,131),(61,151,66,156,71,141,76,146),(62,160,67,145,72,150,77,155),(63,149,68,154,73,159,78,144),(64,158,69,143,74,148,79,153),(65,147,70,152,75,157,80,142)]])
104 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 4A | ··· | 4H | 4I | 4J | 4K | 4L | 5A | 5B | 8A | ··· | 8P | 10A | ··· | 10AD | 20A | ··· | 20AF |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | ··· | 8 | 10 | ··· | 10 | 20 | ··· | 20 |
| size | 1 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 10 | ··· | 10 | 2 | ··· | 2 | 2 | ··· | 2 |
104 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | - | + | - | ||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | D5 | M4(2) | Dic5 | D10 | Dic5 | C4.Dic5 |
| kernel | C22×C4.Dic5 | C22×C5⋊2C8 | C2×C4.Dic5 | C23×C20 | C22×C20 | C23×C10 | C23×C4 | C2×C10 | C22×C4 | C22×C4 | C24 | C22 |
| # reps | 1 | 2 | 12 | 1 | 14 | 2 | 2 | 8 | 14 | 14 | 2 | 32 |
Matrix representation of C22×C4.Dic5 ►in GL5(𝔽41)
| 1 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 40 |
| 40 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 0 | 32 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 19 | 1 | 0 | 0 |
| 0 | 16 | 16 | 0 | 0 |
| 0 | 0 | 0 | 39 | 0 |
| 0 | 0 | 0 | 0 | 21 |
| 40 | 0 | 0 | 0 | 0 |
| 0 | 37 | 15 | 0 | 0 |
| 0 | 18 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 9 | 0 |
G:=sub<GL(5,GF(41))| [1,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,40],[40,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,9,0,0,0,0,0,32],[1,0,0,0,0,0,19,16,0,0,0,1,16,0,0,0,0,0,39,0,0,0,0,0,21],[40,0,0,0,0,0,37,18,0,0,0,15,4,0,0,0,0,0,0,9,0,0,0,1,0] >;
C22×C4.Dic5 in GAP, Magma, Sage, TeX
C_2^2\times C_4.{\rm Dic}_5 % in TeX
G:=Group("C2^2xC4.Dic5"); // GroupNames label
G:=SmallGroup(320,1453);
// by ID
G=gap.SmallGroup(320,1453);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,1123,102,12550]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^4=1,d^10=c^2,e^2=d^5,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,e*c*e^-1=c^-1,e*d*e^-1=d^9>;
// generators/relations