metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C23.34D20, C40⋊6C4⋊6C2, (C2×C4).29D20, (C2×C20).40D4, C22⋊C8.6D5, (C2×C8).107D10, C10.6(C2×SD16), C20.44D4⋊9C2, (C2×C10).13SD16, (C22×C10).48D4, (C22×C4).75D10, C20.279(C4○D4), (C2×C20).738C23, (C2×C40).118C22, C20.48D4.2C2, C22.101(C2×D20), C10.7(C8.C22), C22.8(C40⋊C2), C5⋊1(C23.47D4), C4.103(D4⋊2D5), C2.10(C8.D10), C4⋊Dic5.268C22, (C22×C20).48C22, (C2×Dic10).12C22, C10.14(C22.D4), C2.10(C22.D20), C2.9(C2×C40⋊C2), (C5×C22⋊C8).8C2, (C2×C10).121(C2×D4), (C2×C4⋊Dic5).12C2, (C2×C4).683(C22×D5), SmallGroup(320,348)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C23.34D20
G = < a,b,c,d,e | a2=b2=c2=1, d20=c, e2=cb=bc, dad-1=eae-1=ab=ba, ac=ca, bd=db, be=eb, cd=dc, ce=ec, ede-1=d19 >
Subgroups: 398 in 104 conjugacy classes, 43 normal (25 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C5, C8, C2×C4, C2×C4, Q8, C23, C10, C10, C22⋊C4, C4⋊C4, C2×C8, C22×C4, C22×C4, C2×Q8, Dic5, C20, C20, C2×C10, C2×C10, C2×C10, C22⋊C8, Q8⋊C4, C4.Q8, C2×C4⋊C4, C22⋊Q8, C40, Dic10, C2×Dic5, C2×C20, C2×C20, C22×C10, C23.47D4, C10.D4, C4⋊Dic5, C4⋊Dic5, C4⋊Dic5, C23.D5, C2×C40, C2×Dic10, C22×Dic5, C22×C20, C20.44D4, C40⋊6C4, C5×C22⋊C8, C20.48D4, C2×C4⋊Dic5, C23.34D20
Quotients: C1, C2, C22, D4, C23, D5, SD16, C2×D4, C4○D4, D10, C22.D4, C2×SD16, C8.C22, D20, C22×D5, C23.47D4, C40⋊C2, C2×D20, D4⋊2D5, C22.D20, C2×C40⋊C2, C8.D10, C23.34D20
►On 160 points
Generators in S160
(2 50)(4 52)(6 54)(8 56)(10 58)(12 60)(14 62)(16 64)(18 66)(20 68)(22 70)(24 72)(26 74)(28 76)(30 78)(32 80)(34 42)(36 44)(38 46)(40 48)(81 147)(83 149)(85 151)(87 153)(89 155)(91 157)(93 159)(95 121)(97 123)(99 125)(101 127)(103 129)(105 131)(107 133)(109 135)(111 137)(113 139)(115 141)(117 143)(119 145)
(1 49)(2 50)(3 51)(4 52)(5 53)(6 54)(7 55)(8 56)(9 57)(10 58)(11 59)(12 60)(13 61)(14 62)(15 63)(16 64)(17 65)(18 66)(19 67)(20 68)(21 69)(22 70)(23 71)(24 72)(25 73)(26 74)(27 75)(28 76)(29 77)(30 78)(31 79)(32 80)(33 41)(34 42)(35 43)(36 44)(37 45)(38 46)(39 47)(40 48)(81 147)(82 148)(83 149)(84 150)(85 151)(86 152)(87 153)(88 154)(89 155)(90 156)(91 157)(92 158)(93 159)(94 160)(95 121)(96 122)(97 123)(98 124)(99 125)(100 126)(101 127)(102 128)(103 129)(104 130)(105 131)(106 132)(107 133)(108 134)(109 135)(110 136)(111 137)(112 138)(113 139)(114 140)(115 141)(116 142)(117 143)(118 144)(119 145)(120 146)
(1 21)(2 22)(3 23)(4 24)(5 25)(6 26)(7 27)(8 28)(9 29)(10 30)(11 31)(12 32)(13 33)(14 34)(15 35)(16 36)(17 37)(18 38)(19 39)(20 40)(41 61)(42 62)(43 63)(44 64)(45 65)(46 66)(47 67)(48 68)(49 69)(50 70)(51 71)(52 72)(53 73)(54 74)(55 75)(56 76)(57 77)(58 78)(59 79)(60 80)(81 101)(82 102)(83 103)(84 104)(85 105)(86 106)(87 107)(88 108)(89 109)(90 110)(91 111)(92 112)(93 113)(94 114)(95 115)(96 116)(97 117)(98 118)(99 119)(100 120)(121 141)(122 142)(123 143)(124 144)(125 145)(126 146)(127 147)(128 148)(129 149)(130 150)(131 151)(132 152)(133 153)(134 154)(135 155)(136 156)(137 157)(138 158)(139 159)(140 160)
(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)
(1 115 69 121)(2 94 70 140)(3 113 71 159)(4 92 72 138)(5 111 73 157)(6 90 74 136)(7 109 75 155)(8 88 76 134)(9 107 77 153)(10 86 78 132)(11 105 79 151)(12 84 80 130)(13 103 41 149)(14 82 42 128)(15 101 43 147)(16 120 44 126)(17 99 45 145)(18 118 46 124)(19 97 47 143)(20 116 48 122)(21 95 49 141)(22 114 50 160)(23 93 51 139)(24 112 52 158)(25 91 53 137)(26 110 54 156)(27 89 55 135)(28 108 56 154)(29 87 57 133)(30 106 58 152)(31 85 59 131)(32 104 60 150)(33 83 61 129)(34 102 62 148)(35 81 63 127)(36 100 64 146)(37 119 65 125)(38 98 66 144)(39 117 67 123)(40 96 68 142)
G:=sub<Sym(160)| (2,50)(4,52)(6,54)(8,56)(10,58)(12,60)(14,62)(16,64)(18,66)(20,68)(22,70)(24,72)(26,74)(28,76)(30,78)(32,80)(34,42)(36,44)(38,46)(40,48)(81,147)(83,149)(85,151)(87,153)(89,155)(91,157)(93,159)(95,121)(97,123)(99,125)(101,127)(103,129)(105,131)(107,133)(109,135)(111,137)(113,139)(115,141)(117,143)(119,145), (1,49)(2,50)(3,51)(4,52)(5,53)(6,54)(7,55)(8,56)(9,57)(10,58)(11,59)(12,60)(13,61)(14,62)(15,63)(16,64)(17,65)(18,66)(19,67)(20,68)(21,69)(22,70)(23,71)(24,72)(25,73)(26,74)(27,75)(28,76)(29,77)(30,78)(31,79)(32,80)(33,41)(34,42)(35,43)(36,44)(37,45)(38,46)(39,47)(40,48)(81,147)(82,148)(83,149)(84,150)(85,151)(86,152)(87,153)(88,154)(89,155)(90,156)(91,157)(92,158)(93,159)(94,160)(95,121)(96,122)(97,123)(98,124)(99,125)(100,126)(101,127)(102,128)(103,129)(104,130)(105,131)(106,132)(107,133)(108,134)(109,135)(110,136)(111,137)(112,138)(113,139)(114,140)(115,141)(116,142)(117,143)(118,144)(119,145)(120,146), (1,21)(2,22)(3,23)(4,24)(5,25)(6,26)(7,27)(8,28)(9,29)(10,30)(11,31)(12,32)(13,33)(14,34)(15,35)(16,36)(17,37)(18,38)(19,39)(20,40)(41,61)(42,62)(43,63)(44,64)(45,65)(46,66)(47,67)(48,68)(49,69)(50,70)(51,71)(52,72)(53,73)(54,74)(55,75)(56,76)(57,77)(58,78)(59,79)(60,80)(81,101)(82,102)(83,103)(84,104)(85,105)(86,106)(87,107)(88,108)(89,109)(90,110)(91,111)(92,112)(93,113)(94,114)(95,115)(96,116)(97,117)(98,118)(99,119)(100,120)(121,141)(122,142)(123,143)(124,144)(125,145)(126,146)(127,147)(128,148)(129,149)(130,150)(131,151)(132,152)(133,153)(134,154)(135,155)(136,156)(137,157)(138,158)(139,159)(140,160), (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), (1,115,69,121)(2,94,70,140)(3,113,71,159)(4,92,72,138)(5,111,73,157)(6,90,74,136)(7,109,75,155)(8,88,76,134)(9,107,77,153)(10,86,78,132)(11,105,79,151)(12,84,80,130)(13,103,41,149)(14,82,42,128)(15,101,43,147)(16,120,44,126)(17,99,45,145)(18,118,46,124)(19,97,47,143)(20,116,48,122)(21,95,49,141)(22,114,50,160)(23,93,51,139)(24,112,52,158)(25,91,53,137)(26,110,54,156)(27,89,55,135)(28,108,56,154)(29,87,57,133)(30,106,58,152)(31,85,59,131)(32,104,60,150)(33,83,61,129)(34,102,62,148)(35,81,63,127)(36,100,64,146)(37,119,65,125)(38,98,66,144)(39,117,67,123)(40,96,68,142)>;
G:=Group( (2,50)(4,52)(6,54)(8,56)(10,58)(12,60)(14,62)(16,64)(18,66)(20,68)(22,70)(24,72)(26,74)(28,76)(30,78)(32,80)(34,42)(36,44)(38,46)(40,48)(81,147)(83,149)(85,151)(87,153)(89,155)(91,157)(93,159)(95,121)(97,123)(99,125)(101,127)(103,129)(105,131)(107,133)(109,135)(111,137)(113,139)(115,141)(117,143)(119,145), (1,49)(2,50)(3,51)(4,52)(5,53)(6,54)(7,55)(8,56)(9,57)(10,58)(11,59)(12,60)(13,61)(14,62)(15,63)(16,64)(17,65)(18,66)(19,67)(20,68)(21,69)(22,70)(23,71)(24,72)(25,73)(26,74)(27,75)(28,76)(29,77)(30,78)(31,79)(32,80)(33,41)(34,42)(35,43)(36,44)(37,45)(38,46)(39,47)(40,48)(81,147)(82,148)(83,149)(84,150)(85,151)(86,152)(87,153)(88,154)(89,155)(90,156)(91,157)(92,158)(93,159)(94,160)(95,121)(96,122)(97,123)(98,124)(99,125)(100,126)(101,127)(102,128)(103,129)(104,130)(105,131)(106,132)(107,133)(108,134)(109,135)(110,136)(111,137)(112,138)(113,139)(114,140)(115,141)(116,142)(117,143)(118,144)(119,145)(120,146), (1,21)(2,22)(3,23)(4,24)(5,25)(6,26)(7,27)(8,28)(9,29)(10,30)(11,31)(12,32)(13,33)(14,34)(15,35)(16,36)(17,37)(18,38)(19,39)(20,40)(41,61)(42,62)(43,63)(44,64)(45,65)(46,66)(47,67)(48,68)(49,69)(50,70)(51,71)(52,72)(53,73)(54,74)(55,75)(56,76)(57,77)(58,78)(59,79)(60,80)(81,101)(82,102)(83,103)(84,104)(85,105)(86,106)(87,107)(88,108)(89,109)(90,110)(91,111)(92,112)(93,113)(94,114)(95,115)(96,116)(97,117)(98,118)(99,119)(100,120)(121,141)(122,142)(123,143)(124,144)(125,145)(126,146)(127,147)(128,148)(129,149)(130,150)(131,151)(132,152)(133,153)(134,154)(135,155)(136,156)(137,157)(138,158)(139,159)(140,160), (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), (1,115,69,121)(2,94,70,140)(3,113,71,159)(4,92,72,138)(5,111,73,157)(6,90,74,136)(7,109,75,155)(8,88,76,134)(9,107,77,153)(10,86,78,132)(11,105,79,151)(12,84,80,130)(13,103,41,149)(14,82,42,128)(15,101,43,147)(16,120,44,126)(17,99,45,145)(18,118,46,124)(19,97,47,143)(20,116,48,122)(21,95,49,141)(22,114,50,160)(23,93,51,139)(24,112,52,158)(25,91,53,137)(26,110,54,156)(27,89,55,135)(28,108,56,154)(29,87,57,133)(30,106,58,152)(31,85,59,131)(32,104,60,150)(33,83,61,129)(34,102,62,148)(35,81,63,127)(36,100,64,146)(37,119,65,125)(38,98,66,144)(39,117,67,123)(40,96,68,142) );
G=PermutationGroup([[(2,50),(4,52),(6,54),(8,56),(10,58),(12,60),(14,62),(16,64),(18,66),(20,68),(22,70),(24,72),(26,74),(28,76),(30,78),(32,80),(34,42),(36,44),(38,46),(40,48),(81,147),(83,149),(85,151),(87,153),(89,155),(91,157),(93,159),(95,121),(97,123),(99,125),(101,127),(103,129),(105,131),(107,133),(109,135),(111,137),(113,139),(115,141),(117,143),(119,145)], [(1,49),(2,50),(3,51),(4,52),(5,53),(6,54),(7,55),(8,56),(9,57),(10,58),(11,59),(12,60),(13,61),(14,62),(15,63),(16,64),(17,65),(18,66),(19,67),(20,68),(21,69),(22,70),(23,71),(24,72),(25,73),(26,74),(27,75),(28,76),(29,77),(30,78),(31,79),(32,80),(33,41),(34,42),(35,43),(36,44),(37,45),(38,46),(39,47),(40,48),(81,147),(82,148),(83,149),(84,150),(85,151),(86,152),(87,153),(88,154),(89,155),(90,156),(91,157),(92,158),(93,159),(94,160),(95,121),(96,122),(97,123),(98,124),(99,125),(100,126),(101,127),(102,128),(103,129),(104,130),(105,131),(106,132),(107,133),(108,134),(109,135),(110,136),(111,137),(112,138),(113,139),(114,140),(115,141),(116,142),(117,143),(118,144),(119,145),(120,146)], [(1,21),(2,22),(3,23),(4,24),(5,25),(6,26),(7,27),(8,28),(9,29),(10,30),(11,31),(12,32),(13,33),(14,34),(15,35),(16,36),(17,37),(18,38),(19,39),(20,40),(41,61),(42,62),(43,63),(44,64),(45,65),(46,66),(47,67),(48,68),(49,69),(50,70),(51,71),(52,72),(53,73),(54,74),(55,75),(56,76),(57,77),(58,78),(59,79),(60,80),(81,101),(82,102),(83,103),(84,104),(85,105),(86,106),(87,107),(88,108),(89,109),(90,110),(91,111),(92,112),(93,113),(94,114),(95,115),(96,116),(97,117),(98,118),(99,119),(100,120),(121,141),(122,142),(123,143),(124,144),(125,145),(126,146),(127,147),(128,148),(129,149),(130,150),(131,151),(132,152),(133,153),(134,154),(135,155),(136,156),(137,157),(138,158),(139,159),(140,160)], [(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)], [(1,115,69,121),(2,94,70,140),(3,113,71,159),(4,92,72,138),(5,111,73,157),(6,90,74,136),(7,109,75,155),(8,88,76,134),(9,107,77,153),(10,86,78,132),(11,105,79,151),(12,84,80,130),(13,103,41,149),(14,82,42,128),(15,101,43,147),(16,120,44,126),(17,99,45,145),(18,118,46,124),(19,97,47,143),(20,116,48,122),(21,95,49,141),(22,114,50,160),(23,93,51,139),(24,112,52,158),(25,91,53,137),(26,110,54,156),(27,89,55,135),(28,108,56,154),(29,87,57,133),(30,106,58,152),(31,85,59,131),(32,104,60,150),(33,83,61,129),(34,102,62,148),(35,81,63,127),(36,100,64,146),(37,119,65,125),(38,98,66,144),(39,117,67,123),(40,96,68,142)]])
59 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 5A | 5B | 8A | 8B | 8C | 8D | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 20A | ··· | 20H | 20I | 20J | 20K | 20L | 40A | ··· | 40P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 20 | ··· | 20 | 20 | 20 | 20 | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 20 | 20 | 20 | 20 | 40 | 40 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
59 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | - | - | - | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D5 | C4○D4 | SD16 | D10 | D10 | D20 | D20 | C40⋊C2 | C8.C22 | D4⋊2D5 | C8.D10 |
| kernel | C23.34D20 | C20.44D4 | C40⋊6C4 | C5×C22⋊C8 | C20.48D4 | C2×C4⋊Dic5 | C2×C20 | C22×C10 | C22⋊C8 | C20 | C2×C10 | C2×C8 | C22×C4 | C2×C4 | C23 | C22 | C10 | C4 | C2 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 2 | 4 | 4 | 4 | 2 | 4 | 4 | 16 | 1 | 4 | 4 |
Matrix representation of C23.34D20 ►in GL4(𝔽41) generated by
| 1 | 0 | 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 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 40 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 23 | 16 |
| 0 | 0 | 25 | 12 |
| 0 | 9 | 0 | 0 |
| 9 | 0 | 0 | 0 |
| 0 | 0 | 29 | 27 |
| 0 | 0 | 25 | 12 |
G:=sub<GL(4,GF(41))| [1,0,0,0,0,40,0,0,0,0,1,0,0,0,0,1],[40,0,0,0,0,40,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,40,0,0,0,0,40],[0,1,0,0,1,0,0,0,0,0,23,25,0,0,16,12],[0,9,0,0,9,0,0,0,0,0,29,25,0,0,27,12] >;
C23.34D20 in GAP, Magma, Sage, TeX
C_2^3._{34}D_{20} % in TeX
G:=Group("C2^3.34D20"); // GroupNames label
G:=SmallGroup(320,348);
// by ID
G=gap.SmallGroup(320,348);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,254,219,58,1123,136,12550]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^2=1,d^20=c,e^2=c*b=b*c,d*a*d^-1=e*a*e^-1=a*b=b*a,a*c=c*a,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=d^19>;
// generators/relations