metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C23.10D20, C40⋊5C4⋊3C2, C40⋊6C4⋊7C2, (C2×C8).2D10, C22⋊C8.5D5, C10.7(C4○D8), (C2×C20).239D4, (C2×C4).117D20, (C2×C40).2C22, C20.44D4⋊5C2, (C22×C4).77D10, (C22×C10).50D4, C20.281(C4○D4), C2.9(D40⋊7C2), (C2×C20).740C23, C20.48D4.8C2, C22.103(C2×D20), C10.8(C8.C22), C5⋊1(C23.20D4), C4.105(D4⋊2D5), C2.11(C8.D10), C4⋊Dic5.269C22, (C22×C20).92C22, (C2×Dic10).14C22, C23.21D10.3C2, C10.16(C22.D4), C2.12(C22.D20), (C5×C22⋊C8).7C2, (C2×C10).123(C2×D4), (C2×C4).685(C22×D5), SmallGroup(320,350)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C23.10D20
G = < a,b,c,d,e | a2=b2=c2=1, d20=e2=c, dad-1=ab=ba, ac=ca, eae-1=abc, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede-1=bd19 >
Subgroups: 350 in 96 conjugacy classes, 39 normal (all characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, Q8, C23, C10, C10, C42, C22⋊C4, C4⋊C4, C2×C8, C22×C4, C2×Q8, Dic5, C20, C20, C2×C10, C2×C10, C22⋊C8, Q8⋊C4, C4.Q8, C2.D8, C42⋊C2, C22⋊Q8, C40, Dic10, C2×Dic5, C2×C20, C2×C20, C22×C10, C23.20D4, C4×Dic5, C10.D4, C4⋊Dic5, C23.D5, C2×C40, C2×Dic10, C22×C20, C20.44D4, C40⋊6C4, C40⋊5C4, C5×C22⋊C8, C20.48D4, C23.21D10, C23.10D20
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, C4○D4, D10, C22.D4, C4○D8, C8.C22, D20, C22×D5, C23.20D4, C2×D20, D4⋊2D5, C22.D20, D40⋊7C2, C8.D10, C23.10D20
►On 160 points
Generators in S160
(2 80)(4 42)(6 44)(8 46)(10 48)(12 50)(14 52)(16 54)(18 56)(20 58)(22 60)(24 62)(26 64)(28 66)(30 68)(32 70)(34 72)(36 74)(38 76)(40 78)(81 121)(82 102)(83 123)(84 104)(85 125)(86 106)(87 127)(88 108)(89 129)(90 110)(91 131)(92 112)(93 133)(94 114)(95 135)(96 116)(97 137)(98 118)(99 139)(100 120)(101 141)(103 143)(105 145)(107 147)(109 149)(111 151)(113 153)(115 155)(117 157)(119 159)(122 142)(124 144)(126 146)(128 148)(130 150)(132 152)(134 154)(136 156)(138 158)(140 160)
(1 79)(2 80)(3 41)(4 42)(5 43)(6 44)(7 45)(8 46)(9 47)(10 48)(11 49)(12 50)(13 51)(14 52)(15 53)(16 54)(17 55)(18 56)(19 57)(20 58)(21 59)(22 60)(23 61)(24 62)(25 63)(26 64)(27 65)(28 66)(29 67)(30 68)(31 69)(32 70)(33 71)(34 72)(35 73)(36 74)(37 75)(38 76)(39 77)(40 78)(81 141)(82 142)(83 143)(84 144)(85 145)(86 146)(87 147)(88 148)(89 149)(90 150)(91 151)(92 152)(93 153)(94 154)(95 155)(96 156)(97 157)(98 158)(99 159)(100 160)(101 121)(102 122)(103 123)(104 124)(105 125)(106 126)(107 127)(108 128)(109 129)(110 130)(111 131)(112 132)(113 133)(114 134)(115 135)(116 136)(117 137)(118 138)(119 139)(120 140)
(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 95 21 115)(2 134 22 154)(3 93 23 113)(4 132 24 152)(5 91 25 111)(6 130 26 150)(7 89 27 109)(8 128 28 148)(9 87 29 107)(10 126 30 146)(11 85 31 105)(12 124 32 144)(13 83 33 103)(14 122 34 142)(15 81 35 101)(16 160 36 140)(17 119 37 99)(18 158 38 138)(19 117 39 97)(20 156 40 136)(41 153 61 133)(42 112 62 92)(43 151 63 131)(44 110 64 90)(45 149 65 129)(46 108 66 88)(47 147 67 127)(48 106 68 86)(49 145 69 125)(50 104 70 84)(51 143 71 123)(52 102 72 82)(53 141 73 121)(54 100 74 120)(55 139 75 159)(56 98 76 118)(57 137 77 157)(58 96 78 116)(59 135 79 155)(60 94 80 114)
G:=sub<Sym(160)| (2,80)(4,42)(6,44)(8,46)(10,48)(12,50)(14,52)(16,54)(18,56)(20,58)(22,60)(24,62)(26,64)(28,66)(30,68)(32,70)(34,72)(36,74)(38,76)(40,78)(81,121)(82,102)(83,123)(84,104)(85,125)(86,106)(87,127)(88,108)(89,129)(90,110)(91,131)(92,112)(93,133)(94,114)(95,135)(96,116)(97,137)(98,118)(99,139)(100,120)(101,141)(103,143)(105,145)(107,147)(109,149)(111,151)(113,153)(115,155)(117,157)(119,159)(122,142)(124,144)(126,146)(128,148)(130,150)(132,152)(134,154)(136,156)(138,158)(140,160), (1,79)(2,80)(3,41)(4,42)(5,43)(6,44)(7,45)(8,46)(9,47)(10,48)(11,49)(12,50)(13,51)(14,52)(15,53)(16,54)(17,55)(18,56)(19,57)(20,58)(21,59)(22,60)(23,61)(24,62)(25,63)(26,64)(27,65)(28,66)(29,67)(30,68)(31,69)(32,70)(33,71)(34,72)(35,73)(36,74)(37,75)(38,76)(39,77)(40,78)(81,141)(82,142)(83,143)(84,144)(85,145)(86,146)(87,147)(88,148)(89,149)(90,150)(91,151)(92,152)(93,153)(94,154)(95,155)(96,156)(97,157)(98,158)(99,159)(100,160)(101,121)(102,122)(103,123)(104,124)(105,125)(106,126)(107,127)(108,128)(109,129)(110,130)(111,131)(112,132)(113,133)(114,134)(115,135)(116,136)(117,137)(118,138)(119,139)(120,140), (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), (1,95,21,115)(2,134,22,154)(3,93,23,113)(4,132,24,152)(5,91,25,111)(6,130,26,150)(7,89,27,109)(8,128,28,148)(9,87,29,107)(10,126,30,146)(11,85,31,105)(12,124,32,144)(13,83,33,103)(14,122,34,142)(15,81,35,101)(16,160,36,140)(17,119,37,99)(18,158,38,138)(19,117,39,97)(20,156,40,136)(41,153,61,133)(42,112,62,92)(43,151,63,131)(44,110,64,90)(45,149,65,129)(46,108,66,88)(47,147,67,127)(48,106,68,86)(49,145,69,125)(50,104,70,84)(51,143,71,123)(52,102,72,82)(53,141,73,121)(54,100,74,120)(55,139,75,159)(56,98,76,118)(57,137,77,157)(58,96,78,116)(59,135,79,155)(60,94,80,114)>;
G:=Group( (2,80)(4,42)(6,44)(8,46)(10,48)(12,50)(14,52)(16,54)(18,56)(20,58)(22,60)(24,62)(26,64)(28,66)(30,68)(32,70)(34,72)(36,74)(38,76)(40,78)(81,121)(82,102)(83,123)(84,104)(85,125)(86,106)(87,127)(88,108)(89,129)(90,110)(91,131)(92,112)(93,133)(94,114)(95,135)(96,116)(97,137)(98,118)(99,139)(100,120)(101,141)(103,143)(105,145)(107,147)(109,149)(111,151)(113,153)(115,155)(117,157)(119,159)(122,142)(124,144)(126,146)(128,148)(130,150)(132,152)(134,154)(136,156)(138,158)(140,160), (1,79)(2,80)(3,41)(4,42)(5,43)(6,44)(7,45)(8,46)(9,47)(10,48)(11,49)(12,50)(13,51)(14,52)(15,53)(16,54)(17,55)(18,56)(19,57)(20,58)(21,59)(22,60)(23,61)(24,62)(25,63)(26,64)(27,65)(28,66)(29,67)(30,68)(31,69)(32,70)(33,71)(34,72)(35,73)(36,74)(37,75)(38,76)(39,77)(40,78)(81,141)(82,142)(83,143)(84,144)(85,145)(86,146)(87,147)(88,148)(89,149)(90,150)(91,151)(92,152)(93,153)(94,154)(95,155)(96,156)(97,157)(98,158)(99,159)(100,160)(101,121)(102,122)(103,123)(104,124)(105,125)(106,126)(107,127)(108,128)(109,129)(110,130)(111,131)(112,132)(113,133)(114,134)(115,135)(116,136)(117,137)(118,138)(119,139)(120,140), (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), (1,95,21,115)(2,134,22,154)(3,93,23,113)(4,132,24,152)(5,91,25,111)(6,130,26,150)(7,89,27,109)(8,128,28,148)(9,87,29,107)(10,126,30,146)(11,85,31,105)(12,124,32,144)(13,83,33,103)(14,122,34,142)(15,81,35,101)(16,160,36,140)(17,119,37,99)(18,158,38,138)(19,117,39,97)(20,156,40,136)(41,153,61,133)(42,112,62,92)(43,151,63,131)(44,110,64,90)(45,149,65,129)(46,108,66,88)(47,147,67,127)(48,106,68,86)(49,145,69,125)(50,104,70,84)(51,143,71,123)(52,102,72,82)(53,141,73,121)(54,100,74,120)(55,139,75,159)(56,98,76,118)(57,137,77,157)(58,96,78,116)(59,135,79,155)(60,94,80,114) );
G=PermutationGroup([[(2,80),(4,42),(6,44),(8,46),(10,48),(12,50),(14,52),(16,54),(18,56),(20,58),(22,60),(24,62),(26,64),(28,66),(30,68),(32,70),(34,72),(36,74),(38,76),(40,78),(81,121),(82,102),(83,123),(84,104),(85,125),(86,106),(87,127),(88,108),(89,129),(90,110),(91,131),(92,112),(93,133),(94,114),(95,135),(96,116),(97,137),(98,118),(99,139),(100,120),(101,141),(103,143),(105,145),(107,147),(109,149),(111,151),(113,153),(115,155),(117,157),(119,159),(122,142),(124,144),(126,146),(128,148),(130,150),(132,152),(134,154),(136,156),(138,158),(140,160)], [(1,79),(2,80),(3,41),(4,42),(5,43),(6,44),(7,45),(8,46),(9,47),(10,48),(11,49),(12,50),(13,51),(14,52),(15,53),(16,54),(17,55),(18,56),(19,57),(20,58),(21,59),(22,60),(23,61),(24,62),(25,63),(26,64),(27,65),(28,66),(29,67),(30,68),(31,69),(32,70),(33,71),(34,72),(35,73),(36,74),(37,75),(38,76),(39,77),(40,78),(81,141),(82,142),(83,143),(84,144),(85,145),(86,146),(87,147),(88,148),(89,149),(90,150),(91,151),(92,152),(93,153),(94,154),(95,155),(96,156),(97,157),(98,158),(99,159),(100,160),(101,121),(102,122),(103,123),(104,124),(105,125),(106,126),(107,127),(108,128),(109,129),(110,130),(111,131),(112,132),(113,133),(114,134),(115,135),(116,136),(117,137),(118,138),(119,139),(120,140)], [(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)], [(1,95,21,115),(2,134,22,154),(3,93,23,113),(4,132,24,152),(5,91,25,111),(6,130,26,150),(7,89,27,109),(8,128,28,148),(9,87,29,107),(10,126,30,146),(11,85,31,105),(12,124,32,144),(13,83,33,103),(14,122,34,142),(15,81,35,101),(16,160,36,140),(17,119,37,99),(18,158,38,138),(19,117,39,97),(20,156,40,136),(41,153,61,133),(42,112,62,92),(43,151,63,131),(44,110,64,90),(45,149,65,129),(46,108,66,88),(47,147,67,127),(48,106,68,86),(49,145,69,125),(50,104,70,84),(51,143,71,123),(52,102,72,82),(53,141,73,121),(54,100,74,120),(55,139,75,159),(56,98,76,118),(57,137,77,157),(58,96,78,116),(59,135,79,155),(60,94,80,114)]])
59 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 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 | 4 | 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 | 4 | 2 | 2 | 2 | 2 | 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 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | - | - | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D5 | C4○D4 | D10 | D10 | C4○D8 | D20 | D20 | D40⋊7C2 | C8.C22 | D4⋊2D5 | C8.D10 |
| kernel | C23.10D20 | C20.44D4 | C40⋊6C4 | C40⋊5C4 | C5×C22⋊C8 | C20.48D4 | C23.21D10 | C2×C20 | C22×C10 | C22⋊C8 | C20 | C2×C8 | C22×C4 | C10 | C2×C4 | C23 | C2 | C10 | C4 | C2 |
| # reps | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 4 | 4 | 2 | 4 | 4 | 4 | 16 | 1 | 4 | 4 |
Matrix representation of C23.10D20 ►in GL4(𝔽41) generated by
| 1 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 40 | 40 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 40 |
| 40 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 24 | 0 | 0 | 0 |
| 0 | 29 | 0 | 0 |
| 0 | 0 | 40 | 39 |
| 0 | 0 | 1 | 1 |
| 0 | 29 | 0 | 0 |
| 24 | 0 | 0 | 0 |
| 0 | 0 | 9 | 18 |
| 0 | 0 | 32 | 32 |
G:=sub<GL(4,GF(41))| [1,0,0,0,0,40,0,0,0,0,1,40,0,0,0,40],[1,0,0,0,0,1,0,0,0,0,40,0,0,0,0,40],[40,0,0,0,0,40,0,0,0,0,1,0,0,0,0,1],[24,0,0,0,0,29,0,0,0,0,40,1,0,0,39,1],[0,24,0,0,29,0,0,0,0,0,9,32,0,0,18,32] >;
C23.10D20 in GAP, Magma, Sage, TeX
C_2^3._{10}D_{20} % in TeX
G:=Group("C2^3.10D20"); // GroupNames label
G:=SmallGroup(320,350);
// by ID
G=gap.SmallGroup(320,350);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,232,254,219,142,1123,136,12550]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^2=1,d^20=e^2=c,d*a*d^-1=a*b=b*a,a*c=c*a,e*a*e^-1=a*b*c,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=b*d^19>;
// generators/relations