metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C23.22D20, (C2×C40)⋊20C4, (C2×C8)⋊6Dic5, C40⋊6C4⋊29C2, C40⋊5C4⋊29C2, C20.78(C4⋊C4), (C2×C20).61Q8, C20.75(C2×Q8), C40.114(C2×C4), (C2×C4).169D20, (C2×C20).401D4, (C2×C8).307D10, C8.17(C2×Dic5), (C22×C8).11D5, C10.15(C4○D8), (C22×C40).17C2, C4.23(C4⋊Dic5), (C2×C4).50Dic10, C4.41(C2×Dic10), C22.52(C2×D20), C2.4(D40⋊7C2), (C2×C20).765C23, C20.230(C22×C4), (C2×C40).392C22, (C22×C4).425D10, (C22×C10).137D4, C5⋊5(C23.25D4), C4.25(C22×Dic5), C4⋊Dic5.281C22, C22.13(C4⋊Dic5), (C22×C20).539C22, C23.21D10.5C2, C10.69(C2×C4⋊C4), C2.12(C2×C4⋊Dic5), (C2×C10).79(C4⋊C4), (C2×C20).479(C2×C4), (C2×C10).155(C2×D4), (C2×C4).83(C2×Dic5), (C2×C4).712(C22×D5), SmallGroup(320,733)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C23.22D20
G = < a,b,c,d,e | a2=b2=c2=1, d20=c, e2=cb=bc, ab=ba, eae-1=ac=ca, ad=da, bd=db, be=eb, cd=dc, ce=ec, ede-1=d19 >
Subgroups: 334 in 114 conjugacy classes, 71 normal (25 characteristic)
C1, C2, C2, C2, C4, C4, C4, C22, C22, C22, C5, C8, C2×C4, C2×C4, C2×C4, C23, C10, C10, C10, C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, C22×C4, Dic5, C20, C20, C2×C10, C2×C10, C2×C10, C4.Q8, C2.D8, C42⋊C2, C22×C8, C40, C2×Dic5, C2×C20, C2×C20, C22×C10, C23.25D4, C4×Dic5, C4⋊Dic5, C23.D5, C2×C40, C2×C40, C22×C20, C40⋊6C4, C40⋊5C4, C23.21D10, C22×C40, C23.22D20
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, D5, C4⋊C4, C22×C4, C2×D4, C2×Q8, Dic5, D10, C2×C4⋊C4, C4○D8, Dic10, D20, C2×Dic5, C22×D5, C23.25D4, C4⋊Dic5, C2×Dic10, C2×D20, C22×Dic5, D40⋊7C2, C2×C4⋊Dic5, C23.22D20
►On 160 points
Generators in S160
(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 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)(81 152)(82 153)(83 154)(84 155)(85 156)(86 157)(87 158)(88 159)(89 160)(90 121)(91 122)(92 123)(93 124)(94 125)(95 126)(96 127)(97 128)(98 129)(99 130)(100 131)(101 132)(102 133)(103 134)(104 135)(105 136)(106 137)(107 138)(108 139)(109 140)(110 141)(111 142)(112 143)(113 144)(114 145)(115 146)(116 147)(117 148)(118 149)(119 150)(120 151)
(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)
(1 135 59 84)(2 154 60 103)(3 133 61 82)(4 152 62 101)(5 131 63 120)(6 150 64 99)(7 129 65 118)(8 148 66 97)(9 127 67 116)(10 146 68 95)(11 125 69 114)(12 144 70 93)(13 123 71 112)(14 142 72 91)(15 121 73 110)(16 140 74 89)(17 159 75 108)(18 138 76 87)(19 157 77 106)(20 136 78 85)(21 155 79 104)(22 134 80 83)(23 153 41 102)(24 132 42 81)(25 151 43 100)(26 130 44 119)(27 149 45 98)(28 128 46 117)(29 147 47 96)(30 126 48 115)(31 145 49 94)(32 124 50 113)(33 143 51 92)(34 122 52 111)(35 141 53 90)(36 160 54 109)(37 139 55 88)(38 158 56 107)(39 137 57 86)(40 156 58 105)
G:=sub<Sym(160)| (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), (1,79)(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)(81,152)(82,153)(83,154)(84,155)(85,156)(86,157)(87,158)(88,159)(89,160)(90,121)(91,122)(92,123)(93,124)(94,125)(95,126)(96,127)(97,128)(98,129)(99,130)(100,131)(101,132)(102,133)(103,134)(104,135)(105,136)(106,137)(107,138)(108,139)(109,140)(110,141)(111,142)(112,143)(113,144)(114,145)(115,146)(116,147)(117,148)(118,149)(119,150)(120,151), (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), (1,135,59,84)(2,154,60,103)(3,133,61,82)(4,152,62,101)(5,131,63,120)(6,150,64,99)(7,129,65,118)(8,148,66,97)(9,127,67,116)(10,146,68,95)(11,125,69,114)(12,144,70,93)(13,123,71,112)(14,142,72,91)(15,121,73,110)(16,140,74,89)(17,159,75,108)(18,138,76,87)(19,157,77,106)(20,136,78,85)(21,155,79,104)(22,134,80,83)(23,153,41,102)(24,132,42,81)(25,151,43,100)(26,130,44,119)(27,149,45,98)(28,128,46,117)(29,147,47,96)(30,126,48,115)(31,145,49,94)(32,124,50,113)(33,143,51,92)(34,122,52,111)(35,141,53,90)(36,160,54,109)(37,139,55,88)(38,158,56,107)(39,137,57,86)(40,156,58,105)>;
G:=Group( (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), (1,79)(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)(81,152)(82,153)(83,154)(84,155)(85,156)(86,157)(87,158)(88,159)(89,160)(90,121)(91,122)(92,123)(93,124)(94,125)(95,126)(96,127)(97,128)(98,129)(99,130)(100,131)(101,132)(102,133)(103,134)(104,135)(105,136)(106,137)(107,138)(108,139)(109,140)(110,141)(111,142)(112,143)(113,144)(114,145)(115,146)(116,147)(117,148)(118,149)(119,150)(120,151), (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), (1,135,59,84)(2,154,60,103)(3,133,61,82)(4,152,62,101)(5,131,63,120)(6,150,64,99)(7,129,65,118)(8,148,66,97)(9,127,67,116)(10,146,68,95)(11,125,69,114)(12,144,70,93)(13,123,71,112)(14,142,72,91)(15,121,73,110)(16,140,74,89)(17,159,75,108)(18,138,76,87)(19,157,77,106)(20,136,78,85)(21,155,79,104)(22,134,80,83)(23,153,41,102)(24,132,42,81)(25,151,43,100)(26,130,44,119)(27,149,45,98)(28,128,46,117)(29,147,47,96)(30,126,48,115)(31,145,49,94)(32,124,50,113)(33,143,51,92)(34,122,52,111)(35,141,53,90)(36,160,54,109)(37,139,55,88)(38,158,56,107)(39,137,57,86)(40,156,58,105) );
G=PermutationGroup([[(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)], [(1,79),(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),(81,152),(82,153),(83,154),(84,155),(85,156),(86,157),(87,158),(88,159),(89,160),(90,121),(91,122),(92,123),(93,124),(94,125),(95,126),(96,127),(97,128),(98,129),(99,130),(100,131),(101,132),(102,133),(103,134),(104,135),(105,136),(106,137),(107,138),(108,139),(109,140),(110,141),(111,142),(112,143),(113,144),(114,145),(115,146),(116,147),(117,148),(118,149),(119,150),(120,151)], [(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)], [(1,135,59,84),(2,154,60,103),(3,133,61,82),(4,152,62,101),(5,131,63,120),(6,150,64,99),(7,129,65,118),(8,148,66,97),(9,127,67,116),(10,146,68,95),(11,125,69,114),(12,144,70,93),(13,123,71,112),(14,142,72,91),(15,121,73,110),(16,140,74,89),(17,159,75,108),(18,138,76,87),(19,157,77,106),(20,136,78,85),(21,155,79,104),(22,134,80,83),(23,153,41,102),(24,132,42,81),(25,151,43,100),(26,130,44,119),(27,149,45,98),(28,128,46,117),(29,147,47,96),(30,126,48,115),(31,145,49,94),(32,124,50,113),(33,143,51,92),(34,122,52,111),(35,141,53,90),(36,160,54,109),(37,139,55,88),(38,158,56,107),(39,137,57,86),(40,156,58,105)]])
92 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | ··· | 4N | 5A | 5B | 8A | ··· | 8H | 10A | ··· | 10N | 20A | ··· | 20P | 40A | ··· | 40AF |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 5 | 5 | 8 | ··· | 8 | 10 | ··· | 10 | 20 | ··· | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 20 | ··· | 20 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
92 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | - | + | + | - | + | + | - | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C4 | D4 | Q8 | D4 | D5 | Dic5 | D10 | D10 | C4○D8 | Dic10 | D20 | D20 | D40⋊7C2 |
| kernel | C23.22D20 | C40⋊6C4 | C40⋊5C4 | C23.21D10 | C22×C40 | C2×C40 | C2×C20 | C2×C20 | C22×C10 | C22×C8 | C2×C8 | C2×C8 | C22×C4 | C10 | C2×C4 | C2×C4 | C23 | C2 |
| # reps | 1 | 2 | 2 | 2 | 1 | 8 | 1 | 2 | 1 | 2 | 8 | 4 | 2 | 8 | 8 | 4 | 4 | 32 |
Matrix representation of C23.22D20 ►in GL3(𝔽41) generated by
| 40 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 40 |
| 40 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 1 |
| 1 | 0 | 0 |
| 0 | 40 | 0 |
| 0 | 0 | 40 |
| 40 | 0 | 0 |
| 0 | 19 | 0 |
| 0 | 0 | 28 |
| 32 | 0 | 0 |
| 0 | 0 | 28 |
| 0 | 19 | 0 |
G:=sub<GL(3,GF(41))| [40,0,0,0,1,0,0,0,40],[40,0,0,0,1,0,0,0,1],[1,0,0,0,40,0,0,0,40],[40,0,0,0,19,0,0,0,28],[32,0,0,0,0,19,0,28,0] >;
C23.22D20 in GAP, Magma, Sage, TeX
C_2^3._{22}D_{20} % in TeX
G:=Group("C2^3.22D20"); // GroupNames label
G:=SmallGroup(320,733);
// by ID
G=gap.SmallGroup(320,733);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,232,422,100,1684,102,12550]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^2=1,d^20=c,e^2=c*b=b*c,a*b=b*a,e*a*e^-1=a*c=c*a,a*d=d*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