metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: (C2×C20).56D4, (C2×C4).45D20, (C22×D5).5Q8, C22.51(Q8×D5), (C2×Dic5).70D4, C22.250(D4×D5), C10.63(C4⋊D4), C2.26(C4⋊D20), C2.11(C20⋊7D4), C5⋊3(C23.Q8), C2.9(D10⋊3Q8), C22.130(C2×D20), (C22×C4).106D10, C10.51(C22⋊Q8), C2.23(D10⋊Q8), C2.17(D10⋊2Q8), (C22×C20).69C22, (C23×D5).23C22, C23.383(C22×D5), C10.10C42⋊21C2, C10.30(C42⋊2C2), C2.13(Dic5⋊D4), C22.111(C4○D20), (C22×C10).358C23, C22.54(Q8⋊2D5), C22.105(D4⋊2D5), (C22×Dic5).63C22, (C2×C4⋊C4)⋊12D5, (C10×C4⋊C4)⋊23C2, (C2×C4⋊Dic5)⋊14C2, (C2×C10).86(C2×Q8), (C2×C10).338(C2×D4), (C2×C4).44(C5⋊D4), C2.15(C4⋊C4⋊D5), (C2×C10).87(C4○D4), (C2×C10.D4)⋊42C2, C22.143(C2×C5⋊D4), (C2×D10⋊C4).26C2, SmallGroup(320,621)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for (C2×C20).56D4
G = < a,b,c,d | a2=b20=c4=d2=1, ab=ba, ac=ca, ad=da, cbc-1=b-1, dbd=ab-1, dcd=ac-1 >
Subgroups: 774 in 186 conjugacy classes, 63 normal (51 characteristic)
C1, C2, C2, C4, C22, C22, C5, C2×C4, C2×C4, C23, C23, D5, C10, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C24, Dic5, C20, D10, C2×C10, C2.C42, C2×C22⋊C4, C2×C4⋊C4, C2×C4⋊C4, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C22×D5, C22×D5, C22×C10, C23.Q8, C10.D4, C4⋊Dic5, D10⋊C4, C5×C4⋊C4, C22×Dic5, C22×C20, C23×D5, C10.10C42, C2×C10.D4, C2×C4⋊Dic5, C2×D10⋊C4, C10×C4⋊C4, (C2×C20).56D4
Quotients: C1, C2, C22, D4, Q8, C23, D5, C2×D4, C2×Q8, C4○D4, D10, C4⋊D4, C22⋊Q8, C42⋊2C2, D20, C5⋊D4, C22×D5, C23.Q8, C2×D20, C4○D20, D4×D5, D4⋊2D5, Q8×D5, Q8⋊2D5, C2×C5⋊D4, C4⋊D20, D10⋊Q8, D10⋊2Q8, C4⋊C4⋊D5, C20⋊7D4, Dic5⋊D4, D10⋊3Q8, (C2×C20).56D4
►On 160 points
Generators in S160
(1 60)(2 41)(3 42)(4 43)(5 44)(6 45)(7 46)(8 47)(9 48)(10 49)(11 50)(12 51)(13 52)(14 53)(15 54)(16 55)(17 56)(18 57)(19 58)(20 59)(21 64)(22 65)(23 66)(24 67)(25 68)(26 69)(27 70)(28 71)(29 72)(30 73)(31 74)(32 75)(33 76)(34 77)(35 78)(36 79)(37 80)(38 61)(39 62)(40 63)(81 154)(82 155)(83 156)(84 157)(85 158)(86 159)(87 160)(88 141)(89 142)(90 143)(91 144)(92 145)(93 146)(94 147)(95 148)(96 149)(97 150)(98 151)(99 152)(100 153)(101 137)(102 138)(103 139)(104 140)(105 121)(106 122)(107 123)(108 124)(109 125)(110 126)(111 127)(112 128)(113 129)(114 130)(115 131)(116 132)(117 133)(118 134)(119 135)(120 136)
(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 96 23 119)(2 95 24 118)(3 94 25 117)(4 93 26 116)(5 92 27 115)(6 91 28 114)(7 90 29 113)(8 89 30 112)(9 88 31 111)(10 87 32 110)(11 86 33 109)(12 85 34 108)(13 84 35 107)(14 83 36 106)(15 82 37 105)(16 81 38 104)(17 100 39 103)(18 99 40 102)(19 98 21 101)(20 97 22 120)(41 148 67 134)(42 147 68 133)(43 146 69 132)(44 145 70 131)(45 144 71 130)(46 143 72 129)(47 142 73 128)(48 141 74 127)(49 160 75 126)(50 159 76 125)(51 158 77 124)(52 157 78 123)(53 156 79 122)(54 155 80 121)(55 154 61 140)(56 153 62 139)(57 152 63 138)(58 151 64 137)(59 150 65 136)(60 149 66 135)
(2 59)(3 19)(4 57)(5 17)(6 55)(7 15)(8 53)(9 13)(10 51)(12 49)(14 47)(16 45)(18 43)(20 41)(21 25)(22 67)(24 65)(26 63)(27 39)(28 61)(29 37)(30 79)(31 35)(32 77)(34 75)(36 73)(38 71)(40 69)(42 58)(44 56)(46 54)(48 52)(62 70)(64 68)(72 80)(74 78)(81 114)(82 129)(83 112)(84 127)(85 110)(86 125)(87 108)(88 123)(89 106)(90 121)(91 104)(92 139)(93 102)(94 137)(95 120)(96 135)(97 118)(98 133)(99 116)(100 131)(101 147)(103 145)(105 143)(107 141)(109 159)(111 157)(113 155)(115 153)(117 151)(119 149)(122 142)(124 160)(126 158)(128 156)(130 154)(132 152)(134 150)(136 148)(138 146)(140 144)
G:=sub<Sym(160)| (1,60)(2,41)(3,42)(4,43)(5,44)(6,45)(7,46)(8,47)(9,48)(10,49)(11,50)(12,51)(13,52)(14,53)(15,54)(16,55)(17,56)(18,57)(19,58)(20,59)(21,64)(22,65)(23,66)(24,67)(25,68)(26,69)(27,70)(28,71)(29,72)(30,73)(31,74)(32,75)(33,76)(34,77)(35,78)(36,79)(37,80)(38,61)(39,62)(40,63)(81,154)(82,155)(83,156)(84,157)(85,158)(86,159)(87,160)(88,141)(89,142)(90,143)(91,144)(92,145)(93,146)(94,147)(95,148)(96,149)(97,150)(98,151)(99,152)(100,153)(101,137)(102,138)(103,139)(104,140)(105,121)(106,122)(107,123)(108,124)(109,125)(110,126)(111,127)(112,128)(113,129)(114,130)(115,131)(116,132)(117,133)(118,134)(119,135)(120,136), (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), (1,96,23,119)(2,95,24,118)(3,94,25,117)(4,93,26,116)(5,92,27,115)(6,91,28,114)(7,90,29,113)(8,89,30,112)(9,88,31,111)(10,87,32,110)(11,86,33,109)(12,85,34,108)(13,84,35,107)(14,83,36,106)(15,82,37,105)(16,81,38,104)(17,100,39,103)(18,99,40,102)(19,98,21,101)(20,97,22,120)(41,148,67,134)(42,147,68,133)(43,146,69,132)(44,145,70,131)(45,144,71,130)(46,143,72,129)(47,142,73,128)(48,141,74,127)(49,160,75,126)(50,159,76,125)(51,158,77,124)(52,157,78,123)(53,156,79,122)(54,155,80,121)(55,154,61,140)(56,153,62,139)(57,152,63,138)(58,151,64,137)(59,150,65,136)(60,149,66,135), (2,59)(3,19)(4,57)(5,17)(6,55)(7,15)(8,53)(9,13)(10,51)(12,49)(14,47)(16,45)(18,43)(20,41)(21,25)(22,67)(24,65)(26,63)(27,39)(28,61)(29,37)(30,79)(31,35)(32,77)(34,75)(36,73)(38,71)(40,69)(42,58)(44,56)(46,54)(48,52)(62,70)(64,68)(72,80)(74,78)(81,114)(82,129)(83,112)(84,127)(85,110)(86,125)(87,108)(88,123)(89,106)(90,121)(91,104)(92,139)(93,102)(94,137)(95,120)(96,135)(97,118)(98,133)(99,116)(100,131)(101,147)(103,145)(105,143)(107,141)(109,159)(111,157)(113,155)(115,153)(117,151)(119,149)(122,142)(124,160)(126,158)(128,156)(130,154)(132,152)(134,150)(136,148)(138,146)(140,144)>;
G:=Group( (1,60)(2,41)(3,42)(4,43)(5,44)(6,45)(7,46)(8,47)(9,48)(10,49)(11,50)(12,51)(13,52)(14,53)(15,54)(16,55)(17,56)(18,57)(19,58)(20,59)(21,64)(22,65)(23,66)(24,67)(25,68)(26,69)(27,70)(28,71)(29,72)(30,73)(31,74)(32,75)(33,76)(34,77)(35,78)(36,79)(37,80)(38,61)(39,62)(40,63)(81,154)(82,155)(83,156)(84,157)(85,158)(86,159)(87,160)(88,141)(89,142)(90,143)(91,144)(92,145)(93,146)(94,147)(95,148)(96,149)(97,150)(98,151)(99,152)(100,153)(101,137)(102,138)(103,139)(104,140)(105,121)(106,122)(107,123)(108,124)(109,125)(110,126)(111,127)(112,128)(113,129)(114,130)(115,131)(116,132)(117,133)(118,134)(119,135)(120,136), (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), (1,96,23,119)(2,95,24,118)(3,94,25,117)(4,93,26,116)(5,92,27,115)(6,91,28,114)(7,90,29,113)(8,89,30,112)(9,88,31,111)(10,87,32,110)(11,86,33,109)(12,85,34,108)(13,84,35,107)(14,83,36,106)(15,82,37,105)(16,81,38,104)(17,100,39,103)(18,99,40,102)(19,98,21,101)(20,97,22,120)(41,148,67,134)(42,147,68,133)(43,146,69,132)(44,145,70,131)(45,144,71,130)(46,143,72,129)(47,142,73,128)(48,141,74,127)(49,160,75,126)(50,159,76,125)(51,158,77,124)(52,157,78,123)(53,156,79,122)(54,155,80,121)(55,154,61,140)(56,153,62,139)(57,152,63,138)(58,151,64,137)(59,150,65,136)(60,149,66,135), (2,59)(3,19)(4,57)(5,17)(6,55)(7,15)(8,53)(9,13)(10,51)(12,49)(14,47)(16,45)(18,43)(20,41)(21,25)(22,67)(24,65)(26,63)(27,39)(28,61)(29,37)(30,79)(31,35)(32,77)(34,75)(36,73)(38,71)(40,69)(42,58)(44,56)(46,54)(48,52)(62,70)(64,68)(72,80)(74,78)(81,114)(82,129)(83,112)(84,127)(85,110)(86,125)(87,108)(88,123)(89,106)(90,121)(91,104)(92,139)(93,102)(94,137)(95,120)(96,135)(97,118)(98,133)(99,116)(100,131)(101,147)(103,145)(105,143)(107,141)(109,159)(111,157)(113,155)(115,153)(117,151)(119,149)(122,142)(124,160)(126,158)(128,156)(130,154)(132,152)(134,150)(136,148)(138,146)(140,144) );
G=PermutationGroup([[(1,60),(2,41),(3,42),(4,43),(5,44),(6,45),(7,46),(8,47),(9,48),(10,49),(11,50),(12,51),(13,52),(14,53),(15,54),(16,55),(17,56),(18,57),(19,58),(20,59),(21,64),(22,65),(23,66),(24,67),(25,68),(26,69),(27,70),(28,71),(29,72),(30,73),(31,74),(32,75),(33,76),(34,77),(35,78),(36,79),(37,80),(38,61),(39,62),(40,63),(81,154),(82,155),(83,156),(84,157),(85,158),(86,159),(87,160),(88,141),(89,142),(90,143),(91,144),(92,145),(93,146),(94,147),(95,148),(96,149),(97,150),(98,151),(99,152),(100,153),(101,137),(102,138),(103,139),(104,140),(105,121),(106,122),(107,123),(108,124),(109,125),(110,126),(111,127),(112,128),(113,129),(114,130),(115,131),(116,132),(117,133),(118,134),(119,135),(120,136)], [(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)], [(1,96,23,119),(2,95,24,118),(3,94,25,117),(4,93,26,116),(5,92,27,115),(6,91,28,114),(7,90,29,113),(8,89,30,112),(9,88,31,111),(10,87,32,110),(11,86,33,109),(12,85,34,108),(13,84,35,107),(14,83,36,106),(15,82,37,105),(16,81,38,104),(17,100,39,103),(18,99,40,102),(19,98,21,101),(20,97,22,120),(41,148,67,134),(42,147,68,133),(43,146,69,132),(44,145,70,131),(45,144,71,130),(46,143,72,129),(47,142,73,128),(48,141,74,127),(49,160,75,126),(50,159,76,125),(51,158,77,124),(52,157,78,123),(53,156,79,122),(54,155,80,121),(55,154,61,140),(56,153,62,139),(57,152,63,138),(58,151,64,137),(59,150,65,136),(60,149,66,135)], [(2,59),(3,19),(4,57),(5,17),(6,55),(7,15),(8,53),(9,13),(10,51),(12,49),(14,47),(16,45),(18,43),(20,41),(21,25),(22,67),(24,65),(26,63),(27,39),(28,61),(29,37),(30,79),(31,35),(32,77),(34,75),(36,73),(38,71),(40,69),(42,58),(44,56),(46,54),(48,52),(62,70),(64,68),(72,80),(74,78),(81,114),(82,129),(83,112),(84,127),(85,110),(86,125),(87,108),(88,123),(89,106),(90,121),(91,104),(92,139),(93,102),(94,137),(95,120),(96,135),(97,118),(98,133),(99,116),(100,131),(101,147),(103,145),(105,143),(107,141),(109,159),(111,157),(113,155),(115,153),(117,151),(119,149),(122,142),(124,160),(126,158),(128,156),(130,154),(132,152),(134,150),(136,148),(138,146),(140,144)]])
62 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 4A | ··· | 4F | 4G | ··· | 4L | 5A | 5B | 10A | ··· | 10N | 20A | ··· | 20X |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 5 | 5 | 10 | ··· | 10 | 20 | ··· | 20 |
| size | 1 | 1 | ··· | 1 | 20 | 20 | 4 | ··· | 4 | 20 | ··· | 20 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 |
62 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | - | + | + | + | + | - | - | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | Q8 | D5 | C4○D4 | D10 | D20 | C5⋊D4 | C4○D20 | D4×D5 | D4⋊2D5 | Q8×D5 | Q8⋊2D5 |
| kernel | (C2×C20).56D4 | C10.10C42 | C2×C10.D4 | C2×C4⋊Dic5 | C2×D10⋊C4 | C10×C4⋊C4 | C2×Dic5 | C2×C20 | C22×D5 | C2×C4⋊C4 | C2×C10 | C22×C4 | C2×C4 | C2×C4 | C22 | C22 | C22 | C22 | C22 |
| # reps | 1 | 1 | 1 | 1 | 3 | 1 | 2 | 4 | 2 | 2 | 6 | 6 | 8 | 8 | 8 | 2 | 2 | 2 | 2 |
Matrix representation of (C2×C20).56D4 ►in GL6(𝔽41)
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 35 | 39 | 0 | 0 | 0 | 0 |
| 39 | 6 | 0 | 0 | 0 | 0 |
| 0 | 0 | 14 | 2 | 0 | 0 |
| 0 | 0 | 25 | 30 | 0 | 0 |
| 0 | 0 | 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 0 | 0 | 32 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 18 | 0 | 0 |
| 0 | 0 | 4 | 38 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 34 | 1 | 0 | 0 |
| 0 | 0 | 34 | 7 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[35,39,0,0,0,0,39,6,0,0,0,0,0,0,14,25,0,0,0,0,2,30,0,0,0,0,0,0,9,0,0,0,0,0,0,32],[0,40,0,0,0,0,1,0,0,0,0,0,0,0,3,4,0,0,0,0,18,38,0,0,0,0,0,0,0,40,0,0,0,0,40,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,34,34,0,0,0,0,1,7,0,0,0,0,0,0,1,0,0,0,0,0,0,40] >;
(C2×C20).56D4 in GAP, Magma, Sage, TeX
(C_2\times C_{20})._{56}D_4 % in TeX
G:=Group("(C2xC20).56D4"); // GroupNames label
G:=SmallGroup(320,621);
// by ID
G=gap.SmallGroup(320,621);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,253,344,254,387,184,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^20=c^4=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^-1,d*b*d=a*b^-1,d*c*d=a*c^-1>;
// generators/relations