metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: (C2×C4)⋊6D20, (C2×C20)⋊28D4, (C2×C42)⋊6D5, (C2×D20)⋊18C4, C10.48(C4×D4), C2.19(C4×D20), C20⋊7(C22⋊C4), C4⋊2(D10⋊C4), C2.2(C20⋊4D4), C2.2(C20⋊7D4), (C22×D20).4C2, C22.39(C2×D20), C10.10(C4⋊1D4), C10.56(C4⋊D4), C2.3(C4.D20), (C22×C4).399D10, C10.12(C4.4D4), C22.48(C4○D20), C5⋊2(C24.3C22), (C23×D5).11C22, C23.272(C22×D5), (C22×C20).478C22, (C22×C10).314C23, (C22×Dic5).32C22, (C2×C4×C20)⋊7C2, (C2×C4⋊Dic5)⋊7C2, (C2×C4).111(C4×D5), (C2×D10⋊C4)⋊2C2, C22.119(C2×C4×D5), (C2×C20).398(C2×C4), (C2×C10).428(C2×D4), C2.6(C2×D10⋊C4), C10.74(C2×C22⋊C4), C22.43(C2×C5⋊D4), (C2×C10).73(C4○D4), (C2×C4).239(C5⋊D4), (C22×D5).23(C2×C4), (C2×C10).201(C22×C4), SmallGroup(320,566)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for (C2×C4)⋊6D20
G = < a,b,c,d | a2=b4=c20=d2=1, dbd=ab=ba, ac=ca, ad=da, bc=cb, dcd=c-1 >
Subgroups: 1182 in 258 conjugacy classes, 87 normal (23 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C5, C2×C4, C2×C4, D4, C23, C23, D5, C10, C10, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C22×C4, C2×D4, C24, Dic5, C20, C20, D10, C2×C10, C2×C10, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C22×D4, D20, C2×Dic5, C2×C20, C2×C20, C22×D5, C22×D5, C22×C10, C24.3C22, C4⋊Dic5, D10⋊C4, C4×C20, C2×D20, C2×D20, C22×Dic5, C22×C20, C22×C20, C23×D5, C2×C4⋊Dic5, C2×D10⋊C4, C2×C4×C20, C22×D20, (C2×C4)⋊6D20
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D5, C22⋊C4, C22×C4, C2×D4, C4○D4, D10, C2×C22⋊C4, C4×D4, C4⋊D4, C4.4D4, C4⋊1D4, C4×D5, D20, C5⋊D4, C22×D5, C24.3C22, D10⋊C4, C2×C4×D5, C2×D20, C4○D20, C2×C5⋊D4, C4×D20, C20⋊4D4, C4.D20, C2×D10⋊C4, C20⋊7D4, (C2×C4)⋊6D20
►On 160 points
Generators in S160
(1 44)(2 45)(3 46)(4 47)(5 48)(6 49)(7 50)(8 51)(9 52)(10 53)(11 54)(12 55)(13 56)(14 57)(15 58)(16 59)(17 60)(18 41)(19 42)(20 43)(21 117)(22 118)(23 119)(24 120)(25 101)(26 102)(27 103)(28 104)(29 105)(30 106)(31 107)(32 108)(33 109)(34 110)(35 111)(36 112)(37 113)(38 114)(39 115)(40 116)(61 144)(62 145)(63 146)(64 147)(65 148)(66 149)(67 150)(68 151)(69 152)(70 153)(71 154)(72 155)(73 156)(74 157)(75 158)(76 159)(77 160)(78 141)(79 142)(80 143)(81 134)(82 135)(83 136)(84 137)(85 138)(86 139)(87 140)(88 121)(89 122)(90 123)(91 124)(92 125)(93 126)(94 127)(95 128)(96 129)(97 130)(98 131)(99 132)(100 133)
(1 104 125 149)(2 105 126 150)(3 106 127 151)(4 107 128 152)(5 108 129 153)(6 109 130 154)(7 110 131 155)(8 111 132 156)(9 112 133 157)(10 113 134 158)(11 114 135 159)(12 115 136 160)(13 116 137 141)(14 117 138 142)(15 118 139 143)(16 119 140 144)(17 120 121 145)(18 101 122 146)(19 102 123 147)(20 103 124 148)(21 85 79 57)(22 86 80 58)(23 87 61 59)(24 88 62 60)(25 89 63 41)(26 90 64 42)(27 91 65 43)(28 92 66 44)(29 93 67 45)(30 94 68 46)(31 95 69 47)(32 96 70 48)(33 97 71 49)(34 98 72 50)(35 99 73 51)(36 100 74 52)(37 81 75 53)(38 82 76 54)(39 83 77 55)(40 84 78 56)
(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 48)(2 47)(3 46)(4 45)(5 44)(6 43)(7 42)(8 41)(9 60)(10 59)(11 58)(12 57)(13 56)(14 55)(15 54)(16 53)(17 52)(18 51)(19 50)(20 49)(21 39)(22 38)(23 37)(24 36)(25 35)(26 34)(27 33)(28 32)(29 31)(61 75)(62 74)(63 73)(64 72)(65 71)(66 70)(67 69)(76 80)(77 79)(81 140)(82 139)(83 138)(84 137)(85 136)(86 135)(87 134)(88 133)(89 132)(90 131)(91 130)(92 129)(93 128)(94 127)(95 126)(96 125)(97 124)(98 123)(99 122)(100 121)(101 111)(102 110)(103 109)(104 108)(105 107)(112 120)(113 119)(114 118)(115 117)(142 160)(143 159)(144 158)(145 157)(146 156)(147 155)(148 154)(149 153)(150 152)
G:=sub<Sym(160)| (1,44)(2,45)(3,46)(4,47)(5,48)(6,49)(7,50)(8,51)(9,52)(10,53)(11,54)(12,55)(13,56)(14,57)(15,58)(16,59)(17,60)(18,41)(19,42)(20,43)(21,117)(22,118)(23,119)(24,120)(25,101)(26,102)(27,103)(28,104)(29,105)(30,106)(31,107)(32,108)(33,109)(34,110)(35,111)(36,112)(37,113)(38,114)(39,115)(40,116)(61,144)(62,145)(63,146)(64,147)(65,148)(66,149)(67,150)(68,151)(69,152)(70,153)(71,154)(72,155)(73,156)(74,157)(75,158)(76,159)(77,160)(78,141)(79,142)(80,143)(81,134)(82,135)(83,136)(84,137)(85,138)(86,139)(87,140)(88,121)(89,122)(90,123)(91,124)(92,125)(93,126)(94,127)(95,128)(96,129)(97,130)(98,131)(99,132)(100,133), (1,104,125,149)(2,105,126,150)(3,106,127,151)(4,107,128,152)(5,108,129,153)(6,109,130,154)(7,110,131,155)(8,111,132,156)(9,112,133,157)(10,113,134,158)(11,114,135,159)(12,115,136,160)(13,116,137,141)(14,117,138,142)(15,118,139,143)(16,119,140,144)(17,120,121,145)(18,101,122,146)(19,102,123,147)(20,103,124,148)(21,85,79,57)(22,86,80,58)(23,87,61,59)(24,88,62,60)(25,89,63,41)(26,90,64,42)(27,91,65,43)(28,92,66,44)(29,93,67,45)(30,94,68,46)(31,95,69,47)(32,96,70,48)(33,97,71,49)(34,98,72,50)(35,99,73,51)(36,100,74,52)(37,81,75,53)(38,82,76,54)(39,83,77,55)(40,84,78,56), (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), (1,48)(2,47)(3,46)(4,45)(5,44)(6,43)(7,42)(8,41)(9,60)(10,59)(11,58)(12,57)(13,56)(14,55)(15,54)(16,53)(17,52)(18,51)(19,50)(20,49)(21,39)(22,38)(23,37)(24,36)(25,35)(26,34)(27,33)(28,32)(29,31)(61,75)(62,74)(63,73)(64,72)(65,71)(66,70)(67,69)(76,80)(77,79)(81,140)(82,139)(83,138)(84,137)(85,136)(86,135)(87,134)(88,133)(89,132)(90,131)(91,130)(92,129)(93,128)(94,127)(95,126)(96,125)(97,124)(98,123)(99,122)(100,121)(101,111)(102,110)(103,109)(104,108)(105,107)(112,120)(113,119)(114,118)(115,117)(142,160)(143,159)(144,158)(145,157)(146,156)(147,155)(148,154)(149,153)(150,152)>;
G:=Group( (1,44)(2,45)(3,46)(4,47)(5,48)(6,49)(7,50)(8,51)(9,52)(10,53)(11,54)(12,55)(13,56)(14,57)(15,58)(16,59)(17,60)(18,41)(19,42)(20,43)(21,117)(22,118)(23,119)(24,120)(25,101)(26,102)(27,103)(28,104)(29,105)(30,106)(31,107)(32,108)(33,109)(34,110)(35,111)(36,112)(37,113)(38,114)(39,115)(40,116)(61,144)(62,145)(63,146)(64,147)(65,148)(66,149)(67,150)(68,151)(69,152)(70,153)(71,154)(72,155)(73,156)(74,157)(75,158)(76,159)(77,160)(78,141)(79,142)(80,143)(81,134)(82,135)(83,136)(84,137)(85,138)(86,139)(87,140)(88,121)(89,122)(90,123)(91,124)(92,125)(93,126)(94,127)(95,128)(96,129)(97,130)(98,131)(99,132)(100,133), (1,104,125,149)(2,105,126,150)(3,106,127,151)(4,107,128,152)(5,108,129,153)(6,109,130,154)(7,110,131,155)(8,111,132,156)(9,112,133,157)(10,113,134,158)(11,114,135,159)(12,115,136,160)(13,116,137,141)(14,117,138,142)(15,118,139,143)(16,119,140,144)(17,120,121,145)(18,101,122,146)(19,102,123,147)(20,103,124,148)(21,85,79,57)(22,86,80,58)(23,87,61,59)(24,88,62,60)(25,89,63,41)(26,90,64,42)(27,91,65,43)(28,92,66,44)(29,93,67,45)(30,94,68,46)(31,95,69,47)(32,96,70,48)(33,97,71,49)(34,98,72,50)(35,99,73,51)(36,100,74,52)(37,81,75,53)(38,82,76,54)(39,83,77,55)(40,84,78,56), (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), (1,48)(2,47)(3,46)(4,45)(5,44)(6,43)(7,42)(8,41)(9,60)(10,59)(11,58)(12,57)(13,56)(14,55)(15,54)(16,53)(17,52)(18,51)(19,50)(20,49)(21,39)(22,38)(23,37)(24,36)(25,35)(26,34)(27,33)(28,32)(29,31)(61,75)(62,74)(63,73)(64,72)(65,71)(66,70)(67,69)(76,80)(77,79)(81,140)(82,139)(83,138)(84,137)(85,136)(86,135)(87,134)(88,133)(89,132)(90,131)(91,130)(92,129)(93,128)(94,127)(95,126)(96,125)(97,124)(98,123)(99,122)(100,121)(101,111)(102,110)(103,109)(104,108)(105,107)(112,120)(113,119)(114,118)(115,117)(142,160)(143,159)(144,158)(145,157)(146,156)(147,155)(148,154)(149,153)(150,152) );
G=PermutationGroup([[(1,44),(2,45),(3,46),(4,47),(5,48),(6,49),(7,50),(8,51),(9,52),(10,53),(11,54),(12,55),(13,56),(14,57),(15,58),(16,59),(17,60),(18,41),(19,42),(20,43),(21,117),(22,118),(23,119),(24,120),(25,101),(26,102),(27,103),(28,104),(29,105),(30,106),(31,107),(32,108),(33,109),(34,110),(35,111),(36,112),(37,113),(38,114),(39,115),(40,116),(61,144),(62,145),(63,146),(64,147),(65,148),(66,149),(67,150),(68,151),(69,152),(70,153),(71,154),(72,155),(73,156),(74,157),(75,158),(76,159),(77,160),(78,141),(79,142),(80,143),(81,134),(82,135),(83,136),(84,137),(85,138),(86,139),(87,140),(88,121),(89,122),(90,123),(91,124),(92,125),(93,126),(94,127),(95,128),(96,129),(97,130),(98,131),(99,132),(100,133)], [(1,104,125,149),(2,105,126,150),(3,106,127,151),(4,107,128,152),(5,108,129,153),(6,109,130,154),(7,110,131,155),(8,111,132,156),(9,112,133,157),(10,113,134,158),(11,114,135,159),(12,115,136,160),(13,116,137,141),(14,117,138,142),(15,118,139,143),(16,119,140,144),(17,120,121,145),(18,101,122,146),(19,102,123,147),(20,103,124,148),(21,85,79,57),(22,86,80,58),(23,87,61,59),(24,88,62,60),(25,89,63,41),(26,90,64,42),(27,91,65,43),(28,92,66,44),(29,93,67,45),(30,94,68,46),(31,95,69,47),(32,96,70,48),(33,97,71,49),(34,98,72,50),(35,99,73,51),(36,100,74,52),(37,81,75,53),(38,82,76,54),(39,83,77,55),(40,84,78,56)], [(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)], [(1,48),(2,47),(3,46),(4,45),(5,44),(6,43),(7,42),(8,41),(9,60),(10,59),(11,58),(12,57),(13,56),(14,55),(15,54),(16,53),(17,52),(18,51),(19,50),(20,49),(21,39),(22,38),(23,37),(24,36),(25,35),(26,34),(27,33),(28,32),(29,31),(61,75),(62,74),(63,73),(64,72),(65,71),(66,70),(67,69),(76,80),(77,79),(81,140),(82,139),(83,138),(84,137),(85,136),(86,135),(87,134),(88,133),(89,132),(90,131),(91,130),(92,129),(93,128),(94,127),(95,126),(96,125),(97,124),(98,123),(99,122),(100,121),(101,111),(102,110),(103,109),(104,108),(105,107),(112,120),(113,119),(114,118),(115,117),(142,160),(143,159),(144,158),(145,157),(146,156),(147,155),(148,154),(149,153),(150,152)]])
92 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 4A | ··· | 4L | 4M | 4N | 4O | 4P | 5A | 5B | 10A | ··· | 10N | 20A | ··· | 20AV |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 10 | ··· | 10 | 20 | ··· | 20 |
| size | 1 | 1 | ··· | 1 | 20 | 20 | 20 | 20 | 2 | ··· | 2 | 20 | 20 | 20 | 20 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
92 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | |||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | D4 | D5 | C4○D4 | D10 | C4×D5 | D20 | C5⋊D4 | C4○D20 |
| kernel | (C2×C4)⋊6D20 | C2×C4⋊Dic5 | C2×D10⋊C4 | C2×C4×C20 | C22×D20 | C2×D20 | C2×C20 | C2×C42 | C2×C10 | C22×C4 | C2×C4 | C2×C4 | C2×C4 | C22 |
| # reps | 1 | 1 | 4 | 1 | 1 | 8 | 8 | 2 | 4 | 6 | 8 | 24 | 8 | 16 |
Matrix representation of (C2×C4)⋊6D20 ►in GL6(𝔽41)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 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 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 9 | 0 | 0 | 0 |
| 0 | 0 | 0 | 9 | 0 | 0 |
| 0 | 0 | 0 | 0 | 30 | 32 |
| 0 | 0 | 0 | 0 | 9 | 11 |
| 16 | 30 | 0 | 0 | 0 | 0 |
| 27 | 2 | 0 | 0 | 0 | 0 |
| 0 | 0 | 35 | 1 | 0 | 0 |
| 0 | 0 | 5 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 9 | 11 |
| 0 | 0 | 0 | 0 | 30 | 14 |
| 40 | 40 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 40 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 0 | 0 | 0 | 0 | 40 | 0 |
G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,9,0,0,0,0,0,0,9,0,0,0,0,0,0,30,9,0,0,0,0,32,11],[16,27,0,0,0,0,30,2,0,0,0,0,0,0,35,5,0,0,0,0,1,40,0,0,0,0,0,0,9,30,0,0,0,0,11,14],[40,0,0,0,0,0,40,1,0,0,0,0,0,0,40,0,0,0,0,0,40,1,0,0,0,0,0,0,0,40,0,0,0,0,40,0] >;
(C2×C4)⋊6D20 in GAP, Magma, Sage, TeX
(C_2\times C_4)\rtimes_6D_{20} % in TeX
G:=Group("(C2xC4):6D20"); // GroupNames label
G:=SmallGroup(320,566);
// by ID
G=gap.SmallGroup(320,566);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,253,120,758,58,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^4=c^20=d^2=1,d*b*d=a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*c*d=c^-1>;
// generators/relations