direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×C20.48D4, C23⋊4Dic10, C24.67D10, (C22×C10)⋊7Q8, C20.424(C2×D4), (C2×C20).477D4, C10⋊4(C22⋊Q8), (C23×C4).10D5, (C23×C20).12C2, C4⋊Dic5⋊63C22, C22⋊3(C2×Dic10), C10.19(C22×Q8), (C2×C20).703C23, (C2×C10).282C24, (C22×C4).446D10, C10.130(C22×D4), (C2×Dic10)⋊58C22, (C22×Dic10)⋊12C2, C22.79(C4○D20), C10.D4⋊43C22, C2.20(C22×Dic10), C22.301(C23×D5), C23.231(C22×D5), (C22×C10).411C23, (C23×C10).104C22, (C22×C20).528C22, (C2×Dic5).148C23, C23.D5.129C22, (C22×Dic5).160C22, C5⋊5(C2×C22⋊Q8), (C2×C10)⋊6(C2×Q8), (C2×C4⋊Dic5)⋊28C2, C2.69(C2×C4○D20), C10.59(C2×C4○D4), C4.120(C2×C5⋊D4), C2.5(C22×C5⋊D4), (C2×C10).571(C2×D4), (C2×C10.D4)⋊17C2, (C2×C4).262(C5⋊D4), (C2×C4).656(C22×D5), C22.100(C2×C5⋊D4), (C2×C23.D5).23C2, (C2×C10).110(C4○D4), SmallGroup(320,1456)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×C20.48D4
G = < a,b,c,d | a2=b20=c4=1, d2=b10, ab=ba, ac=ca, ad=da, cbc-1=dbd-1=b-1, dcd-1=b10c-1 >
Subgroups: 878 in 322 conjugacy classes, 143 normal (23 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C5, C2×C4, C2×C4, Q8, C23, C23, C23, C10, C10, C10, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C22×C4, C2×Q8, C24, Dic5, C20, C20, C2×C10, C2×C10, C2×C10, C2×C22⋊C4, C2×C4⋊C4, C22⋊Q8, C23×C4, C22×Q8, Dic10, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C22×C10, C22×C10, C22×C10, C2×C22⋊Q8, C10.D4, C4⋊Dic5, C23.D5, C2×Dic10, C2×Dic10, C22×Dic5, C22×C20, C22×C20, C22×C20, C23×C10, C2×C10.D4, C20.48D4, C2×C4⋊Dic5, C2×C23.D5, C22×Dic10, C23×C20, C2×C20.48D4
Quotients: C1, C2, C22, D4, Q8, C23, D5, C2×D4, C2×Q8, C4○D4, C24, D10, C22⋊Q8, C22×D4, C22×Q8, C2×C4○D4, Dic10, C5⋊D4, C22×D5, C2×C22⋊Q8, C2×Dic10, C4○D20, C2×C5⋊D4, C23×D5, C20.48D4, C22×Dic10, C2×C4○D20, C22×C5⋊D4, C2×C20.48D4
►On 160 points
Generators in S160
(1 135)(2 136)(3 137)(4 138)(5 139)(6 140)(7 121)(8 122)(9 123)(10 124)(11 125)(12 126)(13 127)(14 128)(15 129)(16 130)(17 131)(18 132)(19 133)(20 134)(21 48)(22 49)(23 50)(24 51)(25 52)(26 53)(27 54)(28 55)(29 56)(30 57)(31 58)(32 59)(33 60)(34 41)(35 42)(36 43)(37 44)(38 45)(39 46)(40 47)(61 107)(62 108)(63 109)(64 110)(65 111)(66 112)(67 113)(68 114)(69 115)(70 116)(71 117)(72 118)(73 119)(74 120)(75 101)(76 102)(77 103)(78 104)(79 105)(80 106)(81 153)(82 154)(83 155)(84 156)(85 157)(86 158)(87 159)(88 160)(89 141)(90 142)(91 143)(92 144)(93 145)(94 146)(95 147)(96 148)(97 149)(98 150)(99 151)(100 152)
(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)
(1 117 48 159)(2 116 49 158)(3 115 50 157)(4 114 51 156)(5 113 52 155)(6 112 53 154)(7 111 54 153)(8 110 55 152)(9 109 56 151)(10 108 57 150)(11 107 58 149)(12 106 59 148)(13 105 60 147)(14 104 41 146)(15 103 42 145)(16 102 43 144)(17 101 44 143)(18 120 45 142)(19 119 46 141)(20 118 47 160)(21 87 135 71)(22 86 136 70)(23 85 137 69)(24 84 138 68)(25 83 139 67)(26 82 140 66)(27 81 121 65)(28 100 122 64)(29 99 123 63)(30 98 124 62)(31 97 125 61)(32 96 126 80)(33 95 127 79)(34 94 128 78)(35 93 129 77)(36 92 130 76)(37 91 131 75)(38 90 132 74)(39 89 133 73)(40 88 134 72)
(1 149 11 159)(2 148 12 158)(3 147 13 157)(4 146 14 156)(5 145 15 155)(6 144 16 154)(7 143 17 153)(8 142 18 152)(9 141 19 151)(10 160 20 150)(21 61 31 71)(22 80 32 70)(23 79 33 69)(24 78 34 68)(25 77 35 67)(26 76 36 66)(27 75 37 65)(28 74 38 64)(29 73 39 63)(30 72 40 62)(41 114 51 104)(42 113 52 103)(43 112 53 102)(44 111 54 101)(45 110 55 120)(46 109 56 119)(47 108 57 118)(48 107 58 117)(49 106 59 116)(50 105 60 115)(81 121 91 131)(82 140 92 130)(83 139 93 129)(84 138 94 128)(85 137 95 127)(86 136 96 126)(87 135 97 125)(88 134 98 124)(89 133 99 123)(90 132 100 122)
G:=sub<Sym(160)| (1,135)(2,136)(3,137)(4,138)(5,139)(6,140)(7,121)(8,122)(9,123)(10,124)(11,125)(12,126)(13,127)(14,128)(15,129)(16,130)(17,131)(18,132)(19,133)(20,134)(21,48)(22,49)(23,50)(24,51)(25,52)(26,53)(27,54)(28,55)(29,56)(30,57)(31,58)(32,59)(33,60)(34,41)(35,42)(36,43)(37,44)(38,45)(39,46)(40,47)(61,107)(62,108)(63,109)(64,110)(65,111)(66,112)(67,113)(68,114)(69,115)(70,116)(71,117)(72,118)(73,119)(74,120)(75,101)(76,102)(77,103)(78,104)(79,105)(80,106)(81,153)(82,154)(83,155)(84,156)(85,157)(86,158)(87,159)(88,160)(89,141)(90,142)(91,143)(92,144)(93,145)(94,146)(95,147)(96,148)(97,149)(98,150)(99,151)(100,152), (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), (1,117,48,159)(2,116,49,158)(3,115,50,157)(4,114,51,156)(5,113,52,155)(6,112,53,154)(7,111,54,153)(8,110,55,152)(9,109,56,151)(10,108,57,150)(11,107,58,149)(12,106,59,148)(13,105,60,147)(14,104,41,146)(15,103,42,145)(16,102,43,144)(17,101,44,143)(18,120,45,142)(19,119,46,141)(20,118,47,160)(21,87,135,71)(22,86,136,70)(23,85,137,69)(24,84,138,68)(25,83,139,67)(26,82,140,66)(27,81,121,65)(28,100,122,64)(29,99,123,63)(30,98,124,62)(31,97,125,61)(32,96,126,80)(33,95,127,79)(34,94,128,78)(35,93,129,77)(36,92,130,76)(37,91,131,75)(38,90,132,74)(39,89,133,73)(40,88,134,72), (1,149,11,159)(2,148,12,158)(3,147,13,157)(4,146,14,156)(5,145,15,155)(6,144,16,154)(7,143,17,153)(8,142,18,152)(9,141,19,151)(10,160,20,150)(21,61,31,71)(22,80,32,70)(23,79,33,69)(24,78,34,68)(25,77,35,67)(26,76,36,66)(27,75,37,65)(28,74,38,64)(29,73,39,63)(30,72,40,62)(41,114,51,104)(42,113,52,103)(43,112,53,102)(44,111,54,101)(45,110,55,120)(46,109,56,119)(47,108,57,118)(48,107,58,117)(49,106,59,116)(50,105,60,115)(81,121,91,131)(82,140,92,130)(83,139,93,129)(84,138,94,128)(85,137,95,127)(86,136,96,126)(87,135,97,125)(88,134,98,124)(89,133,99,123)(90,132,100,122)>;
G:=Group( (1,135)(2,136)(3,137)(4,138)(5,139)(6,140)(7,121)(8,122)(9,123)(10,124)(11,125)(12,126)(13,127)(14,128)(15,129)(16,130)(17,131)(18,132)(19,133)(20,134)(21,48)(22,49)(23,50)(24,51)(25,52)(26,53)(27,54)(28,55)(29,56)(30,57)(31,58)(32,59)(33,60)(34,41)(35,42)(36,43)(37,44)(38,45)(39,46)(40,47)(61,107)(62,108)(63,109)(64,110)(65,111)(66,112)(67,113)(68,114)(69,115)(70,116)(71,117)(72,118)(73,119)(74,120)(75,101)(76,102)(77,103)(78,104)(79,105)(80,106)(81,153)(82,154)(83,155)(84,156)(85,157)(86,158)(87,159)(88,160)(89,141)(90,142)(91,143)(92,144)(93,145)(94,146)(95,147)(96,148)(97,149)(98,150)(99,151)(100,152), (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), (1,117,48,159)(2,116,49,158)(3,115,50,157)(4,114,51,156)(5,113,52,155)(6,112,53,154)(7,111,54,153)(8,110,55,152)(9,109,56,151)(10,108,57,150)(11,107,58,149)(12,106,59,148)(13,105,60,147)(14,104,41,146)(15,103,42,145)(16,102,43,144)(17,101,44,143)(18,120,45,142)(19,119,46,141)(20,118,47,160)(21,87,135,71)(22,86,136,70)(23,85,137,69)(24,84,138,68)(25,83,139,67)(26,82,140,66)(27,81,121,65)(28,100,122,64)(29,99,123,63)(30,98,124,62)(31,97,125,61)(32,96,126,80)(33,95,127,79)(34,94,128,78)(35,93,129,77)(36,92,130,76)(37,91,131,75)(38,90,132,74)(39,89,133,73)(40,88,134,72), (1,149,11,159)(2,148,12,158)(3,147,13,157)(4,146,14,156)(5,145,15,155)(6,144,16,154)(7,143,17,153)(8,142,18,152)(9,141,19,151)(10,160,20,150)(21,61,31,71)(22,80,32,70)(23,79,33,69)(24,78,34,68)(25,77,35,67)(26,76,36,66)(27,75,37,65)(28,74,38,64)(29,73,39,63)(30,72,40,62)(41,114,51,104)(42,113,52,103)(43,112,53,102)(44,111,54,101)(45,110,55,120)(46,109,56,119)(47,108,57,118)(48,107,58,117)(49,106,59,116)(50,105,60,115)(81,121,91,131)(82,140,92,130)(83,139,93,129)(84,138,94,128)(85,137,95,127)(86,136,96,126)(87,135,97,125)(88,134,98,124)(89,133,99,123)(90,132,100,122) );
G=PermutationGroup([[(1,135),(2,136),(3,137),(4,138),(5,139),(6,140),(7,121),(8,122),(9,123),(10,124),(11,125),(12,126),(13,127),(14,128),(15,129),(16,130),(17,131),(18,132),(19,133),(20,134),(21,48),(22,49),(23,50),(24,51),(25,52),(26,53),(27,54),(28,55),(29,56),(30,57),(31,58),(32,59),(33,60),(34,41),(35,42),(36,43),(37,44),(38,45),(39,46),(40,47),(61,107),(62,108),(63,109),(64,110),(65,111),(66,112),(67,113),(68,114),(69,115),(70,116),(71,117),(72,118),(73,119),(74,120),(75,101),(76,102),(77,103),(78,104),(79,105),(80,106),(81,153),(82,154),(83,155),(84,156),(85,157),(86,158),(87,159),(88,160),(89,141),(90,142),(91,143),(92,144),(93,145),(94,146),(95,147),(96,148),(97,149),(98,150),(99,151),(100,152)], [(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)], [(1,117,48,159),(2,116,49,158),(3,115,50,157),(4,114,51,156),(5,113,52,155),(6,112,53,154),(7,111,54,153),(8,110,55,152),(9,109,56,151),(10,108,57,150),(11,107,58,149),(12,106,59,148),(13,105,60,147),(14,104,41,146),(15,103,42,145),(16,102,43,144),(17,101,44,143),(18,120,45,142),(19,119,46,141),(20,118,47,160),(21,87,135,71),(22,86,136,70),(23,85,137,69),(24,84,138,68),(25,83,139,67),(26,82,140,66),(27,81,121,65),(28,100,122,64),(29,99,123,63),(30,98,124,62),(31,97,125,61),(32,96,126,80),(33,95,127,79),(34,94,128,78),(35,93,129,77),(36,92,130,76),(37,91,131,75),(38,90,132,74),(39,89,133,73),(40,88,134,72)], [(1,149,11,159),(2,148,12,158),(3,147,13,157),(4,146,14,156),(5,145,15,155),(6,144,16,154),(7,143,17,153),(8,142,18,152),(9,141,19,151),(10,160,20,150),(21,61,31,71),(22,80,32,70),(23,79,33,69),(24,78,34,68),(25,77,35,67),(26,76,36,66),(27,75,37,65),(28,74,38,64),(29,73,39,63),(30,72,40,62),(41,114,51,104),(42,113,52,103),(43,112,53,102),(44,111,54,101),(45,110,55,120),(46,109,56,119),(47,108,57,118),(48,107,58,117),(49,106,59,116),(50,105,60,115),(81,121,91,131),(82,140,92,130),(83,139,93,129),(84,138,94,128),(85,137,95,127),(86,136,96,126),(87,135,97,125),(88,134,98,124),(89,133,99,123),(90,132,100,122)]])
92 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 4A | ··· | 4H | 4I | ··· | 4P | 5A | 5B | 10A | ··· | 10AD | 20A | ··· | 20AF |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 5 | 5 | 10 | ··· | 10 | 20 | ··· | 20 |
| size | 1 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 20 | ··· | 20 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
92 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | - | + | + | + | - | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | Q8 | D5 | C4○D4 | D10 | D10 | C5⋊D4 | Dic10 | C4○D20 |
| kernel | C2×C20.48D4 | C2×C10.D4 | C20.48D4 | C2×C4⋊Dic5 | C2×C23.D5 | C22×Dic10 | C23×C20 | C2×C20 | C22×C10 | C23×C4 | C2×C10 | C22×C4 | C24 | C2×C4 | C23 | C22 |
| # reps | 1 | 2 | 8 | 1 | 2 | 1 | 1 | 4 | 4 | 2 | 4 | 12 | 2 | 16 | 16 | 16 |
Matrix representation of C2×C20.48D4 ►in GL5(𝔽41)
| 40 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 40 |
| 40 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 31 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 40 | 0 |
| 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 32 |
| 0 | 0 | 0 | 32 | 0 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 32 |
| 0 | 0 | 0 | 32 | 0 |
G:=sub<GL(5,GF(41))| [40,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,40,0,0,0,0,0,40],[40,0,0,0,0,0,4,0,0,0,0,0,31,0,0,0,0,0,0,40,0,0,0,1,0],[40,0,0,0,0,0,0,40,0,0,0,1,0,0,0,0,0,0,0,32,0,0,0,32,0],[1,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,0,32,0,0,0,32,0] >;
C2×C20.48D4 in GAP, Magma, Sage, TeX
C_2\times C_{20}._{48}D_4 % in TeX
G:=Group("C2xC20.48D4"); // GroupNames label
G:=SmallGroup(320,1456);
// by ID
G=gap.SmallGroup(320,1456);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,758,184,675,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^20=c^4=1,d^2=b^10,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d^-1=b^-1,d*c*d^-1=b^10*c^-1>;
// generators/relations