direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×C4.D20, C42⋊40D10, C4.43(C2×D20), (C2×C4).98D20, (C2×C42)⋊10D5, (C4×C20)⋊51C22, (C2×C20).389D4, C20.286(C2×D4), C10.4(C22×D4), C2.6(C22×D20), C10⋊1(C4.4D4), (C2×C10).20C24, (C22×D20).8C2, C22.65(C2×D20), (C2×C20).781C23, (C22×Dic10)⋊4C2, (C22×C4).439D10, (C2×Dic5).4C23, D10⋊C4⋊39C22, (C22×D5).2C23, C22.63(C23×D5), (C2×Dic10)⋊46C22, (C2×D20).210C22, C22.69(C4○D20), (C23×D5).28C22, C23.316(C22×D5), (C22×C20).504C22, (C22×C10).382C23, (C22×Dic5).74C22, (C2×C4×C20)⋊9C2, C5⋊1(C2×C4.4D4), C10.7(C2×C4○D4), C2.9(C2×C4○D20), (C2×C10).171(C2×D4), (C2×D10⋊C4)⋊12C2, (C2×C10).97(C4○D4), (C2×C4).649(C22×D5), SmallGroup(320,1148)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×C4.D20
G = < a,b,c,d | a2=b4=c20=1, d2=b2, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=b2c-1 >
Subgroups: 1374 in 330 conjugacy classes, 127 normal (15 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C5, C2×C4, C2×C4, D4, Q8, C23, C23, D5, C10, C10, C42, C22⋊C4, C22×C4, C22×C4, C22×C4, C2×D4, C2×Q8, C24, Dic5, C20, C20, D10, C2×C10, C2×C10, C2×C42, C2×C22⋊C4, C4.4D4, C22×D4, C22×Q8, Dic10, D20, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C22×D5, C22×D5, C22×C10, C2×C4.4D4, D10⋊C4, C4×C20, C2×Dic10, C2×Dic10, C2×D20, C2×D20, C22×Dic5, C22×C20, C22×C20, C23×D5, C4.D20, C2×D10⋊C4, C2×C4×C20, C22×Dic10, C22×D20, C2×C4.D20
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, C4○D4, C24, D10, C4.4D4, C22×D4, C2×C4○D4, D20, C22×D5, C2×C4.4D4, C2×D20, C4○D20, C23×D5, C4.D20, C22×D20, C2×C4○D20, C2×C4.D20
►On 160 points
Generators in S160
(1 34)(2 35)(3 36)(4 37)(5 38)(6 39)(7 40)(8 21)(9 22)(10 23)(11 24)(12 25)(13 26)(14 27)(15 28)(16 29)(17 30)(18 31)(19 32)(20 33)(41 159)(42 160)(43 141)(44 142)(45 143)(46 144)(47 145)(48 146)(49 147)(50 148)(51 149)(52 150)(53 151)(54 152)(55 153)(56 154)(57 155)(58 156)(59 157)(60 158)(61 123)(62 124)(63 125)(64 126)(65 127)(66 128)(67 129)(68 130)(69 131)(70 132)(71 133)(72 134)(73 135)(74 136)(75 137)(76 138)(77 139)(78 140)(79 121)(80 122)(81 119)(82 120)(83 101)(84 102)(85 103)(86 104)(87 105)(88 106)(89 107)(90 108)(91 109)(92 110)(93 111)(94 112)(95 113)(96 114)(97 115)(98 116)(99 117)(100 118)
(1 100 65 148)(2 81 66 149)(3 82 67 150)(4 83 68 151)(5 84 69 152)(6 85 70 153)(7 86 71 154)(8 87 72 155)(9 88 73 156)(10 89 74 157)(11 90 75 158)(12 91 76 159)(13 92 77 160)(14 93 78 141)(15 94 79 142)(16 95 80 143)(17 96 61 144)(18 97 62 145)(19 98 63 146)(20 99 64 147)(21 105 134 57)(22 106 135 58)(23 107 136 59)(24 108 137 60)(25 109 138 41)(26 110 139 42)(27 111 140 43)(28 112 121 44)(29 113 122 45)(30 114 123 46)(31 115 124 47)(32 116 125 48)(33 117 126 49)(34 118 127 50)(35 119 128 51)(36 120 129 52)(37 101 130 53)(38 102 131 54)(39 103 132 55)(40 104 133 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 49 65 117)(2 116 66 48)(3 47 67 115)(4 114 68 46)(5 45 69 113)(6 112 70 44)(7 43 71 111)(8 110 72 42)(9 41 73 109)(10 108 74 60)(11 59 75 107)(12 106 76 58)(13 57 77 105)(14 104 78 56)(15 55 79 103)(16 102 80 54)(17 53 61 101)(18 120 62 52)(19 51 63 119)(20 118 64 50)(21 92 134 160)(22 159 135 91)(23 90 136 158)(24 157 137 89)(25 88 138 156)(26 155 139 87)(27 86 140 154)(28 153 121 85)(29 84 122 152)(30 151 123 83)(31 82 124 150)(32 149 125 81)(33 100 126 148)(34 147 127 99)(35 98 128 146)(36 145 129 97)(37 96 130 144)(38 143 131 95)(39 94 132 142)(40 141 133 93)
G:=sub<Sym(160)| (1,34)(2,35)(3,36)(4,37)(5,38)(6,39)(7,40)(8,21)(9,22)(10,23)(11,24)(12,25)(13,26)(14,27)(15,28)(16,29)(17,30)(18,31)(19,32)(20,33)(41,159)(42,160)(43,141)(44,142)(45,143)(46,144)(47,145)(48,146)(49,147)(50,148)(51,149)(52,150)(53,151)(54,152)(55,153)(56,154)(57,155)(58,156)(59,157)(60,158)(61,123)(62,124)(63,125)(64,126)(65,127)(66,128)(67,129)(68,130)(69,131)(70,132)(71,133)(72,134)(73,135)(74,136)(75,137)(76,138)(77,139)(78,140)(79,121)(80,122)(81,119)(82,120)(83,101)(84,102)(85,103)(86,104)(87,105)(88,106)(89,107)(90,108)(91,109)(92,110)(93,111)(94,112)(95,113)(96,114)(97,115)(98,116)(99,117)(100,118), (1,100,65,148)(2,81,66,149)(3,82,67,150)(4,83,68,151)(5,84,69,152)(6,85,70,153)(7,86,71,154)(8,87,72,155)(9,88,73,156)(10,89,74,157)(11,90,75,158)(12,91,76,159)(13,92,77,160)(14,93,78,141)(15,94,79,142)(16,95,80,143)(17,96,61,144)(18,97,62,145)(19,98,63,146)(20,99,64,147)(21,105,134,57)(22,106,135,58)(23,107,136,59)(24,108,137,60)(25,109,138,41)(26,110,139,42)(27,111,140,43)(28,112,121,44)(29,113,122,45)(30,114,123,46)(31,115,124,47)(32,116,125,48)(33,117,126,49)(34,118,127,50)(35,119,128,51)(36,120,129,52)(37,101,130,53)(38,102,131,54)(39,103,132,55)(40,104,133,56), (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), (1,49,65,117)(2,116,66,48)(3,47,67,115)(4,114,68,46)(5,45,69,113)(6,112,70,44)(7,43,71,111)(8,110,72,42)(9,41,73,109)(10,108,74,60)(11,59,75,107)(12,106,76,58)(13,57,77,105)(14,104,78,56)(15,55,79,103)(16,102,80,54)(17,53,61,101)(18,120,62,52)(19,51,63,119)(20,118,64,50)(21,92,134,160)(22,159,135,91)(23,90,136,158)(24,157,137,89)(25,88,138,156)(26,155,139,87)(27,86,140,154)(28,153,121,85)(29,84,122,152)(30,151,123,83)(31,82,124,150)(32,149,125,81)(33,100,126,148)(34,147,127,99)(35,98,128,146)(36,145,129,97)(37,96,130,144)(38,143,131,95)(39,94,132,142)(40,141,133,93)>;
G:=Group( (1,34)(2,35)(3,36)(4,37)(5,38)(6,39)(7,40)(8,21)(9,22)(10,23)(11,24)(12,25)(13,26)(14,27)(15,28)(16,29)(17,30)(18,31)(19,32)(20,33)(41,159)(42,160)(43,141)(44,142)(45,143)(46,144)(47,145)(48,146)(49,147)(50,148)(51,149)(52,150)(53,151)(54,152)(55,153)(56,154)(57,155)(58,156)(59,157)(60,158)(61,123)(62,124)(63,125)(64,126)(65,127)(66,128)(67,129)(68,130)(69,131)(70,132)(71,133)(72,134)(73,135)(74,136)(75,137)(76,138)(77,139)(78,140)(79,121)(80,122)(81,119)(82,120)(83,101)(84,102)(85,103)(86,104)(87,105)(88,106)(89,107)(90,108)(91,109)(92,110)(93,111)(94,112)(95,113)(96,114)(97,115)(98,116)(99,117)(100,118), (1,100,65,148)(2,81,66,149)(3,82,67,150)(4,83,68,151)(5,84,69,152)(6,85,70,153)(7,86,71,154)(8,87,72,155)(9,88,73,156)(10,89,74,157)(11,90,75,158)(12,91,76,159)(13,92,77,160)(14,93,78,141)(15,94,79,142)(16,95,80,143)(17,96,61,144)(18,97,62,145)(19,98,63,146)(20,99,64,147)(21,105,134,57)(22,106,135,58)(23,107,136,59)(24,108,137,60)(25,109,138,41)(26,110,139,42)(27,111,140,43)(28,112,121,44)(29,113,122,45)(30,114,123,46)(31,115,124,47)(32,116,125,48)(33,117,126,49)(34,118,127,50)(35,119,128,51)(36,120,129,52)(37,101,130,53)(38,102,131,54)(39,103,132,55)(40,104,133,56), (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), (1,49,65,117)(2,116,66,48)(3,47,67,115)(4,114,68,46)(5,45,69,113)(6,112,70,44)(7,43,71,111)(8,110,72,42)(9,41,73,109)(10,108,74,60)(11,59,75,107)(12,106,76,58)(13,57,77,105)(14,104,78,56)(15,55,79,103)(16,102,80,54)(17,53,61,101)(18,120,62,52)(19,51,63,119)(20,118,64,50)(21,92,134,160)(22,159,135,91)(23,90,136,158)(24,157,137,89)(25,88,138,156)(26,155,139,87)(27,86,140,154)(28,153,121,85)(29,84,122,152)(30,151,123,83)(31,82,124,150)(32,149,125,81)(33,100,126,148)(34,147,127,99)(35,98,128,146)(36,145,129,97)(37,96,130,144)(38,143,131,95)(39,94,132,142)(40,141,133,93) );
G=PermutationGroup([[(1,34),(2,35),(3,36),(4,37),(5,38),(6,39),(7,40),(8,21),(9,22),(10,23),(11,24),(12,25),(13,26),(14,27),(15,28),(16,29),(17,30),(18,31),(19,32),(20,33),(41,159),(42,160),(43,141),(44,142),(45,143),(46,144),(47,145),(48,146),(49,147),(50,148),(51,149),(52,150),(53,151),(54,152),(55,153),(56,154),(57,155),(58,156),(59,157),(60,158),(61,123),(62,124),(63,125),(64,126),(65,127),(66,128),(67,129),(68,130),(69,131),(70,132),(71,133),(72,134),(73,135),(74,136),(75,137),(76,138),(77,139),(78,140),(79,121),(80,122),(81,119),(82,120),(83,101),(84,102),(85,103),(86,104),(87,105),(88,106),(89,107),(90,108),(91,109),(92,110),(93,111),(94,112),(95,113),(96,114),(97,115),(98,116),(99,117),(100,118)], [(1,100,65,148),(2,81,66,149),(3,82,67,150),(4,83,68,151),(5,84,69,152),(6,85,70,153),(7,86,71,154),(8,87,72,155),(9,88,73,156),(10,89,74,157),(11,90,75,158),(12,91,76,159),(13,92,77,160),(14,93,78,141),(15,94,79,142),(16,95,80,143),(17,96,61,144),(18,97,62,145),(19,98,63,146),(20,99,64,147),(21,105,134,57),(22,106,135,58),(23,107,136,59),(24,108,137,60),(25,109,138,41),(26,110,139,42),(27,111,140,43),(28,112,121,44),(29,113,122,45),(30,114,123,46),(31,115,124,47),(32,116,125,48),(33,117,126,49),(34,118,127,50),(35,119,128,51),(36,120,129,52),(37,101,130,53),(38,102,131,54),(39,103,132,55),(40,104,133,56)], [(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)], [(1,49,65,117),(2,116,66,48),(3,47,67,115),(4,114,68,46),(5,45,69,113),(6,112,70,44),(7,43,71,111),(8,110,72,42),(9,41,73,109),(10,108,74,60),(11,59,75,107),(12,106,76,58),(13,57,77,105),(14,104,78,56),(15,55,79,103),(16,102,80,54),(17,53,61,101),(18,120,62,52),(19,51,63,119),(20,118,64,50),(21,92,134,160),(22,159,135,91),(23,90,136,158),(24,157,137,89),(25,88,138,156),(26,155,139,87),(27,86,140,154),(28,153,121,85),(29,84,122,152),(30,151,123,83),(31,82,124,150),(32,149,125,81),(33,100,126,148),(34,147,127,99),(35,98,128,146),(36,145,129,97),(37,96,130,144),(38,143,131,95),(39,94,132,142),(40,141,133,93)]])
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 |
| type | + | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | D5 | C4○D4 | D10 | D10 | D20 | C4○D20 |
| kernel | C2×C4.D20 | C4.D20 | C2×D10⋊C4 | C2×C4×C20 | C22×Dic10 | C22×D20 | C2×C20 | C2×C42 | C2×C10 | C42 | C22×C4 | C2×C4 | C22 |
| # reps | 1 | 8 | 4 | 1 | 1 | 1 | 4 | 2 | 8 | 8 | 6 | 16 | 32 |
Matrix representation of C2×C4.D20 ►in GL5(𝔽41)
| 40 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 40 | 0 | 0 | 0 | 0 |
| 0 | 30 | 32 | 0 | 0 |
| 0 | 9 | 11 | 0 | 0 |
| 0 | 0 | 0 | 11 | 19 |
| 0 | 0 | 0 | 13 | 30 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 22 | 9 | 0 | 0 |
| 0 | 32 | 0 | 0 | 0 |
| 0 | 0 | 0 | 3 | 21 |
| 0 | 0 | 0 | 23 | 24 |
| 40 | 0 | 0 | 0 | 0 |
| 0 | 38 | 17 | 0 | 0 |
| 0 | 38 | 3 | 0 | 0 |
| 0 | 0 | 0 | 0 | 22 |
| 0 | 0 | 0 | 13 | 0 |
G:=sub<GL(5,GF(41))| [40,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[40,0,0,0,0,0,30,9,0,0,0,32,11,0,0,0,0,0,11,13,0,0,0,19,30],[1,0,0,0,0,0,22,32,0,0,0,9,0,0,0,0,0,0,3,23,0,0,0,21,24],[40,0,0,0,0,0,38,38,0,0,0,17,3,0,0,0,0,0,0,13,0,0,0,22,0] >;
C2×C4.D20 in GAP, Magma, Sage, TeX
C_2\times C_4.D_{20} % in TeX
G:=Group("C2xC4.D20"); // GroupNames label
G:=SmallGroup(320,1148);
// by ID
G=gap.SmallGroup(320,1148);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,758,100,675,136,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^4=c^20=1,d^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=b^2*c^-1>;
// generators/relations