direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×C4⋊D20, C4⋊3(C2×D20), (C2×C20)⋊8D4, C20⋊4(C2×D4), C4⋊C4⋊38D10, D10⋊2(C2×D4), (C2×C4)⋊10D20, C10⋊2(C4⋊D4), (C22×D20)⋊7C2, (C22×D5)⋊10D4, C10.9(C22×D4), (C2×D20)⋊45C22, (C2×C10).50C24, C22.67(C2×D20), C22.133(D4×D5), C2.11(C22×D20), (C2×C20).487C23, (C22×C4).360D10, D10⋊C4⋊50C22, C22.84(C23×D5), C23.328(C22×D5), (C22×C10).399C23, (C22×C20).217C22, C22.36(Q8⋊2D5), (C2×Dic5).198C23, (C23×D5).112C22, (C22×D5).166C23, (C22×Dic5).236C22, C5⋊2(C2×C4⋊D4), C2.15(C2×D4×D5), (C2×C4⋊C4)⋊15D5, (C10×C4⋊C4)⋊12C2, (D5×C22×C4)⋊1C2, (C2×C4×D5)⋊55C22, (C5×C4⋊C4)⋊46C22, C2.7(C2×Q8⋊2D5), C10.109(C2×C4○D4), (C2×C10).174(C2×D4), (C2×D10⋊C4)⋊20C2, (C2×C4).141(C22×D5), (C2×C10).197(C4○D4), SmallGroup(320,1178)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×C4⋊D20
G = < a,b,c,d | a2=b4=c20=d2=1, ab=ba, ac=ca, ad=da, cbc-1=dbd=b-1, dcd=c-1 >
Subgroups: 1902 in 426 conjugacy classes, 135 normal (21 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C5, C2×C4, C2×C4, D4, C23, C23, D5, C10, C10, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C22×C4, C2×D4, C24, Dic5, C20, C20, D10, D10, C2×C10, C2×C10, C2×C22⋊C4, C2×C4⋊C4, C4⋊D4, C23×C4, C22×D4, C4×D5, D20, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C22×D5, C22×D5, C22×C10, C2×C4⋊D4, D10⋊C4, C5×C4⋊C4, C2×C4×D5, C2×C4×D5, C2×D20, C2×D20, C22×Dic5, C22×C20, C22×C20, C23×D5, C23×D5, C4⋊D20, C2×D10⋊C4, C10×C4⋊C4, D5×C22×C4, C22×D20, C22×D20, C2×C4⋊D20
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, C4○D4, C24, D10, C4⋊D4, C22×D4, C2×C4○D4, D20, C22×D5, C2×C4⋊D4, C2×D20, D4×D5, Q8⋊2D5, C23×D5, C4⋊D20, C22×D20, C2×D4×D5, C2×Q8⋊2D5, C2×C4⋊D20
►On 160 points
Generators in S160
(1 141)(2 142)(3 143)(4 144)(5 145)(6 146)(7 147)(8 148)(9 149)(10 150)(11 151)(12 152)(13 153)(14 154)(15 155)(16 156)(17 157)(18 158)(19 159)(20 160)(21 97)(22 98)(23 99)(24 100)(25 81)(26 82)(27 83)(28 84)(29 85)(30 86)(31 87)(32 88)(33 89)(34 90)(35 91)(36 92)(37 93)(38 94)(39 95)(40 96)(41 125)(42 126)(43 127)(44 128)(45 129)(46 130)(47 131)(48 132)(49 133)(50 134)(51 135)(52 136)(53 137)(54 138)(55 139)(56 140)(57 121)(58 122)(59 123)(60 124)(61 102)(62 103)(63 104)(64 105)(65 106)(66 107)(67 108)(68 109)(69 110)(70 111)(71 112)(72 113)(73 114)(74 115)(75 116)(76 117)(77 118)(78 119)(79 120)(80 101)
(1 119 90 52)(2 53 91 120)(3 101 92 54)(4 55 93 102)(5 103 94 56)(6 57 95 104)(7 105 96 58)(8 59 97 106)(9 107 98 60)(10 41 99 108)(11 109 100 42)(12 43 81 110)(13 111 82 44)(14 45 83 112)(15 113 84 46)(16 47 85 114)(17 115 86 48)(18 49 87 116)(19 117 88 50)(20 51 89 118)(21 65 148 123)(22 124 149 66)(23 67 150 125)(24 126 151 68)(25 69 152 127)(26 128 153 70)(27 71 154 129)(28 130 155 72)(29 73 156 131)(30 132 157 74)(31 75 158 133)(32 134 159 76)(33 77 160 135)(34 136 141 78)(35 79 142 137)(36 138 143 80)(37 61 144 139)(38 140 145 62)(39 63 146 121)(40 122 147 64)
(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 28)(2 27)(3 26)(4 25)(5 24)(6 23)(7 22)(8 21)(9 40)(10 39)(11 38)(12 37)(13 36)(14 35)(15 34)(16 33)(17 32)(18 31)(19 30)(20 29)(41 121)(42 140)(43 139)(44 138)(45 137)(46 136)(47 135)(48 134)(49 133)(50 132)(51 131)(52 130)(53 129)(54 128)(55 127)(56 126)(57 125)(58 124)(59 123)(60 122)(61 110)(62 109)(63 108)(64 107)(65 106)(66 105)(67 104)(68 103)(69 102)(70 101)(71 120)(72 119)(73 118)(74 117)(75 116)(76 115)(77 114)(78 113)(79 112)(80 111)(81 144)(82 143)(83 142)(84 141)(85 160)(86 159)(87 158)(88 157)(89 156)(90 155)(91 154)(92 153)(93 152)(94 151)(95 150)(96 149)(97 148)(98 147)(99 146)(100 145)
G:=sub<Sym(160)| (1,141)(2,142)(3,143)(4,144)(5,145)(6,146)(7,147)(8,148)(9,149)(10,150)(11,151)(12,152)(13,153)(14,154)(15,155)(16,156)(17,157)(18,158)(19,159)(20,160)(21,97)(22,98)(23,99)(24,100)(25,81)(26,82)(27,83)(28,84)(29,85)(30,86)(31,87)(32,88)(33,89)(34,90)(35,91)(36,92)(37,93)(38,94)(39,95)(40,96)(41,125)(42,126)(43,127)(44,128)(45,129)(46,130)(47,131)(48,132)(49,133)(50,134)(51,135)(52,136)(53,137)(54,138)(55,139)(56,140)(57,121)(58,122)(59,123)(60,124)(61,102)(62,103)(63,104)(64,105)(65,106)(66,107)(67,108)(68,109)(69,110)(70,111)(71,112)(72,113)(73,114)(74,115)(75,116)(76,117)(77,118)(78,119)(79,120)(80,101), (1,119,90,52)(2,53,91,120)(3,101,92,54)(4,55,93,102)(5,103,94,56)(6,57,95,104)(7,105,96,58)(8,59,97,106)(9,107,98,60)(10,41,99,108)(11,109,100,42)(12,43,81,110)(13,111,82,44)(14,45,83,112)(15,113,84,46)(16,47,85,114)(17,115,86,48)(18,49,87,116)(19,117,88,50)(20,51,89,118)(21,65,148,123)(22,124,149,66)(23,67,150,125)(24,126,151,68)(25,69,152,127)(26,128,153,70)(27,71,154,129)(28,130,155,72)(29,73,156,131)(30,132,157,74)(31,75,158,133)(32,134,159,76)(33,77,160,135)(34,136,141,78)(35,79,142,137)(36,138,143,80)(37,61,144,139)(38,140,145,62)(39,63,146,121)(40,122,147,64), (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), (1,28)(2,27)(3,26)(4,25)(5,24)(6,23)(7,22)(8,21)(9,40)(10,39)(11,38)(12,37)(13,36)(14,35)(15,34)(16,33)(17,32)(18,31)(19,30)(20,29)(41,121)(42,140)(43,139)(44,138)(45,137)(46,136)(47,135)(48,134)(49,133)(50,132)(51,131)(52,130)(53,129)(54,128)(55,127)(56,126)(57,125)(58,124)(59,123)(60,122)(61,110)(62,109)(63,108)(64,107)(65,106)(66,105)(67,104)(68,103)(69,102)(70,101)(71,120)(72,119)(73,118)(74,117)(75,116)(76,115)(77,114)(78,113)(79,112)(80,111)(81,144)(82,143)(83,142)(84,141)(85,160)(86,159)(87,158)(88,157)(89,156)(90,155)(91,154)(92,153)(93,152)(94,151)(95,150)(96,149)(97,148)(98,147)(99,146)(100,145)>;
G:=Group( (1,141)(2,142)(3,143)(4,144)(5,145)(6,146)(7,147)(8,148)(9,149)(10,150)(11,151)(12,152)(13,153)(14,154)(15,155)(16,156)(17,157)(18,158)(19,159)(20,160)(21,97)(22,98)(23,99)(24,100)(25,81)(26,82)(27,83)(28,84)(29,85)(30,86)(31,87)(32,88)(33,89)(34,90)(35,91)(36,92)(37,93)(38,94)(39,95)(40,96)(41,125)(42,126)(43,127)(44,128)(45,129)(46,130)(47,131)(48,132)(49,133)(50,134)(51,135)(52,136)(53,137)(54,138)(55,139)(56,140)(57,121)(58,122)(59,123)(60,124)(61,102)(62,103)(63,104)(64,105)(65,106)(66,107)(67,108)(68,109)(69,110)(70,111)(71,112)(72,113)(73,114)(74,115)(75,116)(76,117)(77,118)(78,119)(79,120)(80,101), (1,119,90,52)(2,53,91,120)(3,101,92,54)(4,55,93,102)(5,103,94,56)(6,57,95,104)(7,105,96,58)(8,59,97,106)(9,107,98,60)(10,41,99,108)(11,109,100,42)(12,43,81,110)(13,111,82,44)(14,45,83,112)(15,113,84,46)(16,47,85,114)(17,115,86,48)(18,49,87,116)(19,117,88,50)(20,51,89,118)(21,65,148,123)(22,124,149,66)(23,67,150,125)(24,126,151,68)(25,69,152,127)(26,128,153,70)(27,71,154,129)(28,130,155,72)(29,73,156,131)(30,132,157,74)(31,75,158,133)(32,134,159,76)(33,77,160,135)(34,136,141,78)(35,79,142,137)(36,138,143,80)(37,61,144,139)(38,140,145,62)(39,63,146,121)(40,122,147,64), (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), (1,28)(2,27)(3,26)(4,25)(5,24)(6,23)(7,22)(8,21)(9,40)(10,39)(11,38)(12,37)(13,36)(14,35)(15,34)(16,33)(17,32)(18,31)(19,30)(20,29)(41,121)(42,140)(43,139)(44,138)(45,137)(46,136)(47,135)(48,134)(49,133)(50,132)(51,131)(52,130)(53,129)(54,128)(55,127)(56,126)(57,125)(58,124)(59,123)(60,122)(61,110)(62,109)(63,108)(64,107)(65,106)(66,105)(67,104)(68,103)(69,102)(70,101)(71,120)(72,119)(73,118)(74,117)(75,116)(76,115)(77,114)(78,113)(79,112)(80,111)(81,144)(82,143)(83,142)(84,141)(85,160)(86,159)(87,158)(88,157)(89,156)(90,155)(91,154)(92,153)(93,152)(94,151)(95,150)(96,149)(97,148)(98,147)(99,146)(100,145) );
G=PermutationGroup([[(1,141),(2,142),(3,143),(4,144),(5,145),(6,146),(7,147),(8,148),(9,149),(10,150),(11,151),(12,152),(13,153),(14,154),(15,155),(16,156),(17,157),(18,158),(19,159),(20,160),(21,97),(22,98),(23,99),(24,100),(25,81),(26,82),(27,83),(28,84),(29,85),(30,86),(31,87),(32,88),(33,89),(34,90),(35,91),(36,92),(37,93),(38,94),(39,95),(40,96),(41,125),(42,126),(43,127),(44,128),(45,129),(46,130),(47,131),(48,132),(49,133),(50,134),(51,135),(52,136),(53,137),(54,138),(55,139),(56,140),(57,121),(58,122),(59,123),(60,124),(61,102),(62,103),(63,104),(64,105),(65,106),(66,107),(67,108),(68,109),(69,110),(70,111),(71,112),(72,113),(73,114),(74,115),(75,116),(76,117),(77,118),(78,119),(79,120),(80,101)], [(1,119,90,52),(2,53,91,120),(3,101,92,54),(4,55,93,102),(5,103,94,56),(6,57,95,104),(7,105,96,58),(8,59,97,106),(9,107,98,60),(10,41,99,108),(11,109,100,42),(12,43,81,110),(13,111,82,44),(14,45,83,112),(15,113,84,46),(16,47,85,114),(17,115,86,48),(18,49,87,116),(19,117,88,50),(20,51,89,118),(21,65,148,123),(22,124,149,66),(23,67,150,125),(24,126,151,68),(25,69,152,127),(26,128,153,70),(27,71,154,129),(28,130,155,72),(29,73,156,131),(30,132,157,74),(31,75,158,133),(32,134,159,76),(33,77,160,135),(34,136,141,78),(35,79,142,137),(36,138,143,80),(37,61,144,139),(38,140,145,62),(39,63,146,121),(40,122,147,64)], [(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)], [(1,28),(2,27),(3,26),(4,25),(5,24),(6,23),(7,22),(8,21),(9,40),(10,39),(11,38),(12,37),(13,36),(14,35),(15,34),(16,33),(17,32),(18,31),(19,30),(20,29),(41,121),(42,140),(43,139),(44,138),(45,137),(46,136),(47,135),(48,134),(49,133),(50,132),(51,131),(52,130),(53,129),(54,128),(55,127),(56,126),(57,125),(58,124),(59,123),(60,122),(61,110),(62,109),(63,108),(64,107),(65,106),(66,105),(67,104),(68,103),(69,102),(70,101),(71,120),(72,119),(73,118),(74,117),(75,116),(76,115),(77,114),(78,113),(79,112),(80,111),(81,144),(82,143),(83,142),(84,141),(85,160),(86,159),(87,158),(88,157),(89,156),(90,155),(91,154),(92,153),(93,152),(94,151),(95,150),(96,149),(97,148),(98,147),(99,146),(100,145)]])
68 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 2L | 2M | 2N | 2O | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 5A | 5B | 10A | ··· | 10N | 20A | ··· | 20X |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 10 | ··· | 10 | 20 | ··· | 20 |
| size | 1 | 1 | ··· | 1 | 10 | 10 | 10 | 10 | 20 | 20 | 20 | 20 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 10 | 10 | 10 | 10 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 |
68 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D5 | C4○D4 | D10 | D10 | D20 | D4×D5 | Q8⋊2D5 |
| kernel | C2×C4⋊D20 | C4⋊D20 | C2×D10⋊C4 | C10×C4⋊C4 | D5×C22×C4 | C22×D20 | C2×C20 | C22×D5 | C2×C4⋊C4 | C2×C10 | C4⋊C4 | C22×C4 | C2×C4 | C22 | C22 |
| # reps | 1 | 8 | 2 | 1 | 1 | 3 | 4 | 4 | 2 | 4 | 8 | 6 | 16 | 4 | 4 |
Matrix representation of C2×C4⋊D20 ►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 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 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 | 29 | 36 |
| 0 | 0 | 0 | 0 | 29 | 12 |
| 34 | 40 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 25 | 39 | 0 | 0 |
| 0 | 0 | 2 | 13 | 0 | 0 |
| 0 | 0 | 0 | 0 | 15 | 37 |
| 0 | 0 | 0 | 0 | 36 | 26 |
| 7 | 1 | 0 | 0 | 0 | 0 |
| 34 | 34 | 0 | 0 | 0 | 0 |
| 0 | 0 | 35 | 1 | 0 | 0 |
| 0 | 0 | 6 | 6 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 28 | 40 |
G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,29,29,0,0,0,0,36,12],[34,1,0,0,0,0,40,0,0,0,0,0,0,0,25,2,0,0,0,0,39,13,0,0,0,0,0,0,15,36,0,0,0,0,37,26],[7,34,0,0,0,0,1,34,0,0,0,0,0,0,35,6,0,0,0,0,1,6,0,0,0,0,0,0,1,28,0,0,0,0,0,40] >;
C2×C4⋊D20 in GAP, Magma, Sage, TeX
C_2\times C_4\rtimes D_{20} % in TeX
G:=Group("C2xC4:D20"); // GroupNames label
G:=SmallGroup(320,1178);
// by ID
G=gap.SmallGroup(320,1178);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,184,675,297,80,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^4=c^20=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations