metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C20⋊Q8⋊18C2, C20⋊9(C4○D4), C4⋊D4⋊25D5, C20⋊D4⋊14C2, C20⋊7D4⋊31C2, C4⋊3(D4⋊2D5), C22.1(D4×D5), C4⋊C4.175D10, (D4×Dic5)⋊14C2, (C2×Dic5)⋊13D4, D20⋊8C4⋊19C2, Dic5⋊2(C4○D4), Dic5⋊D4⋊7C2, Dic5⋊4D4⋊5C2, (C2×D4).150D10, (C2×C20).34C23, C22⋊C4.45D10, Dic5.17(C2×D4), C10.59(C22×D4), (C2×C10).140C24, (C22×C4).366D10, Dic5.5D4⋊16C2, (D4×C10).114C22, (C2×D20).147C22, C4⋊Dic5.203C22, (C22×C10).11C23, C5⋊3(C22.26C24), (C22×D5).59C23, C22.161(C23×D5), C23.178(C22×D5), C23.D5.18C22, D10⋊C4.57C22, (C22×C20).235C22, (C4×Dic5).285C22, (C2×Dic5).233C23, C10.D4.12C22, (C2×Dic10).156C22, (C22×Dic5).244C22, C2.32(C2×D4×D5), (C2×C4×Dic5)⋊7C2, (C5×C4⋊D4)⋊5C2, C2.33(D5×C4○D4), (C2×C10).3(C2×D4), (C2×D4⋊2D5)⋊8C2, C10.79(C2×C4○D4), (C2×C4×D5).88C22, C2.30(C2×D4⋊2D5), (C2×C4).34(C22×D5), (C5×C4⋊C4).136C22, (C2×C5⋊D4).23C22, (C5×C22⋊C4).5C22, SmallGroup(320,1268)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C20⋊(C4○D4)
G = < a,b,c,d | a20=b4=d2=1, c2=b2, bab-1=a9, cac-1=a11, ad=da, bc=cb, bd=db, dcd=b2c >
Subgroups: 1102 in 310 conjugacy classes, 109 normal (43 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C5, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, C23, D5, C10, C10, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×D4, C2×Q8, C4○D4, Dic5, Dic5, C20, C20, D10, C2×C10, C2×C10, C2×C10, C2×C42, C4×D4, C4⋊D4, C4⋊D4, C4.4D4, C4⋊1D4, C4⋊Q8, C2×C4○D4, Dic10, C4×D5, D20, C2×Dic5, C2×Dic5, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C2×C20, C5×D4, C22×D5, C22×C10, C22×C10, C22.26C24, C4×Dic5, C10.D4, C4⋊Dic5, D10⋊C4, C23.D5, C5×C22⋊C4, C5×C4⋊C4, C2×Dic10, C2×C4×D5, C2×D20, D4⋊2D5, C22×Dic5, C22×Dic5, C2×C5⋊D4, C22×C20, D4×C10, D4×C10, Dic5⋊4D4, Dic5.5D4, C20⋊Q8, D20⋊8C4, C2×C4×Dic5, C20⋊7D4, D4×Dic5, Dic5⋊D4, C20⋊D4, C5×C4⋊D4, C2×D4⋊2D5, C20⋊(C4○D4)
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, C4○D4, C24, D10, C22×D4, C2×C4○D4, C22×D5, C22.26C24, D4×D5, D4⋊2D5, C23×D5, C2×D4×D5, C2×D4⋊2D5, D5×C4○D4, C20⋊(C4○D4)
►On 160 points
Generators in S160
(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 95 46 26)(2 84 47 35)(3 93 48 24)(4 82 49 33)(5 91 50 22)(6 100 51 31)(7 89 52 40)(8 98 53 29)(9 87 54 38)(10 96 55 27)(11 85 56 36)(12 94 57 25)(13 83 58 34)(14 92 59 23)(15 81 60 32)(16 90 41 21)(17 99 42 30)(18 88 43 39)(19 97 44 28)(20 86 45 37)(61 123 114 159)(62 132 115 148)(63 121 116 157)(64 130 117 146)(65 139 118 155)(66 128 119 144)(67 137 120 153)(68 126 101 142)(69 135 102 151)(70 124 103 160)(71 133 104 149)(72 122 105 158)(73 131 106 147)(74 140 107 156)(75 129 108 145)(76 138 109 154)(77 127 110 143)(78 136 111 152)(79 125 112 141)(80 134 113 150)
(1 26 46 95)(2 37 47 86)(3 28 48 97)(4 39 49 88)(5 30 50 99)(6 21 51 90)(7 32 52 81)(8 23 53 92)(9 34 54 83)(10 25 55 94)(11 36 56 85)(12 27 57 96)(13 38 58 87)(14 29 59 98)(15 40 60 89)(16 31 41 100)(17 22 42 91)(18 33 43 82)(19 24 44 93)(20 35 45 84)(61 131 114 147)(62 122 115 158)(63 133 116 149)(64 124 117 160)(65 135 118 151)(66 126 119 142)(67 137 120 153)(68 128 101 144)(69 139 102 155)(70 130 103 146)(71 121 104 157)(72 132 105 148)(73 123 106 159)(74 134 107 150)(75 125 108 141)(76 136 109 152)(77 127 110 143)(78 138 111 154)(79 129 112 145)(80 140 113 156)
(1 137)(2 138)(3 139)(4 140)(5 121)(6 122)(7 123)(8 124)(9 125)(10 126)(11 127)(12 128)(13 129)(14 130)(15 131)(16 132)(17 133)(18 134)(19 135)(20 136)(21 62)(22 63)(23 64)(24 65)(25 66)(26 67)(27 68)(28 69)(29 70)(30 71)(31 72)(32 73)(33 74)(34 75)(35 76)(36 77)(37 78)(38 79)(39 80)(40 61)(41 148)(42 149)(43 150)(44 151)(45 152)(46 153)(47 154)(48 155)(49 156)(50 157)(51 158)(52 159)(53 160)(54 141)(55 142)(56 143)(57 144)(58 145)(59 146)(60 147)(81 106)(82 107)(83 108)(84 109)(85 110)(86 111)(87 112)(88 113)(89 114)(90 115)(91 116)(92 117)(93 118)(94 119)(95 120)(96 101)(97 102)(98 103)(99 104)(100 105)
G:=sub<Sym(160)| (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), (1,95,46,26)(2,84,47,35)(3,93,48,24)(4,82,49,33)(5,91,50,22)(6,100,51,31)(7,89,52,40)(8,98,53,29)(9,87,54,38)(10,96,55,27)(11,85,56,36)(12,94,57,25)(13,83,58,34)(14,92,59,23)(15,81,60,32)(16,90,41,21)(17,99,42,30)(18,88,43,39)(19,97,44,28)(20,86,45,37)(61,123,114,159)(62,132,115,148)(63,121,116,157)(64,130,117,146)(65,139,118,155)(66,128,119,144)(67,137,120,153)(68,126,101,142)(69,135,102,151)(70,124,103,160)(71,133,104,149)(72,122,105,158)(73,131,106,147)(74,140,107,156)(75,129,108,145)(76,138,109,154)(77,127,110,143)(78,136,111,152)(79,125,112,141)(80,134,113,150), (1,26,46,95)(2,37,47,86)(3,28,48,97)(4,39,49,88)(5,30,50,99)(6,21,51,90)(7,32,52,81)(8,23,53,92)(9,34,54,83)(10,25,55,94)(11,36,56,85)(12,27,57,96)(13,38,58,87)(14,29,59,98)(15,40,60,89)(16,31,41,100)(17,22,42,91)(18,33,43,82)(19,24,44,93)(20,35,45,84)(61,131,114,147)(62,122,115,158)(63,133,116,149)(64,124,117,160)(65,135,118,151)(66,126,119,142)(67,137,120,153)(68,128,101,144)(69,139,102,155)(70,130,103,146)(71,121,104,157)(72,132,105,148)(73,123,106,159)(74,134,107,150)(75,125,108,141)(76,136,109,152)(77,127,110,143)(78,138,111,154)(79,129,112,145)(80,140,113,156), (1,137)(2,138)(3,139)(4,140)(5,121)(6,122)(7,123)(8,124)(9,125)(10,126)(11,127)(12,128)(13,129)(14,130)(15,131)(16,132)(17,133)(18,134)(19,135)(20,136)(21,62)(22,63)(23,64)(24,65)(25,66)(26,67)(27,68)(28,69)(29,70)(30,71)(31,72)(32,73)(33,74)(34,75)(35,76)(36,77)(37,78)(38,79)(39,80)(40,61)(41,148)(42,149)(43,150)(44,151)(45,152)(46,153)(47,154)(48,155)(49,156)(50,157)(51,158)(52,159)(53,160)(54,141)(55,142)(56,143)(57,144)(58,145)(59,146)(60,147)(81,106)(82,107)(83,108)(84,109)(85,110)(86,111)(87,112)(88,113)(89,114)(90,115)(91,116)(92,117)(93,118)(94,119)(95,120)(96,101)(97,102)(98,103)(99,104)(100,105)>;
G:=Group( (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), (1,95,46,26)(2,84,47,35)(3,93,48,24)(4,82,49,33)(5,91,50,22)(6,100,51,31)(7,89,52,40)(8,98,53,29)(9,87,54,38)(10,96,55,27)(11,85,56,36)(12,94,57,25)(13,83,58,34)(14,92,59,23)(15,81,60,32)(16,90,41,21)(17,99,42,30)(18,88,43,39)(19,97,44,28)(20,86,45,37)(61,123,114,159)(62,132,115,148)(63,121,116,157)(64,130,117,146)(65,139,118,155)(66,128,119,144)(67,137,120,153)(68,126,101,142)(69,135,102,151)(70,124,103,160)(71,133,104,149)(72,122,105,158)(73,131,106,147)(74,140,107,156)(75,129,108,145)(76,138,109,154)(77,127,110,143)(78,136,111,152)(79,125,112,141)(80,134,113,150), (1,26,46,95)(2,37,47,86)(3,28,48,97)(4,39,49,88)(5,30,50,99)(6,21,51,90)(7,32,52,81)(8,23,53,92)(9,34,54,83)(10,25,55,94)(11,36,56,85)(12,27,57,96)(13,38,58,87)(14,29,59,98)(15,40,60,89)(16,31,41,100)(17,22,42,91)(18,33,43,82)(19,24,44,93)(20,35,45,84)(61,131,114,147)(62,122,115,158)(63,133,116,149)(64,124,117,160)(65,135,118,151)(66,126,119,142)(67,137,120,153)(68,128,101,144)(69,139,102,155)(70,130,103,146)(71,121,104,157)(72,132,105,148)(73,123,106,159)(74,134,107,150)(75,125,108,141)(76,136,109,152)(77,127,110,143)(78,138,111,154)(79,129,112,145)(80,140,113,156), (1,137)(2,138)(3,139)(4,140)(5,121)(6,122)(7,123)(8,124)(9,125)(10,126)(11,127)(12,128)(13,129)(14,130)(15,131)(16,132)(17,133)(18,134)(19,135)(20,136)(21,62)(22,63)(23,64)(24,65)(25,66)(26,67)(27,68)(28,69)(29,70)(30,71)(31,72)(32,73)(33,74)(34,75)(35,76)(36,77)(37,78)(38,79)(39,80)(40,61)(41,148)(42,149)(43,150)(44,151)(45,152)(46,153)(47,154)(48,155)(49,156)(50,157)(51,158)(52,159)(53,160)(54,141)(55,142)(56,143)(57,144)(58,145)(59,146)(60,147)(81,106)(82,107)(83,108)(84,109)(85,110)(86,111)(87,112)(88,113)(89,114)(90,115)(91,116)(92,117)(93,118)(94,119)(95,120)(96,101)(97,102)(98,103)(99,104)(100,105) );
G=PermutationGroup([[(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)], [(1,95,46,26),(2,84,47,35),(3,93,48,24),(4,82,49,33),(5,91,50,22),(6,100,51,31),(7,89,52,40),(8,98,53,29),(9,87,54,38),(10,96,55,27),(11,85,56,36),(12,94,57,25),(13,83,58,34),(14,92,59,23),(15,81,60,32),(16,90,41,21),(17,99,42,30),(18,88,43,39),(19,97,44,28),(20,86,45,37),(61,123,114,159),(62,132,115,148),(63,121,116,157),(64,130,117,146),(65,139,118,155),(66,128,119,144),(67,137,120,153),(68,126,101,142),(69,135,102,151),(70,124,103,160),(71,133,104,149),(72,122,105,158),(73,131,106,147),(74,140,107,156),(75,129,108,145),(76,138,109,154),(77,127,110,143),(78,136,111,152),(79,125,112,141),(80,134,113,150)], [(1,26,46,95),(2,37,47,86),(3,28,48,97),(4,39,49,88),(5,30,50,99),(6,21,51,90),(7,32,52,81),(8,23,53,92),(9,34,54,83),(10,25,55,94),(11,36,56,85),(12,27,57,96),(13,38,58,87),(14,29,59,98),(15,40,60,89),(16,31,41,100),(17,22,42,91),(18,33,43,82),(19,24,44,93),(20,35,45,84),(61,131,114,147),(62,122,115,158),(63,133,116,149),(64,124,117,160),(65,135,118,151),(66,126,119,142),(67,137,120,153),(68,128,101,144),(69,139,102,155),(70,130,103,146),(71,121,104,157),(72,132,105,148),(73,123,106,159),(74,134,107,150),(75,125,108,141),(76,136,109,152),(77,127,110,143),(78,138,111,154),(79,129,112,145),(80,140,113,156)], [(1,137),(2,138),(3,139),(4,140),(5,121),(6,122),(7,123),(8,124),(9,125),(10,126),(11,127),(12,128),(13,129),(14,130),(15,131),(16,132),(17,133),(18,134),(19,135),(20,136),(21,62),(22,63),(23,64),(24,65),(25,66),(26,67),(27,68),(28,69),(29,70),(30,71),(31,72),(32,73),(33,74),(34,75),(35,76),(36,77),(37,78),(38,79),(39,80),(40,61),(41,148),(42,149),(43,150),(44,151),(45,152),(46,153),(47,154),(48,155),(49,156),(50,157),(51,158),(52,159),(53,160),(54,141),(55,142),(56,143),(57,144),(58,145),(59,146),(60,147),(81,106),(82,107),(83,108),(84,109),(85,110),(86,111),(87,112),(88,113),(89,114),(90,115),(91,116),(92,117),(93,118),(94,119),(95,120),(96,101),(97,102),(98,103),(99,104),(100,105)]])
56 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | ··· | 4P | 4Q | 4R | 5A | 5B | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 10K | 10L | 10M | 10N | 20A | ··· | 20H | 20I | 20J | 20K | 20L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 5 | 5 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 20 | ··· | 20 | 20 | 20 | 20 | 20 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 20 | 20 | 2 | 2 | 2 | 2 | 4 | 4 | 5 | 5 | 5 | 5 | 10 | ··· | 10 | 20 | 20 | 2 | 2 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 4 | ··· | 4 | 8 | 8 | 8 | 8 |
56 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D5 | C4○D4 | C4○D4 | D10 | D10 | D10 | D10 | D4⋊2D5 | D4×D5 | D5×C4○D4 |
| kernel | C20⋊(C4○D4) | Dic5⋊4D4 | Dic5.5D4 | C20⋊Q8 | D20⋊8C4 | C2×C4×Dic5 | C20⋊7D4 | D4×Dic5 | Dic5⋊D4 | C20⋊D4 | C5×C4⋊D4 | C2×D4⋊2D5 | C2×Dic5 | C4⋊D4 | Dic5 | C20 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×D4 | C4 | C22 | C2 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 2 | 4 | 2 | 4 | 4 | 4 | 2 | 2 | 6 | 4 | 4 | 4 |
Matrix representation of C20⋊(C4○D4) ►in GL6(𝔽41)
| 0 | 1 | 0 | 0 | 0 | 0 |
| 40 | 35 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 39 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 35 | 6 | 0 | 0 | 0 | 0 |
| 1 | 6 | 0 | 0 | 0 | 0 |
| 0 | 0 | 9 | 0 | 0 | 0 |
| 0 | 0 | 0 | 9 | 0 | 0 |
| 0 | 0 | 0 | 0 | 32 | 0 |
| 0 | 0 | 0 | 0 | 0 | 32 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 32 | 0 | 0 | 0 |
| 0 | 0 | 9 | 9 | 0 | 0 |
| 0 | 0 | 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 0 | 1 | 32 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 9 | 18 | 0 | 0 |
| 0 | 0 | 32 | 32 | 0 | 0 |
| 0 | 0 | 0 | 0 | 9 | 2 |
| 0 | 0 | 0 | 0 | 1 | 32 |
G:=sub<GL(6,GF(41))| [0,40,0,0,0,0,1,35,0,0,0,0,0,0,40,1,0,0,0,0,39,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[35,1,0,0,0,0,6,6,0,0,0,0,0,0,9,0,0,0,0,0,0,9,0,0,0,0,0,0,32,0,0,0,0,0,0,32],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,32,9,0,0,0,0,0,9,0,0,0,0,0,0,9,1,0,0,0,0,0,32],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,9,32,0,0,0,0,18,32,0,0,0,0,0,0,9,1,0,0,0,0,2,32] >;
C20⋊(C4○D4) in GAP, Magma, Sage, TeX
C_{20}\rtimes (C_4\circ D_4) % in TeX
G:=Group("C20:(C4oD4)"); // GroupNames label
G:=SmallGroup(320,1268);
// by ID
G=gap.SmallGroup(320,1268);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,387,570,185,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^20=b^4=d^2=1,c^2=b^2,b*a*b^-1=a^9,c*a*c^-1=a^11,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=b^2*c>;
// generators/relations