direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×C20.17D4, C24.37D10, C20.251(C2×D4), (C2×C20).209D4, (C2×D4).229D10, C10⋊3(C4.4D4), (C22×D4).11D5, (C2×C20).540C23, (C2×C10).292C24, (C4×Dic5)⋊67C22, C10.140(C22×D4), (C22×C4).378D10, C23.D5⋊58C22, (C2×Dic10)⋊67C22, (C22×Dic10)⋊20C2, (D4×C10).269C22, (C23×C10).74C22, C23.134(C22×D5), C22.306(C23×D5), C22.78(D4⋊2D5), (C22×C10).228C23, (C22×C20).273C22, (C2×Dic5).292C23, (C22×Dic5).254C22, (D4×C2×C10).8C2, C5⋊4(C2×C4.4D4), (C2×C4×Dic5)⋊11C2, C4.23(C2×C5⋊D4), C10.104(C2×C4○D4), (C2×C10).579(C2×D4), C2.68(C2×D4⋊2D5), (C2×C23.D5)⋊25C2, C2.13(C22×C5⋊D4), (C2×C4).153(C5⋊D4), (C2×C4).623(C22×D5), C22.109(C2×C5⋊D4), (C2×C10).176(C4○D4), SmallGroup(320,1469)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×C20.17D4
G = < a,b,c,d | a2=b20=c4=1, d2=b10, ab=ba, ac=ca, ad=da, cbc-1=b9, dbd-1=b-1, dcd-1=b10c-1 >
Subgroups: 958 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, C23, C10, C10, C10, C42, C22⋊C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C24, Dic5, C20, C2×C10, C2×C10, C2×C10, C2×C42, C2×C22⋊C4, C4.4D4, C22×D4, C22×Q8, Dic10, C2×Dic5, C2×Dic5, C2×C20, C5×D4, C22×C10, C22×C10, C22×C10, C2×C4.4D4, C4×Dic5, C23.D5, C2×Dic10, C2×Dic10, C22×Dic5, C22×C20, D4×C10, D4×C10, C23×C10, C2×C4×Dic5, C20.17D4, C2×C23.D5, C22×Dic10, D4×C2×C10, C2×C20.17D4
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, C4○D4, C24, D10, C4.4D4, C22×D4, C2×C4○D4, C5⋊D4, C22×D5, C2×C4.4D4, D4⋊2D5, C2×C5⋊D4, C23×D5, C20.17D4, C2×D4⋊2D5, C22×C5⋊D4, C2×C20.17D4
►On 160 points
Generators in S160
(1 38)(2 39)(3 40)(4 21)(5 22)(6 23)(7 24)(8 25)(9 26)(10 27)(11 28)(12 29)(13 30)(14 31)(15 32)(16 33)(17 34)(18 35)(19 36)(20 37)(41 110)(42 111)(43 112)(44 113)(45 114)(46 115)(47 116)(48 117)(49 118)(50 119)(51 120)(52 101)(53 102)(54 103)(55 104)(56 105)(57 106)(58 107)(59 108)(60 109)(61 138)(62 139)(63 140)(64 121)(65 122)(66 123)(67 124)(68 125)(69 126)(70 127)(71 128)(72 129)(73 130)(74 131)(75 132)(76 133)(77 134)(78 135)(79 136)(80 137)(81 155)(82 156)(83 157)(84 158)(85 159)(86 160)(87 141)(88 142)(89 143)(90 144)(91 145)(92 146)(93 147)(94 148)(95 149)(96 150)(97 151)(98 152)(99 153)(100 154)
(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)
(1 151 119 139)(2 160 120 128)(3 149 101 137)(4 158 102 126)(5 147 103 135)(6 156 104 124)(7 145 105 133)(8 154 106 122)(9 143 107 131)(10 152 108 140)(11 141 109 129)(12 150 110 138)(13 159 111 127)(14 148 112 136)(15 157 113 125)(16 146 114 134)(17 155 115 123)(18 144 116 132)(19 153 117 121)(20 142 118 130)(21 84 53 69)(22 93 54 78)(23 82 55 67)(24 91 56 76)(25 100 57 65)(26 89 58 74)(27 98 59 63)(28 87 60 72)(29 96 41 61)(30 85 42 70)(31 94 43 79)(32 83 44 68)(33 92 45 77)(34 81 46 66)(35 90 47 75)(36 99 48 64)(37 88 49 73)(38 97 50 62)(39 86 51 71)(40 95 52 80)
(1 97 11 87)(2 96 12 86)(3 95 13 85)(4 94 14 84)(5 93 15 83)(6 92 16 82)(7 91 17 81)(8 90 18 100)(9 89 19 99)(10 88 20 98)(21 148 31 158)(22 147 32 157)(23 146 33 156)(24 145 34 155)(25 144 35 154)(26 143 36 153)(27 142 37 152)(28 141 38 151)(29 160 39 150)(30 159 40 149)(41 128 51 138)(42 127 52 137)(43 126 53 136)(44 125 54 135)(45 124 55 134)(46 123 56 133)(47 122 57 132)(48 121 58 131)(49 140 59 130)(50 139 60 129)(61 110 71 120)(62 109 72 119)(63 108 73 118)(64 107 74 117)(65 106 75 116)(66 105 76 115)(67 104 77 114)(68 103 78 113)(69 102 79 112)(70 101 80 111)
G:=sub<Sym(160)| (1,38)(2,39)(3,40)(4,21)(5,22)(6,23)(7,24)(8,25)(9,26)(10,27)(11,28)(12,29)(13,30)(14,31)(15,32)(16,33)(17,34)(18,35)(19,36)(20,37)(41,110)(42,111)(43,112)(44,113)(45,114)(46,115)(47,116)(48,117)(49,118)(50,119)(51,120)(52,101)(53,102)(54,103)(55,104)(56,105)(57,106)(58,107)(59,108)(60,109)(61,138)(62,139)(63,140)(64,121)(65,122)(66,123)(67,124)(68,125)(69,126)(70,127)(71,128)(72,129)(73,130)(74,131)(75,132)(76,133)(77,134)(78,135)(79,136)(80,137)(81,155)(82,156)(83,157)(84,158)(85,159)(86,160)(87,141)(88,142)(89,143)(90,144)(91,145)(92,146)(93,147)(94,148)(95,149)(96,150)(97,151)(98,152)(99,153)(100,154), (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), (1,151,119,139)(2,160,120,128)(3,149,101,137)(4,158,102,126)(5,147,103,135)(6,156,104,124)(7,145,105,133)(8,154,106,122)(9,143,107,131)(10,152,108,140)(11,141,109,129)(12,150,110,138)(13,159,111,127)(14,148,112,136)(15,157,113,125)(16,146,114,134)(17,155,115,123)(18,144,116,132)(19,153,117,121)(20,142,118,130)(21,84,53,69)(22,93,54,78)(23,82,55,67)(24,91,56,76)(25,100,57,65)(26,89,58,74)(27,98,59,63)(28,87,60,72)(29,96,41,61)(30,85,42,70)(31,94,43,79)(32,83,44,68)(33,92,45,77)(34,81,46,66)(35,90,47,75)(36,99,48,64)(37,88,49,73)(38,97,50,62)(39,86,51,71)(40,95,52,80), (1,97,11,87)(2,96,12,86)(3,95,13,85)(4,94,14,84)(5,93,15,83)(6,92,16,82)(7,91,17,81)(8,90,18,100)(9,89,19,99)(10,88,20,98)(21,148,31,158)(22,147,32,157)(23,146,33,156)(24,145,34,155)(25,144,35,154)(26,143,36,153)(27,142,37,152)(28,141,38,151)(29,160,39,150)(30,159,40,149)(41,128,51,138)(42,127,52,137)(43,126,53,136)(44,125,54,135)(45,124,55,134)(46,123,56,133)(47,122,57,132)(48,121,58,131)(49,140,59,130)(50,139,60,129)(61,110,71,120)(62,109,72,119)(63,108,73,118)(64,107,74,117)(65,106,75,116)(66,105,76,115)(67,104,77,114)(68,103,78,113)(69,102,79,112)(70,101,80,111)>;
G:=Group( (1,38)(2,39)(3,40)(4,21)(5,22)(6,23)(7,24)(8,25)(9,26)(10,27)(11,28)(12,29)(13,30)(14,31)(15,32)(16,33)(17,34)(18,35)(19,36)(20,37)(41,110)(42,111)(43,112)(44,113)(45,114)(46,115)(47,116)(48,117)(49,118)(50,119)(51,120)(52,101)(53,102)(54,103)(55,104)(56,105)(57,106)(58,107)(59,108)(60,109)(61,138)(62,139)(63,140)(64,121)(65,122)(66,123)(67,124)(68,125)(69,126)(70,127)(71,128)(72,129)(73,130)(74,131)(75,132)(76,133)(77,134)(78,135)(79,136)(80,137)(81,155)(82,156)(83,157)(84,158)(85,159)(86,160)(87,141)(88,142)(89,143)(90,144)(91,145)(92,146)(93,147)(94,148)(95,149)(96,150)(97,151)(98,152)(99,153)(100,154), (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), (1,151,119,139)(2,160,120,128)(3,149,101,137)(4,158,102,126)(5,147,103,135)(6,156,104,124)(7,145,105,133)(8,154,106,122)(9,143,107,131)(10,152,108,140)(11,141,109,129)(12,150,110,138)(13,159,111,127)(14,148,112,136)(15,157,113,125)(16,146,114,134)(17,155,115,123)(18,144,116,132)(19,153,117,121)(20,142,118,130)(21,84,53,69)(22,93,54,78)(23,82,55,67)(24,91,56,76)(25,100,57,65)(26,89,58,74)(27,98,59,63)(28,87,60,72)(29,96,41,61)(30,85,42,70)(31,94,43,79)(32,83,44,68)(33,92,45,77)(34,81,46,66)(35,90,47,75)(36,99,48,64)(37,88,49,73)(38,97,50,62)(39,86,51,71)(40,95,52,80), (1,97,11,87)(2,96,12,86)(3,95,13,85)(4,94,14,84)(5,93,15,83)(6,92,16,82)(7,91,17,81)(8,90,18,100)(9,89,19,99)(10,88,20,98)(21,148,31,158)(22,147,32,157)(23,146,33,156)(24,145,34,155)(25,144,35,154)(26,143,36,153)(27,142,37,152)(28,141,38,151)(29,160,39,150)(30,159,40,149)(41,128,51,138)(42,127,52,137)(43,126,53,136)(44,125,54,135)(45,124,55,134)(46,123,56,133)(47,122,57,132)(48,121,58,131)(49,140,59,130)(50,139,60,129)(61,110,71,120)(62,109,72,119)(63,108,73,118)(64,107,74,117)(65,106,75,116)(66,105,76,115)(67,104,77,114)(68,103,78,113)(69,102,79,112)(70,101,80,111) );
G=PermutationGroup([[(1,38),(2,39),(3,40),(4,21),(5,22),(6,23),(7,24),(8,25),(9,26),(10,27),(11,28),(12,29),(13,30),(14,31),(15,32),(16,33),(17,34),(18,35),(19,36),(20,37),(41,110),(42,111),(43,112),(44,113),(45,114),(46,115),(47,116),(48,117),(49,118),(50,119),(51,120),(52,101),(53,102),(54,103),(55,104),(56,105),(57,106),(58,107),(59,108),(60,109),(61,138),(62,139),(63,140),(64,121),(65,122),(66,123),(67,124),(68,125),(69,126),(70,127),(71,128),(72,129),(73,130),(74,131),(75,132),(76,133),(77,134),(78,135),(79,136),(80,137),(81,155),(82,156),(83,157),(84,158),(85,159),(86,160),(87,141),(88,142),(89,143),(90,144),(91,145),(92,146),(93,147),(94,148),(95,149),(96,150),(97,151),(98,152),(99,153),(100,154)], [(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)], [(1,151,119,139),(2,160,120,128),(3,149,101,137),(4,158,102,126),(5,147,103,135),(6,156,104,124),(7,145,105,133),(8,154,106,122),(9,143,107,131),(10,152,108,140),(11,141,109,129),(12,150,110,138),(13,159,111,127),(14,148,112,136),(15,157,113,125),(16,146,114,134),(17,155,115,123),(18,144,116,132),(19,153,117,121),(20,142,118,130),(21,84,53,69),(22,93,54,78),(23,82,55,67),(24,91,56,76),(25,100,57,65),(26,89,58,74),(27,98,59,63),(28,87,60,72),(29,96,41,61),(30,85,42,70),(31,94,43,79),(32,83,44,68),(33,92,45,77),(34,81,46,66),(35,90,47,75),(36,99,48,64),(37,88,49,73),(38,97,50,62),(39,86,51,71),(40,95,52,80)], [(1,97,11,87),(2,96,12,86),(3,95,13,85),(4,94,14,84),(5,93,15,83),(6,92,16,82),(7,91,17,81),(8,90,18,100),(9,89,19,99),(10,88,20,98),(21,148,31,158),(22,147,32,157),(23,146,33,156),(24,145,34,155),(25,144,35,154),(26,143,36,153),(27,142,37,152),(28,141,38,151),(29,160,39,150),(30,159,40,149),(41,128,51,138),(42,127,52,137),(43,126,53,136),(44,125,54,135),(45,124,55,134),(46,123,56,133),(47,122,57,132),(48,121,58,131),(49,140,59,130),(50,139,60,129),(61,110,71,120),(62,109,72,119),(63,108,73,118),(64,107,74,117),(65,106,75,116),(66,105,76,115),(67,104,77,114),(68,103,78,113),(69,102,79,112),(70,101,80,111)]])
68 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 4A | 4B | 4C | 4D | 4E | ··· | 4L | 4M | 4N | 4O | 4P | 5A | 5B | 10A | ··· | 10N | 10O | ··· | 10AD | 20A | ··· | 20H |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | ··· | 20 |
| size | 1 | 1 | ··· | 1 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 10 | ··· | 10 | 20 | 20 | 20 | 20 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
68 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | D5 | C4○D4 | D10 | D10 | D10 | C5⋊D4 | D4⋊2D5 |
| kernel | C2×C20.17D4 | C2×C4×Dic5 | C20.17D4 | C2×C23.D5 | C22×Dic10 | D4×C2×C10 | C2×C20 | C22×D4 | C2×C10 | C22×C4 | C2×D4 | C24 | C2×C4 | C22 |
| # reps | 1 | 1 | 8 | 4 | 1 | 1 | 4 | 2 | 8 | 2 | 8 | 4 | 16 | 8 |
Matrix representation of C2×C20.17D4 ►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 | 0 | 1 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 23 | 0 |
| 0 | 0 | 0 | 18 | 25 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 | 0 |
| 0 | 0 | 9 | 0 | 0 |
| 0 | 0 | 0 | 23 | 39 |
| 0 | 0 | 0 | 19 | 18 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 | 0 |
| 0 | 0 | 32 | 0 | 0 |
| 0 | 0 | 0 | 23 | 39 |
| 0 | 0 | 0 | 18 | 18 |
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],[40,0,0,0,0,0,0,40,0,0,0,1,0,0,0,0,0,0,23,18,0,0,0,0,25],[1,0,0,0,0,0,9,0,0,0,0,0,9,0,0,0,0,0,23,19,0,0,0,39,18],[1,0,0,0,0,0,9,0,0,0,0,0,32,0,0,0,0,0,23,18,0,0,0,39,18] >;
C2×C20.17D4 in GAP, Magma, Sage, TeX
C_2\times C_{20}._{17}D_4 % in TeX
G:=Group("C2xC20.17D4"); // GroupNames label
G:=SmallGroup(320,1469);
// by ID
G=gap.SmallGroup(320,1469);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,100,1571,185,12550]);
// 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=b^9,d*b*d^-1=b^-1,d*c*d^-1=b^10*c^-1>;
// generators/relations