metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C40:14D4, D10:4SD16, C5:7(C8:8D4), C8:11(C5:D4), C40:6C4:27C2, D10:3Q8:4C2, (C10xSD16):8C2, (C2xSD16):12D5, (C2xD4).73D10, (C2xC8).263D10, C20:2D4.9C2, C20.176(C2xD4), (C2xQ8).54D10, C2.30(D5xSD16), C10.63(C4oD8), Q8:Dic5:29C2, D4:Dic5:34C2, C10.47(C2xSD16), (C22xD5).90D4, C22.268(D4xD5), C20.101(C4oD4), C4.32(D4:2D5), C2.19(C20:2D4), (C2xC40).164C22, (C2xC20).448C23, (C2xDic5).159D4, (D4xC10).97C22, (Q8xC10).77C22, C10.116(C4:D4), C4:Dic5.175C22, C2.29(SD16:3D5), (D5xC2xC8):8C2, C4.82(C2xC5:D4), (C2xC10).360(C2xD4), (C2xC4xD5).310C22, (C2xC4).537(C22xD5), (C2xC5:2C8).282C22, SmallGroup(320,798)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C40:14D4
G = < a,b,c | a40=b4=c2=1, bab-1=a19, cac=a9, cbc=b-1 >
Subgroups: 486 in 124 conjugacy classes, 43 normal (37 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C8, C2xC4, C2xC4, D4, Q8, C23, D5, C10, C10, C22:C4, C4:C4, C2xC8, C2xC8, SD16, C22xC4, C2xD4, C2xD4, C2xQ8, Dic5, C20, C20, D10, D10, C2xC10, C2xC10, D4:C4, Q8:C4, C4.Q8, C4:D4, C22:Q8, C22xC8, C2xSD16, C5:2C8, C40, C4xD5, C2xDic5, C2xDic5, C5:D4, C2xC20, C2xC20, C5xD4, C5xQ8, C22xD5, C22xC10, C8:8D4, C8xD5, C2xC5:2C8, C10.D4, C4:Dic5, D10:C4, C23.D5, C2xC40, C5xSD16, C2xC4xD5, C2xC5:D4, D4xC10, Q8xC10, C40:6C4, D4:Dic5, Q8:Dic5, D5xC2xC8, C20:2D4, D10:3Q8, C10xSD16, C40:14D4
Quotients: C1, C2, C22, D4, C23, D5, SD16, C2xD4, C4oD4, D10, C4:D4, C2xSD16, C4oD8, C5:D4, C22xD5, C8:8D4, D4xD5, D4:2D5, C2xC5:D4, D5xSD16, SD16:3D5, C20:2D4, C40:14D4
►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)
(1 104 61 131)(2 83 62 150)(3 102 63 129)(4 81 64 148)(5 100 65 127)(6 119 66 146)(7 98 67 125)(8 117 68 144)(9 96 69 123)(10 115 70 142)(11 94 71 121)(12 113 72 140)(13 92 73 159)(14 111 74 138)(15 90 75 157)(16 109 76 136)(17 88 77 155)(18 107 78 134)(19 86 79 153)(20 105 80 132)(21 84 41 151)(22 103 42 130)(23 82 43 149)(24 101 44 128)(25 120 45 147)(26 99 46 126)(27 118 47 145)(28 97 48 124)(29 116 49 143)(30 95 50 122)(31 114 51 141)(32 93 52 160)(33 112 53 139)(34 91 54 158)(35 110 55 137)(36 89 56 156)(37 108 57 135)(38 87 58 154)(39 106 59 133)(40 85 60 152)
(2 10)(3 19)(4 28)(5 37)(7 15)(8 24)(9 33)(12 20)(13 29)(14 38)(17 25)(18 34)(22 30)(23 39)(27 35)(32 40)(42 50)(43 59)(44 68)(45 77)(47 55)(48 64)(49 73)(52 60)(53 69)(54 78)(57 65)(58 74)(62 70)(63 79)(67 75)(72 80)(81 124)(82 133)(83 142)(84 151)(85 160)(86 129)(87 138)(88 147)(89 156)(90 125)(91 134)(92 143)(93 152)(94 121)(95 130)(96 139)(97 148)(98 157)(99 126)(100 135)(101 144)(102 153)(103 122)(104 131)(105 140)(106 149)(107 158)(108 127)(109 136)(110 145)(111 154)(112 123)(113 132)(114 141)(115 150)(116 159)(117 128)(118 137)(119 146)(120 155)
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), (1,104,61,131)(2,83,62,150)(3,102,63,129)(4,81,64,148)(5,100,65,127)(6,119,66,146)(7,98,67,125)(8,117,68,144)(9,96,69,123)(10,115,70,142)(11,94,71,121)(12,113,72,140)(13,92,73,159)(14,111,74,138)(15,90,75,157)(16,109,76,136)(17,88,77,155)(18,107,78,134)(19,86,79,153)(20,105,80,132)(21,84,41,151)(22,103,42,130)(23,82,43,149)(24,101,44,128)(25,120,45,147)(26,99,46,126)(27,118,47,145)(28,97,48,124)(29,116,49,143)(30,95,50,122)(31,114,51,141)(32,93,52,160)(33,112,53,139)(34,91,54,158)(35,110,55,137)(36,89,56,156)(37,108,57,135)(38,87,58,154)(39,106,59,133)(40,85,60,152), (2,10)(3,19)(4,28)(5,37)(7,15)(8,24)(9,33)(12,20)(13,29)(14,38)(17,25)(18,34)(22,30)(23,39)(27,35)(32,40)(42,50)(43,59)(44,68)(45,77)(47,55)(48,64)(49,73)(52,60)(53,69)(54,78)(57,65)(58,74)(62,70)(63,79)(67,75)(72,80)(81,124)(82,133)(83,142)(84,151)(85,160)(86,129)(87,138)(88,147)(89,156)(90,125)(91,134)(92,143)(93,152)(94,121)(95,130)(96,139)(97,148)(98,157)(99,126)(100,135)(101,144)(102,153)(103,122)(104,131)(105,140)(106,149)(107,158)(108,127)(109,136)(110,145)(111,154)(112,123)(113,132)(114,141)(115,150)(116,159)(117,128)(118,137)(119,146)(120,155)>;
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), (1,104,61,131)(2,83,62,150)(3,102,63,129)(4,81,64,148)(5,100,65,127)(6,119,66,146)(7,98,67,125)(8,117,68,144)(9,96,69,123)(10,115,70,142)(11,94,71,121)(12,113,72,140)(13,92,73,159)(14,111,74,138)(15,90,75,157)(16,109,76,136)(17,88,77,155)(18,107,78,134)(19,86,79,153)(20,105,80,132)(21,84,41,151)(22,103,42,130)(23,82,43,149)(24,101,44,128)(25,120,45,147)(26,99,46,126)(27,118,47,145)(28,97,48,124)(29,116,49,143)(30,95,50,122)(31,114,51,141)(32,93,52,160)(33,112,53,139)(34,91,54,158)(35,110,55,137)(36,89,56,156)(37,108,57,135)(38,87,58,154)(39,106,59,133)(40,85,60,152), (2,10)(3,19)(4,28)(5,37)(7,15)(8,24)(9,33)(12,20)(13,29)(14,38)(17,25)(18,34)(22,30)(23,39)(27,35)(32,40)(42,50)(43,59)(44,68)(45,77)(47,55)(48,64)(49,73)(52,60)(53,69)(54,78)(57,65)(58,74)(62,70)(63,79)(67,75)(72,80)(81,124)(82,133)(83,142)(84,151)(85,160)(86,129)(87,138)(88,147)(89,156)(90,125)(91,134)(92,143)(93,152)(94,121)(95,130)(96,139)(97,148)(98,157)(99,126)(100,135)(101,144)(102,153)(103,122)(104,131)(105,140)(106,149)(107,158)(108,127)(109,136)(110,145)(111,154)(112,123)(113,132)(114,141)(115,150)(116,159)(117,128)(118,137)(119,146)(120,155) );
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)], [(1,104,61,131),(2,83,62,150),(3,102,63,129),(4,81,64,148),(5,100,65,127),(6,119,66,146),(7,98,67,125),(8,117,68,144),(9,96,69,123),(10,115,70,142),(11,94,71,121),(12,113,72,140),(13,92,73,159),(14,111,74,138),(15,90,75,157),(16,109,76,136),(17,88,77,155),(18,107,78,134),(19,86,79,153),(20,105,80,132),(21,84,41,151),(22,103,42,130),(23,82,43,149),(24,101,44,128),(25,120,45,147),(26,99,46,126),(27,118,47,145),(28,97,48,124),(29,116,49,143),(30,95,50,122),(31,114,51,141),(32,93,52,160),(33,112,53,139),(34,91,54,158),(35,110,55,137),(36,89,56,156),(37,108,57,135),(38,87,58,154),(39,106,59,133),(40,85,60,152)], [(2,10),(3,19),(4,28),(5,37),(7,15),(8,24),(9,33),(12,20),(13,29),(14,38),(17,25),(18,34),(22,30),(23,39),(27,35),(32,40),(42,50),(43,59),(44,68),(45,77),(47,55),(48,64),(49,73),(52,60),(53,69),(54,78),(57,65),(58,74),(62,70),(63,79),(67,75),(72,80),(81,124),(82,133),(83,142),(84,151),(85,160),(86,129),(87,138),(88,147),(89,156),(90,125),(91,134),(92,143),(93,152),(94,121),(95,130),(96,139),(97,148),(98,157),(99,126),(100,135),(101,144),(102,153),(103,122),(104,131),(105,140),(106,149),(107,158),(108,127),(109,136),(110,145),(111,154),(112,123),(113,132),(114,141),(115,150),(116,159),(117,128),(118,137),(119,146),(120,155)]])
50 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 5A | 5B | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 20A | 20B | 20C | 20D | 20E | 20F | 20G | 20H | 40A | ··· | 40H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 8 | 10 | 10 | 2 | 2 | 8 | 10 | 10 | 40 | 40 | 2 | 2 | 2 | 2 | 2 | 2 | 10 | 10 | 10 | 10 | 2 | ··· | 2 | 8 | 8 | 8 | 8 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 4 | ··· | 4 |
50 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | ||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | D5 | C4oD4 | SD16 | D10 | D10 | D10 | C4oD8 | C5:D4 | D4:2D5 | D4xD5 | D5xSD16 | SD16:3D5 |
| kernel | C40:14D4 | C40:6C4 | D4:Dic5 | Q8:Dic5 | D5xC2xC8 | C20:2D4 | D10:3Q8 | C10xSD16 | C40 | C2xDic5 | C22xD5 | C2xSD16 | C20 | D10 | C2xC8 | C2xD4 | C2xQ8 | C10 | C8 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 2 | 2 | 4 | 2 | 2 | 2 | 4 | 8 | 2 | 2 | 4 | 4 |
Matrix representation of C40:14D4 ►in GL4(F41) generated by
| 3 | 20 | 0 | 0 |
| 0 | 27 | 0 | 0 |
| 0 | 0 | 7 | 34 |
| 0 | 0 | 7 | 40 |
| 12 | 11 | 0 | 0 |
| 39 | 29 | 0 | 0 |
| 0 | 0 | 3 | 3 |
| 0 | 0 | 24 | 38 |
| 1 | 12 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 34 | 7 |
| 0 | 0 | 40 | 7 |
G:=sub<GL(4,GF(41))| [3,0,0,0,20,27,0,0,0,0,7,7,0,0,34,40],[12,39,0,0,11,29,0,0,0,0,3,24,0,0,3,38],[1,0,0,0,12,40,0,0,0,0,34,40,0,0,7,7] >;
C40:14D4 in GAP, Magma, Sage, TeX
C_{40}\rtimes_{14}D_4 % in TeX
G:=Group("C40:14D4"); // GroupNames label
G:=SmallGroup(320,798);
// by ID
G=gap.SmallGroup(320,798);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,120,254,555,438,102,12550]);
// Polycyclic
G:=Group<a,b,c|a^40=b^4=c^2=1,b*a*b^-1=a^19,c*a*c=a^9,c*b*c=b^-1>;
// generators/relations