metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C40.37D4, (C2×Q16)⋊9D5, C4.28(D4×D5), C40⋊8C4⋊10C2, (C2×C8).96D10, C5⋊2C8.10D4, C8.6(C5⋊D4), C5⋊5(C8.2D4), (C10×Q16)⋊10C2, C20.188(C2×D4), (C2×Q8).66D10, Dic5⋊Q8⋊6C2, (C2×Dic5).86D4, C22.281(D4×D5), C2.25(C20⋊D4), C10.34(C4⋊1D4), (C2×C40).151C22, (C2×C20).464C23, C20.23D4.8C2, (Q8×C10).93C22, C2.31(Q16⋊D5), (C2×D20).130C22, C10.81(C8.C22), (C4×Dic5).62C22, (C2×Dic10).137C22, C4.15(C2×C5⋊D4), (C2×Q8⋊D5).9C2, (C2×C5⋊Q16)⋊21C2, (C2×C40⋊C2).8C2, (C2×C10).375(C2×D4), (C2×C4).552(C22×D5), (C2×C5⋊2C8).168C22, SmallGroup(320,817)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C40.37D4
G = < a,b,c | a40=b4=c2=1, bab-1=a29, cac=a19, cbc=a20b-1 >
Subgroups: 526 in 124 conjugacy classes, 43 normal (31 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C5, C8, C8, C2×C4, C2×C4, D4, Q8, C23, D5, C10, C10, C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, SD16, Q16, C2×D4, C2×Q8, C2×Q8, Dic5, C20, C20, D10, C2×C10, C8⋊C4, C4.4D4, C4⋊Q8, C2×SD16, C2×Q16, C2×Q16, C5⋊2C8, C40, Dic10, D20, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C5×Q8, C22×D5, C8.2D4, C40⋊C2, C2×C5⋊2C8, C4×Dic5, C10.D4, D10⋊C4, Q8⋊D5, C5⋊Q16, C2×C40, C5×Q16, C2×Dic10, C2×D20, Q8×C10, C40⋊8C4, C2×C40⋊C2, C2×Q8⋊D5, C2×C5⋊Q16, Dic5⋊Q8, C20.23D4, C10×Q16, C40.37D4
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, D10, C4⋊1D4, C8.C22, C5⋊D4, C22×D5, C8.2D4, D4×D5, C2×C5⋊D4, Q16⋊D5, C20⋊D4, C40.37D4
►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)
(1 137 78 115)(2 126 79 104)(3 155 80 93)(4 144 41 82)(5 133 42 111)(6 122 43 100)(7 151 44 89)(8 140 45 118)(9 129 46 107)(10 158 47 96)(11 147 48 85)(12 136 49 114)(13 125 50 103)(14 154 51 92)(15 143 52 81)(16 132 53 110)(17 121 54 99)(18 150 55 88)(19 139 56 117)(20 128 57 106)(21 157 58 95)(22 146 59 84)(23 135 60 113)(24 124 61 102)(25 153 62 91)(26 142 63 120)(27 131 64 109)(28 160 65 98)(29 149 66 87)(30 138 67 116)(31 127 68 105)(32 156 69 94)(33 145 70 83)(34 134 71 112)(35 123 72 101)(36 152 73 90)(37 141 74 119)(38 130 75 108)(39 159 76 97)(40 148 77 86)
(2 20)(3 39)(4 18)(5 37)(6 16)(7 35)(8 14)(9 33)(10 12)(11 31)(13 29)(15 27)(17 25)(19 23)(22 40)(24 38)(26 36)(28 34)(30 32)(41 55)(42 74)(43 53)(44 72)(45 51)(46 70)(47 49)(48 68)(50 66)(52 64)(54 62)(56 60)(57 79)(59 77)(61 75)(63 73)(65 71)(67 69)(76 80)(81 151)(82 130)(83 149)(84 128)(85 147)(86 126)(87 145)(88 124)(89 143)(90 122)(91 141)(92 160)(93 139)(94 158)(95 137)(96 156)(97 135)(98 154)(99 133)(100 152)(101 131)(102 150)(103 129)(104 148)(105 127)(106 146)(107 125)(108 144)(109 123)(110 142)(111 121)(112 140)(113 159)(114 138)(115 157)(116 136)(117 155)(118 134)(119 153)(120 132)
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), (1,137,78,115)(2,126,79,104)(3,155,80,93)(4,144,41,82)(5,133,42,111)(6,122,43,100)(7,151,44,89)(8,140,45,118)(9,129,46,107)(10,158,47,96)(11,147,48,85)(12,136,49,114)(13,125,50,103)(14,154,51,92)(15,143,52,81)(16,132,53,110)(17,121,54,99)(18,150,55,88)(19,139,56,117)(20,128,57,106)(21,157,58,95)(22,146,59,84)(23,135,60,113)(24,124,61,102)(25,153,62,91)(26,142,63,120)(27,131,64,109)(28,160,65,98)(29,149,66,87)(30,138,67,116)(31,127,68,105)(32,156,69,94)(33,145,70,83)(34,134,71,112)(35,123,72,101)(36,152,73,90)(37,141,74,119)(38,130,75,108)(39,159,76,97)(40,148,77,86), (2,20)(3,39)(4,18)(5,37)(6,16)(7,35)(8,14)(9,33)(10,12)(11,31)(13,29)(15,27)(17,25)(19,23)(22,40)(24,38)(26,36)(28,34)(30,32)(41,55)(42,74)(43,53)(44,72)(45,51)(46,70)(47,49)(48,68)(50,66)(52,64)(54,62)(56,60)(57,79)(59,77)(61,75)(63,73)(65,71)(67,69)(76,80)(81,151)(82,130)(83,149)(84,128)(85,147)(86,126)(87,145)(88,124)(89,143)(90,122)(91,141)(92,160)(93,139)(94,158)(95,137)(96,156)(97,135)(98,154)(99,133)(100,152)(101,131)(102,150)(103,129)(104,148)(105,127)(106,146)(107,125)(108,144)(109,123)(110,142)(111,121)(112,140)(113,159)(114,138)(115,157)(116,136)(117,155)(118,134)(119,153)(120,132)>;
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), (1,137,78,115)(2,126,79,104)(3,155,80,93)(4,144,41,82)(5,133,42,111)(6,122,43,100)(7,151,44,89)(8,140,45,118)(9,129,46,107)(10,158,47,96)(11,147,48,85)(12,136,49,114)(13,125,50,103)(14,154,51,92)(15,143,52,81)(16,132,53,110)(17,121,54,99)(18,150,55,88)(19,139,56,117)(20,128,57,106)(21,157,58,95)(22,146,59,84)(23,135,60,113)(24,124,61,102)(25,153,62,91)(26,142,63,120)(27,131,64,109)(28,160,65,98)(29,149,66,87)(30,138,67,116)(31,127,68,105)(32,156,69,94)(33,145,70,83)(34,134,71,112)(35,123,72,101)(36,152,73,90)(37,141,74,119)(38,130,75,108)(39,159,76,97)(40,148,77,86), (2,20)(3,39)(4,18)(5,37)(6,16)(7,35)(8,14)(9,33)(10,12)(11,31)(13,29)(15,27)(17,25)(19,23)(22,40)(24,38)(26,36)(28,34)(30,32)(41,55)(42,74)(43,53)(44,72)(45,51)(46,70)(47,49)(48,68)(50,66)(52,64)(54,62)(56,60)(57,79)(59,77)(61,75)(63,73)(65,71)(67,69)(76,80)(81,151)(82,130)(83,149)(84,128)(85,147)(86,126)(87,145)(88,124)(89,143)(90,122)(91,141)(92,160)(93,139)(94,158)(95,137)(96,156)(97,135)(98,154)(99,133)(100,152)(101,131)(102,150)(103,129)(104,148)(105,127)(106,146)(107,125)(108,144)(109,123)(110,142)(111,121)(112,140)(113,159)(114,138)(115,157)(116,136)(117,155)(118,134)(119,153)(120,132) );
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)], [(1,137,78,115),(2,126,79,104),(3,155,80,93),(4,144,41,82),(5,133,42,111),(6,122,43,100),(7,151,44,89),(8,140,45,118),(9,129,46,107),(10,158,47,96),(11,147,48,85),(12,136,49,114),(13,125,50,103),(14,154,51,92),(15,143,52,81),(16,132,53,110),(17,121,54,99),(18,150,55,88),(19,139,56,117),(20,128,57,106),(21,157,58,95),(22,146,59,84),(23,135,60,113),(24,124,61,102),(25,153,62,91),(26,142,63,120),(27,131,64,109),(28,160,65,98),(29,149,66,87),(30,138,67,116),(31,127,68,105),(32,156,69,94),(33,145,70,83),(34,134,71,112),(35,123,72,101),(36,152,73,90),(37,141,74,119),(38,130,75,108),(39,159,76,97),(40,148,77,86)], [(2,20),(3,39),(4,18),(5,37),(6,16),(7,35),(8,14),(9,33),(10,12),(11,31),(13,29),(15,27),(17,25),(19,23),(22,40),(24,38),(26,36),(28,34),(30,32),(41,55),(42,74),(43,53),(44,72),(45,51),(46,70),(47,49),(48,68),(50,66),(52,64),(54,62),(56,60),(57,79),(59,77),(61,75),(63,73),(65,71),(67,69),(76,80),(81,151),(82,130),(83,149),(84,128),(85,147),(86,126),(87,145),(88,124),(89,143),(90,122),(91,141),(92,160),(93,139),(94,158),(95,137),(96,156),(97,135),(98,154),(99,133),(100,152),(101,131),(102,150),(103,129),(104,148),(105,127),(106,146),(107,125),(108,144),(109,123),(110,142),(111,121),(112,140),(113,159),(114,138),(115,157),(116,136),(117,155),(118,134),(119,153),(120,132)]])
44 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 5A | 5B | 8A | 8B | 8C | 8D | 10A | ··· | 10F | 20A | 20B | 20C | 20D | 20E | ··· | 20L | 40A | ··· | 40H |
| order | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 20 | 20 | 20 | 20 | 20 | ··· | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 40 | 2 | 2 | 8 | 8 | 20 | 20 | 40 | 2 | 2 | 4 | 4 | 20 | 20 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 8 | ··· | 8 | 4 | ··· | 4 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 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 | D10 | D10 | C5⋊D4 | C8.C22 | D4×D5 | D4×D5 | Q16⋊D5 |
| kernel | C40.37D4 | C40⋊8C4 | C2×C40⋊C2 | C2×Q8⋊D5 | C2×C5⋊Q16 | Dic5⋊Q8 | C20.23D4 | C10×Q16 | C5⋊2C8 | C40 | C2×Dic5 | C2×Q16 | C2×C8 | C2×Q8 | C8 | C10 | C4 | C22 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 8 | 2 | 2 | 2 | 8 |
Matrix representation of C40.37D4 ►in GL8(𝔽41)
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 35 | 7 | 0 | 0 | 0 | 0 |
| 0 | 0 | 35 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 17 | 36 | 7 | 7 |
| 0 | 0 | 0 | 0 | 10 | 18 | 0 | 24 |
| 0 | 0 | 0 | 0 | 19 | 21 | 3 | 3 |
| 0 | 0 | 0 | 0 | 29 | 21 | 21 | 3 |
| 0 | 40 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 35 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 35 | 6 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 37 | 0 |
| 0 | 0 | 0 | 0 | 21 | 0 | 40 | 40 |
| 0 | 0 | 0 | 0 | 21 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 40 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 6 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 6 | 35 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 21 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 21 | 0 | 0 | 40 |
G:=sub<GL(8,GF(41))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,35,35,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,17,10,19,29,0,0,0,0,36,18,21,21,0,0,0,0,7,0,3,21,0,0,0,0,7,24,3,3],[0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,35,35,0,0,0,0,0,0,40,6,0,0,0,0,0,0,0,0,1,21,21,0,0,0,0,0,0,0,0,1,0,0,0,0,37,40,40,40,0,0,0,0,0,40,0,0],[1,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,6,6,0,0,0,0,0,0,1,35,0,0,0,0,0,0,0,0,1,21,0,21,0,0,0,0,0,40,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,40] >;
C40.37D4 in GAP, Magma, Sage, TeX
C_{40}._{37}D_4 % in TeX
G:=Group("C40.37D4"); // GroupNames label
G:=SmallGroup(320,817);
// by ID
G=gap.SmallGroup(320,817);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,253,120,254,219,184,1684,438,102,12550]);
// Polycyclic
G:=Group<a,b,c|a^40=b^4=c^2=1,b*a*b^-1=a^29,c*a*c=a^19,c*b*c=a^20*b^-1>;
// generators/relations