metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C40⋊12D4, (C2×D8)⋊9D5, C8⋊4(C5⋊D4), C5⋊5(C8⋊2D4), (C10×D8)⋊10C2, C20⋊2D4⋊6C2, C40⋊6C4⋊21C2, (C2×C8).86D10, (C2×D4).66D10, C20.168(C2×D4), C20.94(C4○D4), D4⋊Dic5⋊31C2, (C2×Dic5).74D4, (C22×D5).44D4, C22.259(D4×D5), C4.29(D4⋊2D5), C2.31(D8⋊D5), C2.17(C20⋊2D4), C10.52(C8⋊C22), (C2×C40).148C22, (C2×C20).436C23, (D4×C10).85C22, C10.110(C4⋊D4), C4⋊Dic5.167C22, (C2×C8⋊D5)⋊7C2, C4.80(C2×C5⋊D4), (C2×C4×D5).50C22, (C2×C10).349(C2×D4), (C2×C4).526(C22×D5), (C2×C5⋊2C8).150C22, SmallGroup(320,786)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C40⋊12D4
G = < a,b,c | a40=b4=c2=1, bab-1=a19, cac=a29, cbc=b-1 >
Subgroups: 566 in 130 conjugacy classes, 41 normal (23 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C5, C8, C8, C2×C4, C2×C4, D4, C23, D5, C10, C10, C10, C22⋊C4, C4⋊C4, C2×C8, C2×C8, M4(2), D8, C22×C4, C2×D4, C2×D4, Dic5, C20, D10, C2×C10, C2×C10, D4⋊C4, C4.Q8, C4⋊D4, C2×M4(2), C2×D8, C5⋊2C8, C40, C4×D5, C2×Dic5, C2×Dic5, C5⋊D4, C2×C20, C5×D4, C22×D5, C22×C10, C8⋊2D4, C8⋊D5, C2×C5⋊2C8, C4⋊Dic5, C23.D5, C2×C40, C5×D8, C2×C4×D5, C2×C5⋊D4, D4×C10, C40⋊6C4, D4⋊Dic5, C2×C8⋊D5, C20⋊2D4, C10×D8, C40⋊12D4
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, C4○D4, D10, C4⋊D4, C8⋊C22, C5⋊D4, C22×D5, C8⋊2D4, D4×D5, D4⋊2D5, C2×C5⋊D4, D8⋊D5, C20⋊2D4, C40⋊12D4
►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)
(1 145 64 97)(2 124 65 116)(3 143 66 95)(4 122 67 114)(5 141 68 93)(6 160 69 112)(7 139 70 91)(8 158 71 110)(9 137 72 89)(10 156 73 108)(11 135 74 87)(12 154 75 106)(13 133 76 85)(14 152 77 104)(15 131 78 83)(16 150 79 102)(17 129 80 81)(18 148 41 100)(19 127 42 119)(20 146 43 98)(21 125 44 117)(22 144 45 96)(23 123 46 115)(24 142 47 94)(25 121 48 113)(26 140 49 92)(27 159 50 111)(28 138 51 90)(29 157 52 109)(30 136 53 88)(31 155 54 107)(32 134 55 86)(33 153 56 105)(34 132 57 84)(35 151 58 103)(36 130 59 82)(37 149 60 101)(38 128 61 120)(39 147 62 99)(40 126 63 118)
(2 30)(3 19)(4 8)(5 37)(6 26)(7 15)(9 33)(10 22)(12 40)(13 29)(14 18)(16 36)(17 25)(20 32)(23 39)(24 28)(27 35)(34 38)(41 77)(42 66)(43 55)(45 73)(46 62)(47 51)(48 80)(49 69)(50 58)(52 76)(53 65)(56 72)(57 61)(59 79)(60 68)(63 75)(67 71)(70 78)(81 121)(82 150)(83 139)(84 128)(85 157)(86 146)(87 135)(88 124)(89 153)(90 142)(91 131)(92 160)(93 149)(94 138)(95 127)(96 156)(97 145)(98 134)(99 123)(100 152)(101 141)(102 130)(103 159)(104 148)(105 137)(106 126)(107 155)(108 144)(109 133)(110 122)(111 151)(112 140)(113 129)(114 158)(115 147)(116 136)(117 125)(118 154)(119 143)(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), (1,145,64,97)(2,124,65,116)(3,143,66,95)(4,122,67,114)(5,141,68,93)(6,160,69,112)(7,139,70,91)(8,158,71,110)(9,137,72,89)(10,156,73,108)(11,135,74,87)(12,154,75,106)(13,133,76,85)(14,152,77,104)(15,131,78,83)(16,150,79,102)(17,129,80,81)(18,148,41,100)(19,127,42,119)(20,146,43,98)(21,125,44,117)(22,144,45,96)(23,123,46,115)(24,142,47,94)(25,121,48,113)(26,140,49,92)(27,159,50,111)(28,138,51,90)(29,157,52,109)(30,136,53,88)(31,155,54,107)(32,134,55,86)(33,153,56,105)(34,132,57,84)(35,151,58,103)(36,130,59,82)(37,149,60,101)(38,128,61,120)(39,147,62,99)(40,126,63,118), (2,30)(3,19)(4,8)(5,37)(6,26)(7,15)(9,33)(10,22)(12,40)(13,29)(14,18)(16,36)(17,25)(20,32)(23,39)(24,28)(27,35)(34,38)(41,77)(42,66)(43,55)(45,73)(46,62)(47,51)(48,80)(49,69)(50,58)(52,76)(53,65)(56,72)(57,61)(59,79)(60,68)(63,75)(67,71)(70,78)(81,121)(82,150)(83,139)(84,128)(85,157)(86,146)(87,135)(88,124)(89,153)(90,142)(91,131)(92,160)(93,149)(94,138)(95,127)(96,156)(97,145)(98,134)(99,123)(100,152)(101,141)(102,130)(103,159)(104,148)(105,137)(106,126)(107,155)(108,144)(109,133)(110,122)(111,151)(112,140)(113,129)(114,158)(115,147)(116,136)(117,125)(118,154)(119,143)(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), (1,145,64,97)(2,124,65,116)(3,143,66,95)(4,122,67,114)(5,141,68,93)(6,160,69,112)(7,139,70,91)(8,158,71,110)(9,137,72,89)(10,156,73,108)(11,135,74,87)(12,154,75,106)(13,133,76,85)(14,152,77,104)(15,131,78,83)(16,150,79,102)(17,129,80,81)(18,148,41,100)(19,127,42,119)(20,146,43,98)(21,125,44,117)(22,144,45,96)(23,123,46,115)(24,142,47,94)(25,121,48,113)(26,140,49,92)(27,159,50,111)(28,138,51,90)(29,157,52,109)(30,136,53,88)(31,155,54,107)(32,134,55,86)(33,153,56,105)(34,132,57,84)(35,151,58,103)(36,130,59,82)(37,149,60,101)(38,128,61,120)(39,147,62,99)(40,126,63,118), (2,30)(3,19)(4,8)(5,37)(6,26)(7,15)(9,33)(10,22)(12,40)(13,29)(14,18)(16,36)(17,25)(20,32)(23,39)(24,28)(27,35)(34,38)(41,77)(42,66)(43,55)(45,73)(46,62)(47,51)(48,80)(49,69)(50,58)(52,76)(53,65)(56,72)(57,61)(59,79)(60,68)(63,75)(67,71)(70,78)(81,121)(82,150)(83,139)(84,128)(85,157)(86,146)(87,135)(88,124)(89,153)(90,142)(91,131)(92,160)(93,149)(94,138)(95,127)(96,156)(97,145)(98,134)(99,123)(100,152)(101,141)(102,130)(103,159)(104,148)(105,137)(106,126)(107,155)(108,144)(109,133)(110,122)(111,151)(112,140)(113,129)(114,158)(115,147)(116,136)(117,125)(118,154)(119,143)(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)], [(1,145,64,97),(2,124,65,116),(3,143,66,95),(4,122,67,114),(5,141,68,93),(6,160,69,112),(7,139,70,91),(8,158,71,110),(9,137,72,89),(10,156,73,108),(11,135,74,87),(12,154,75,106),(13,133,76,85),(14,152,77,104),(15,131,78,83),(16,150,79,102),(17,129,80,81),(18,148,41,100),(19,127,42,119),(20,146,43,98),(21,125,44,117),(22,144,45,96),(23,123,46,115),(24,142,47,94),(25,121,48,113),(26,140,49,92),(27,159,50,111),(28,138,51,90),(29,157,52,109),(30,136,53,88),(31,155,54,107),(32,134,55,86),(33,153,56,105),(34,132,57,84),(35,151,58,103),(36,130,59,82),(37,149,60,101),(38,128,61,120),(39,147,62,99),(40,126,63,118)], [(2,30),(3,19),(4,8),(5,37),(6,26),(7,15),(9,33),(10,22),(12,40),(13,29),(14,18),(16,36),(17,25),(20,32),(23,39),(24,28),(27,35),(34,38),(41,77),(42,66),(43,55),(45,73),(46,62),(47,51),(48,80),(49,69),(50,58),(52,76),(53,65),(56,72),(57,61),(59,79),(60,68),(63,75),(67,71),(70,78),(81,121),(82,150),(83,139),(84,128),(85,157),(86,146),(87,135),(88,124),(89,153),(90,142),(91,131),(92,160),(93,149),(94,138),(95,127),(96,156),(97,145),(98,134),(99,123),(100,152),(101,141),(102,130),(103,159),(104,148),(105,137),(106,126),(107,155),(108,144),(109,133),(110,122),(111,151),(112,140),(113,129),(114,158),(115,147),(116,136),(117,125),(118,154),(119,143),(120,132)]])
44 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | 4B | 4C | 4D | 4E | 5A | 5B | 8A | 8B | 8C | 8D | 10A | ··· | 10F | 10G | ··· | 10N | 20A | 20B | 20C | 20D | 40A | ··· | 40H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | 20 | 20 | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 8 | 8 | 20 | 2 | 2 | 20 | 40 | 40 | 2 | 2 | 4 | 4 | 20 | 20 | 2 | ··· | 2 | 8 | ··· | 8 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | D5 | C4○D4 | D10 | D10 | C5⋊D4 | C8⋊C22 | D4⋊2D5 | D4×D5 | D8⋊D5 |
| kernel | C40⋊12D4 | C40⋊6C4 | D4⋊Dic5 | C2×C8⋊D5 | C20⋊2D4 | C10×D8 | C40 | C2×Dic5 | C22×D5 | C2×D8 | C20 | C2×C8 | C2×D4 | C8 | C10 | C4 | C22 | C2 |
| # reps | 1 | 1 | 2 | 1 | 2 | 1 | 2 | 1 | 1 | 2 | 2 | 2 | 4 | 8 | 2 | 2 | 2 | 8 |
Matrix representation of C40⋊12D4 ►in GL6(𝔽41)
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 31 | 9 | 0 | 0 |
| 0 | 0 | 32 | 9 | 0 | 0 |
| 0 | 0 | 4 | 4 | 2 | 1 |
| 0 | 0 | 20 | 0 | 5 | 7 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 33 | 27 | 3 | 3 |
| 0 | 0 | 1 | 8 | 23 | 20 |
| 0 | 0 | 36 | 11 | 7 | 40 |
| 0 | 0 | 16 | 30 | 40 | 34 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 34 | 40 | 0 | 0 |
| 0 | 0 | 13 | 22 | 6 | 7 |
| 0 | 0 | 9 | 37 | 36 | 35 |
G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,31,32,4,20,0,0,9,9,4,0,0,0,0,0,2,5,0,0,0,0,1,7],[0,40,0,0,0,0,1,0,0,0,0,0,0,0,33,1,36,16,0,0,27,8,11,30,0,0,3,23,7,40,0,0,3,20,40,34],[40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,34,13,9,0,0,0,40,22,37,0,0,0,0,6,36,0,0,0,0,7,35] >;
C40⋊12D4 in GAP, Magma, Sage, TeX
C_{40}\rtimes_{12}D_4 % in TeX
G:=Group("C40:12D4"); // GroupNames label
G:=SmallGroup(320,786);
// by ID
G=gap.SmallGroup(320,786);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,253,1094,135,570,297,136,12550]);
// Polycyclic
G:=Group<a,b,c|a^40=b^4=c^2=1,b*a*b^-1=a^19,c*a*c=a^29,c*b*c=b^-1>;
// generators/relations