metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C40⋊6C4⋊13C2, D4⋊C4⋊12D5, C4⋊C4.137D10, (C2×D4).27D10, C10.D8⋊7C2, C20⋊2D4.5C2, (C2×C8).116D10, D4⋊Dic5⋊9C2, D10⋊1C8⋊11C2, C4.53(C4○D20), C10.41(C4○D8), (C22×D5).19D4, C22.177(D4×D5), C20.151(C4○D4), C2.15(D8⋊D5), C4.80(D4⋊2D5), C10.33(C8⋊C22), (C2×C40).127C22, (C2×C20).219C23, (C2×Dic5).197D4, (D4×C10).40C22, C5⋊2(C23.19D4), C4⋊Dic5.73C22, C2.11(SD16⋊3D5), C2.14(D10.12D4), C10.22(C22.D4), C4⋊C4⋊7D5⋊3C2, (C5×D4⋊C4)⋊12C2, (C2×C4×D5).15C22, (C2×C10).232(C2×D4), (C5×C4⋊C4).20C22, (C2×C5⋊2C8).17C22, (C2×C4).326(C22×D5), SmallGroup(320,406)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C40⋊6C4⋊C2
G = < a,b,c | a40=b4=c2=1, bab-1=a19, cac=a31b2, cbc=b-1 >
Subgroups: 446 in 106 conjugacy classes, 37 normal (all characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, C23, D5, C10, C10, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, C22×C4, C2×D4, C2×D4, Dic5, C20, C20, D10, C2×C10, C2×C10, C22⋊C8, D4⋊C4, D4⋊C4, C4.Q8, C2.D8, C42⋊C2, C4⋊D4, C5⋊2C8, C40, C4×D5, C2×Dic5, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C5×D4, C22×D5, C22×C10, C23.19D4, C2×C5⋊2C8, C4×Dic5, C4⋊Dic5, D10⋊C4, C23.D5, C5×C4⋊C4, C2×C40, C2×C4×D5, C2×C5⋊D4, D4×C10, C10.D8, C40⋊6C4, D10⋊1C8, D4⋊Dic5, C5×D4⋊C4, C4⋊C4⋊7D5, C20⋊2D4, C40⋊6C4⋊C2
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, C4○D4, D10, C22.D4, C4○D8, C8⋊C22, C22×D5, C23.19D4, C4○D20, D4×D5, D4⋊2D5, D10.12D4, D8⋊D5, SD16⋊3D5, C40⋊6C4⋊C2
►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 74 124 105)(2 53 125 84)(3 72 126 103)(4 51 127 82)(5 70 128 101)(6 49 129 120)(7 68 130 99)(8 47 131 118)(9 66 132 97)(10 45 133 116)(11 64 134 95)(12 43 135 114)(13 62 136 93)(14 41 137 112)(15 60 138 91)(16 79 139 110)(17 58 140 89)(18 77 141 108)(19 56 142 87)(20 75 143 106)(21 54 144 85)(22 73 145 104)(23 52 146 83)(24 71 147 102)(25 50 148 81)(26 69 149 100)(27 48 150 119)(28 67 151 98)(29 46 152 117)(30 65 153 96)(31 44 154 115)(32 63 155 94)(33 42 156 113)(34 61 157 92)(35 80 158 111)(36 59 159 90)(37 78 160 109)(38 57 121 88)(39 76 122 107)(40 55 123 86)
(1 74)(2 96)(3 56)(4 118)(5 78)(6 100)(7 60)(8 82)(9 42)(10 104)(11 64)(12 86)(13 46)(14 108)(15 68)(16 90)(17 50)(18 112)(19 72)(20 94)(21 54)(22 116)(23 76)(24 98)(25 58)(26 120)(27 80)(28 102)(29 62)(30 84)(31 44)(32 106)(33 66)(34 88)(35 48)(36 110)(37 70)(38 92)(39 52)(40 114)(41 141)(43 123)(45 145)(47 127)(49 149)(51 131)(53 153)(55 135)(57 157)(59 139)(61 121)(63 143)(65 125)(67 147)(69 129)(71 151)(73 133)(75 155)(77 137)(79 159)(81 140)(83 122)(85 144)(87 126)(89 148)(91 130)(93 152)(95 134)(97 156)(99 138)(101 160)(103 142)(105 124)(107 146)(109 128)(111 150)(113 132)(115 154)(117 136)(119 158)
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), (1,74,124,105)(2,53,125,84)(3,72,126,103)(4,51,127,82)(5,70,128,101)(6,49,129,120)(7,68,130,99)(8,47,131,118)(9,66,132,97)(10,45,133,116)(11,64,134,95)(12,43,135,114)(13,62,136,93)(14,41,137,112)(15,60,138,91)(16,79,139,110)(17,58,140,89)(18,77,141,108)(19,56,142,87)(20,75,143,106)(21,54,144,85)(22,73,145,104)(23,52,146,83)(24,71,147,102)(25,50,148,81)(26,69,149,100)(27,48,150,119)(28,67,151,98)(29,46,152,117)(30,65,153,96)(31,44,154,115)(32,63,155,94)(33,42,156,113)(34,61,157,92)(35,80,158,111)(36,59,159,90)(37,78,160,109)(38,57,121,88)(39,76,122,107)(40,55,123,86), (1,74)(2,96)(3,56)(4,118)(5,78)(6,100)(7,60)(8,82)(9,42)(10,104)(11,64)(12,86)(13,46)(14,108)(15,68)(16,90)(17,50)(18,112)(19,72)(20,94)(21,54)(22,116)(23,76)(24,98)(25,58)(26,120)(27,80)(28,102)(29,62)(30,84)(31,44)(32,106)(33,66)(34,88)(35,48)(36,110)(37,70)(38,92)(39,52)(40,114)(41,141)(43,123)(45,145)(47,127)(49,149)(51,131)(53,153)(55,135)(57,157)(59,139)(61,121)(63,143)(65,125)(67,147)(69,129)(71,151)(73,133)(75,155)(77,137)(79,159)(81,140)(83,122)(85,144)(87,126)(89,148)(91,130)(93,152)(95,134)(97,156)(99,138)(101,160)(103,142)(105,124)(107,146)(109,128)(111,150)(113,132)(115,154)(117,136)(119,158)>;
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), (1,74,124,105)(2,53,125,84)(3,72,126,103)(4,51,127,82)(5,70,128,101)(6,49,129,120)(7,68,130,99)(8,47,131,118)(9,66,132,97)(10,45,133,116)(11,64,134,95)(12,43,135,114)(13,62,136,93)(14,41,137,112)(15,60,138,91)(16,79,139,110)(17,58,140,89)(18,77,141,108)(19,56,142,87)(20,75,143,106)(21,54,144,85)(22,73,145,104)(23,52,146,83)(24,71,147,102)(25,50,148,81)(26,69,149,100)(27,48,150,119)(28,67,151,98)(29,46,152,117)(30,65,153,96)(31,44,154,115)(32,63,155,94)(33,42,156,113)(34,61,157,92)(35,80,158,111)(36,59,159,90)(37,78,160,109)(38,57,121,88)(39,76,122,107)(40,55,123,86), (1,74)(2,96)(3,56)(4,118)(5,78)(6,100)(7,60)(8,82)(9,42)(10,104)(11,64)(12,86)(13,46)(14,108)(15,68)(16,90)(17,50)(18,112)(19,72)(20,94)(21,54)(22,116)(23,76)(24,98)(25,58)(26,120)(27,80)(28,102)(29,62)(30,84)(31,44)(32,106)(33,66)(34,88)(35,48)(36,110)(37,70)(38,92)(39,52)(40,114)(41,141)(43,123)(45,145)(47,127)(49,149)(51,131)(53,153)(55,135)(57,157)(59,139)(61,121)(63,143)(65,125)(67,147)(69,129)(71,151)(73,133)(75,155)(77,137)(79,159)(81,140)(83,122)(85,144)(87,126)(89,148)(91,130)(93,152)(95,134)(97,156)(99,138)(101,160)(103,142)(105,124)(107,146)(109,128)(111,150)(113,132)(115,154)(117,136)(119,158) );
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)], [(1,74,124,105),(2,53,125,84),(3,72,126,103),(4,51,127,82),(5,70,128,101),(6,49,129,120),(7,68,130,99),(8,47,131,118),(9,66,132,97),(10,45,133,116),(11,64,134,95),(12,43,135,114),(13,62,136,93),(14,41,137,112),(15,60,138,91),(16,79,139,110),(17,58,140,89),(18,77,141,108),(19,56,142,87),(20,75,143,106),(21,54,144,85),(22,73,145,104),(23,52,146,83),(24,71,147,102),(25,50,148,81),(26,69,149,100),(27,48,150,119),(28,67,151,98),(29,46,152,117),(30,65,153,96),(31,44,154,115),(32,63,155,94),(33,42,156,113),(34,61,157,92),(35,80,158,111),(36,59,159,90),(37,78,160,109),(38,57,121,88),(39,76,122,107),(40,55,123,86)], [(1,74),(2,96),(3,56),(4,118),(5,78),(6,100),(7,60),(8,82),(9,42),(10,104),(11,64),(12,86),(13,46),(14,108),(15,68),(16,90),(17,50),(18,112),(19,72),(20,94),(21,54),(22,116),(23,76),(24,98),(25,58),(26,120),(27,80),(28,102),(29,62),(30,84),(31,44),(32,106),(33,66),(34,88),(35,48),(36,110),(37,70),(38,92),(39,52),(40,114),(41,141),(43,123),(45,145),(47,127),(49,149),(51,131),(53,153),(55,135),(57,157),(59,139),(61,121),(63,143),(65,125),(67,147),(69,129),(71,151),(73,133),(75,155),(77,137),(79,159),(81,140),(83,122),(85,144),(87,126),(89,148),(91,130),(93,152),(95,134),(97,156),(99,138),(101,160),(103,142),(105,124),(107,146),(109,128),(111,150),(113,132),(115,154),(117,136),(119,158)]])
47 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 5A | 5B | 8A | 8B | 8C | 8D | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 20A | 20B | 20C | 20D | 20E | 20F | 20G | 20H | 40A | ··· | 40H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 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 | 20 | 2 | 2 | 4 | 4 | 10 | 10 | 20 | 20 | 40 | 2 | 2 | 4 | 4 | 20 | 20 | 2 | ··· | 2 | 8 | 8 | 8 | 8 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 4 | ··· | 4 |
47 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | |||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D5 | C4○D4 | D10 | D10 | D10 | C4○D8 | C4○D20 | C8⋊C22 | D4⋊2D5 | D4×D5 | D8⋊D5 | SD16⋊3D5 |
| kernel | C40⋊6C4⋊C2 | C10.D8 | C40⋊6C4 | D10⋊1C8 | D4⋊Dic5 | C5×D4⋊C4 | C4⋊C4⋊7D5 | C20⋊2D4 | C2×Dic5 | C22×D5 | D4⋊C4 | C20 | C4⋊C4 | C2×C8 | C2×D4 | C10 | C4 | C10 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 4 | 2 | 2 | 2 | 4 | 8 | 1 | 2 | 2 | 4 | 4 |
Matrix representation of C40⋊6C4⋊C2 ►in GL4(𝔽41) generated by
| 14 | 0 | 0 | 0 |
| 27 | 38 | 0 | 0 |
| 0 | 0 | 9 | 16 |
| 0 | 0 | 36 | 14 |
| 20 | 36 | 0 | 0 |
| 31 | 21 | 0 | 0 |
| 0 | 0 | 38 | 23 |
| 0 | 0 | 37 | 3 |
| 21 | 5 | 0 | 0 |
| 35 | 20 | 0 | 0 |
| 0 | 0 | 24 | 35 |
| 0 | 0 | 7 | 17 |
G:=sub<GL(4,GF(41))| [14,27,0,0,0,38,0,0,0,0,9,36,0,0,16,14],[20,31,0,0,36,21,0,0,0,0,38,37,0,0,23,3],[21,35,0,0,5,20,0,0,0,0,24,7,0,0,35,17] >;
C40⋊6C4⋊C2 in GAP, Magma, Sage, TeX
C_{40}\rtimes_6C_4\rtimes C_2 % in TeX
G:=Group("C40:6C4:C2"); // GroupNames label
G:=SmallGroup(320,406);
// by ID
G=gap.SmallGroup(320,406);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,477,64,254,219,851,438,102,12550]);
// Polycyclic
G:=Group<a,b,c|a^40=b^4=c^2=1,b*a*b^-1=a^19,c*a*c=a^31*b^2,c*b*c=b^-1>;
// generators/relations