metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C8⋊6(C4×D5), C40⋊21(C2×C4), C40⋊C2⋊7C4, C40⋊8C4⋊7C2, C2.D8⋊12D5, (C2×C8).67D10, C10.84(C4×D4), C4⋊C4.172D10, D20.23(C2×C4), C22.92(D4×D5), Dic10⋊16(C2×C4), Dic5⋊3Q8⋊7C2, D20⋊6C4.7C2, D20⋊8C4.7C2, C20.44(C4○D4), C2.6(D8⋊D5), C10.Q16⋊23C2, C5⋊5(SD16⋊C4), C10.44(C8⋊C22), (C2×C20).305C23, C20.110(C22×C4), (C2×C40).145C22, C4.12(Q8⋊2D5), C2.6(Q16⋊D5), (C2×Dic5).226D4, (C2×D20).89C22, C2.14(D20⋊8C4), C10.73(C8.C22), (C4×Dic5).43C22, (C2×Dic10).97C22, C4.45(C2×C4×D5), (C5×C2.D8)⋊9C2, (C2×C40⋊C2).7C2, (C2×C10).310(C2×D4), (C5×C4⋊C4).98C22, (C2×C5⋊2C8).74C22, (C2×C4).408(C22×D5), SmallGroup(320,516)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C40⋊21(C2×C4)
G = < a,b,c | a40=b2=c4=1, bab=a19, cac-1=a31, bc=cb >
Subgroups: 502 in 120 conjugacy classes, 49 normal (37 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C8, C2×C4, C2×C4, D4, Q8, C23, D5, C10, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, SD16, C22×C4, C2×D4, C2×Q8, Dic5, C20, C20, D10, C2×C10, C8⋊C4, D4⋊C4, Q8⋊C4, C2.D8, C4×D4, C4×Q8, C2×SD16, C5⋊2C8, C40, Dic10, Dic10, C4×D5, D20, D20, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C22×D5, SD16⋊C4, C40⋊C2, C2×C5⋊2C8, C4×Dic5, C4×Dic5, C10.D4, D10⋊C4, C5×C4⋊C4, C2×C40, C2×Dic10, C2×C4×D5, C2×D20, D20⋊6C4, C10.Q16, C40⋊8C4, C5×C2.D8, Dic5⋊3Q8, D20⋊8C4, C2×C40⋊C2, C40⋊21(C2×C4)
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D5, C22×C4, C2×D4, C4○D4, D10, C4×D4, C8⋊C22, C8.C22, C4×D5, C22×D5, SD16⋊C4, C2×C4×D5, D4×D5, Q8⋊2D5, D20⋊8C4, D8⋊D5, Q16⋊D5, C40⋊21(C2×C4)
►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)
(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 59)(42 78)(43 57)(44 76)(45 55)(46 74)(47 53)(48 72)(49 51)(50 70)(52 68)(54 66)(56 64)(58 62)(61 79)(63 77)(65 75)(67 73)(69 71)(81 119)(82 98)(83 117)(84 96)(85 115)(86 94)(87 113)(88 92)(89 111)(91 109)(93 107)(95 105)(97 103)(99 101)(100 120)(102 118)(104 116)(106 114)(108 112)(121 139)(122 158)(123 137)(124 156)(125 135)(126 154)(127 133)(128 152)(129 131)(130 150)(132 148)(134 146)(136 144)(138 142)(141 159)(143 157)(145 155)(147 153)(149 151)
(1 80 90 140)(2 71 91 131)(3 62 92 122)(4 53 93 153)(5 44 94 144)(6 75 95 135)(7 66 96 126)(8 57 97 157)(9 48 98 148)(10 79 99 139)(11 70 100 130)(12 61 101 121)(13 52 102 152)(14 43 103 143)(15 74 104 134)(16 65 105 125)(17 56 106 156)(18 47 107 147)(19 78 108 138)(20 69 109 129)(21 60 110 160)(22 51 111 151)(23 42 112 142)(24 73 113 133)(25 64 114 124)(26 55 115 155)(27 46 116 146)(28 77 117 137)(29 68 118 128)(30 59 119 159)(31 50 120 150)(32 41 81 141)(33 72 82 132)(34 63 83 123)(35 54 84 154)(36 45 85 145)(37 76 86 136)(38 67 87 127)(39 58 88 158)(40 49 89 149)
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), (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)(41,59)(42,78)(43,57)(44,76)(45,55)(46,74)(47,53)(48,72)(49,51)(50,70)(52,68)(54,66)(56,64)(58,62)(61,79)(63,77)(65,75)(67,73)(69,71)(81,119)(82,98)(83,117)(84,96)(85,115)(86,94)(87,113)(88,92)(89,111)(91,109)(93,107)(95,105)(97,103)(99,101)(100,120)(102,118)(104,116)(106,114)(108,112)(121,139)(122,158)(123,137)(124,156)(125,135)(126,154)(127,133)(128,152)(129,131)(130,150)(132,148)(134,146)(136,144)(138,142)(141,159)(143,157)(145,155)(147,153)(149,151), (1,80,90,140)(2,71,91,131)(3,62,92,122)(4,53,93,153)(5,44,94,144)(6,75,95,135)(7,66,96,126)(8,57,97,157)(9,48,98,148)(10,79,99,139)(11,70,100,130)(12,61,101,121)(13,52,102,152)(14,43,103,143)(15,74,104,134)(16,65,105,125)(17,56,106,156)(18,47,107,147)(19,78,108,138)(20,69,109,129)(21,60,110,160)(22,51,111,151)(23,42,112,142)(24,73,113,133)(25,64,114,124)(26,55,115,155)(27,46,116,146)(28,77,117,137)(29,68,118,128)(30,59,119,159)(31,50,120,150)(32,41,81,141)(33,72,82,132)(34,63,83,123)(35,54,84,154)(36,45,85,145)(37,76,86,136)(38,67,87,127)(39,58,88,158)(40,49,89,149)>;
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), (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)(41,59)(42,78)(43,57)(44,76)(45,55)(46,74)(47,53)(48,72)(49,51)(50,70)(52,68)(54,66)(56,64)(58,62)(61,79)(63,77)(65,75)(67,73)(69,71)(81,119)(82,98)(83,117)(84,96)(85,115)(86,94)(87,113)(88,92)(89,111)(91,109)(93,107)(95,105)(97,103)(99,101)(100,120)(102,118)(104,116)(106,114)(108,112)(121,139)(122,158)(123,137)(124,156)(125,135)(126,154)(127,133)(128,152)(129,131)(130,150)(132,148)(134,146)(136,144)(138,142)(141,159)(143,157)(145,155)(147,153)(149,151), (1,80,90,140)(2,71,91,131)(3,62,92,122)(4,53,93,153)(5,44,94,144)(6,75,95,135)(7,66,96,126)(8,57,97,157)(9,48,98,148)(10,79,99,139)(11,70,100,130)(12,61,101,121)(13,52,102,152)(14,43,103,143)(15,74,104,134)(16,65,105,125)(17,56,106,156)(18,47,107,147)(19,78,108,138)(20,69,109,129)(21,60,110,160)(22,51,111,151)(23,42,112,142)(24,73,113,133)(25,64,114,124)(26,55,115,155)(27,46,116,146)(28,77,117,137)(29,68,118,128)(30,59,119,159)(31,50,120,150)(32,41,81,141)(33,72,82,132)(34,63,83,123)(35,54,84,154)(36,45,85,145)(37,76,86,136)(38,67,87,127)(39,58,88,158)(40,49,89,149) );
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)], [(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),(41,59),(42,78),(43,57),(44,76),(45,55),(46,74),(47,53),(48,72),(49,51),(50,70),(52,68),(54,66),(56,64),(58,62),(61,79),(63,77),(65,75),(67,73),(69,71),(81,119),(82,98),(83,117),(84,96),(85,115),(86,94),(87,113),(88,92),(89,111),(91,109),(93,107),(95,105),(97,103),(99,101),(100,120),(102,118),(104,116),(106,114),(108,112),(121,139),(122,158),(123,137),(124,156),(125,135),(126,154),(127,133),(128,152),(129,131),(130,150),(132,148),(134,146),(136,144),(138,142),(141,159),(143,157),(145,155),(147,153),(149,151)], [(1,80,90,140),(2,71,91,131),(3,62,92,122),(4,53,93,153),(5,44,94,144),(6,75,95,135),(7,66,96,126),(8,57,97,157),(9,48,98,148),(10,79,99,139),(11,70,100,130),(12,61,101,121),(13,52,102,152),(14,43,103,143),(15,74,104,134),(16,65,105,125),(17,56,106,156),(18,47,107,147),(19,78,108,138),(20,69,109,129),(21,60,110,160),(22,51,111,151),(23,42,112,142),(24,73,113,133),(25,64,114,124),(26,55,115,155),(27,46,116,146),(28,77,117,137),(29,68,118,128),(30,59,119,159),(31,50,120,150),(32,41,81,141),(33,72,82,132),(34,63,83,123),(35,54,84,154),(36,45,85,145),(37,76,86,136),(38,67,87,127),(39,58,88,158),(40,49,89,149)]])
50 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 5A | 5B | 8A | 8B | 8C | 8D | 10A | ··· | 10F | 20A | 20B | 20C | 20D | 20E | ··· | 20L | 40A | ··· | 40H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 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 | 20 | 20 | 2 | 2 | 4 | 4 | 4 | 4 | 10 | 10 | 10 | 10 | 20 | 20 | 2 | 2 | 4 | 4 | 20 | 20 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 8 | ··· | 8 | 4 | ··· | 4 |
50 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | + | |||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | D4 | D5 | C4○D4 | D10 | D10 | C4×D5 | C8⋊C22 | C8.C22 | Q8⋊2D5 | D4×D5 | D8⋊D5 | Q16⋊D5 |
| kernel | C40⋊21(C2×C4) | D20⋊6C4 | C10.Q16 | C40⋊8C4 | C5×C2.D8 | Dic5⋊3Q8 | D20⋊8C4 | C2×C40⋊C2 | C40⋊C2 | C2×Dic5 | C2.D8 | C20 | C4⋊C4 | C2×C8 | C8 | C10 | C10 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 8 | 2 | 2 | 2 | 4 | 2 | 8 | 1 | 1 | 2 | 2 | 4 | 4 |
Matrix representation of C40⋊21(C2×C4) ►in GL6(𝔽41)
| 24 | 25 | 0 | 0 | 0 | 0 |
| 13 | 17 | 0 | 0 | 0 | 0 |
| 0 | 0 | 2 | 31 | 39 | 10 |
| 0 | 0 | 10 | 31 | 31 | 10 |
| 0 | 0 | 1 | 36 | 0 | 0 |
| 0 | 0 | 5 | 36 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 3 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 34 | 40 | 0 | 0 |
| 0 | 0 | 1 | 0 | 40 | 0 |
| 0 | 0 | 34 | 40 | 7 | 1 |
| 32 | 0 | 0 | 0 | 0 | 0 |
| 14 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 15 | 14 | 25 | 13 |
| 0 | 0 | 27 | 40 | 28 | 16 |
| 0 | 0 | 7 | 0 | 26 | 27 |
| 0 | 0 | 0 | 7 | 14 | 1 |
G:=sub<GL(6,GF(41))| [24,13,0,0,0,0,25,17,0,0,0,0,0,0,2,10,1,5,0,0,31,31,36,36,0,0,39,31,0,0,0,0,10,10,0,0],[1,3,0,0,0,0,0,40,0,0,0,0,0,0,1,34,1,34,0,0,0,40,0,40,0,0,0,0,40,7,0,0,0,0,0,1],[32,14,0,0,0,0,0,9,0,0,0,0,0,0,15,27,7,0,0,0,14,40,0,7,0,0,25,28,26,14,0,0,13,16,27,1] >;
C40⋊21(C2×C4) in GAP, Magma, Sage, TeX
C_{40}\rtimes_{21}(C_2\times C_4) % in TeX
G:=Group("C40:21(C2xC4)"); // GroupNames label
G:=SmallGroup(320,516);
// by ID
G=gap.SmallGroup(320,516);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,253,120,219,58,1684,438,102,12550]);
// Polycyclic
G:=Group<a,b,c|a^40=b^2=c^4=1,b*a*b=a^19,c*a*c^-1=a^31,b*c=c*b>;
// generators/relations