direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×C40.C4, (C8×D5).5C4, (C2×C8).15F5, C8.28(C2×F5), (C2×C40).15C4, C40.29(C2×C4), (C4×D5).84D4, C4.14(C4⋊F5), C20.21(C4⋊C4), (C4×D5).26Q8, D10.12(C2×Q8), C10⋊1(C8.C4), D10.30(C4⋊C4), C4.38(C22×F5), C4.F5.7C22, C20.78(C22×C4), Dic5.31(C2×D4), (C8×D5).57C22, (C4×D5).78C23, C22.24(C4⋊F5), (C22×D5).18Q8, Dic5.31(C4⋊C4), (C2×Dic5).175D4, C5⋊1(C2×C8.C4), (D5×C2×C8).25C2, C2.17(C2×C4⋊F5), C10.14(C2×C4⋊C4), (C2×C5⋊2C8).24C4, (C2×C4.F5).9C2, C5⋊2C8.48(C2×C4), (C4×D5).87(C2×C4), (C2×C4).139(C2×F5), (C2×C10).22(C4⋊C4), (C2×C20).128(C2×C4), (C2×C4×D5).396C22, SmallGroup(320,1060)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×C40.C4
G = < a,b,c | a2=b40=1, c4=b20, ab=ba, ac=ca, cbc-1=b3 >
Subgroups: 346 in 106 conjugacy classes, 52 normal (28 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C5, C8, C8, C2×C4, C2×C4, C23, D5, C10, C10, C2×C8, C2×C8, M4(2), C22×C4, Dic5, C20, D10, D10, C2×C10, C8.C4, C22×C8, C2×M4(2), C5⋊2C8, C40, C5⋊C8, C4×D5, C2×Dic5, C2×C20, C22×D5, C2×C8.C4, C8×D5, C2×C5⋊2C8, C2×C40, C4.F5, C4.F5, C2×C5⋊C8, C2×C4×D5, C40.C4, D5×C2×C8, C2×C4.F5, C2×C40.C4
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, C4⋊C4, C22×C4, C2×D4, C2×Q8, F5, C8.C4, C2×C4⋊C4, C2×F5, C2×C8.C4, C4⋊F5, C22×F5, C40.C4, C2×C4⋊F5, C2×C40.C4
►On 160 points
Generators in S160
(1 53)(2 54)(3 55)(4 56)(5 57)(6 58)(7 59)(8 60)(9 61)(10 62)(11 63)(12 64)(13 65)(14 66)(15 67)(16 68)(17 69)(18 70)(19 71)(20 72)(21 73)(22 74)(23 75)(24 76)(25 77)(26 78)(27 79)(28 80)(29 41)(30 42)(31 43)(32 44)(33 45)(34 46)(35 47)(36 48)(37 49)(38 50)(39 51)(40 52)(81 153)(82 154)(83 155)(84 156)(85 157)(86 158)(87 159)(88 160)(89 121)(90 122)(91 123)(92 124)(93 125)(94 126)(95 127)(96 128)(97 129)(98 130)(99 131)(100 132)(101 133)(102 134)(103 135)(104 136)(105 137)(106 138)(107 139)(108 140)(109 141)(110 142)(111 143)(112 144)(113 145)(114 146)(115 147)(116 148)(117 149)(118 150)(119 151)(120 152)
(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)
(1 107 43 149 21 87 63 129)(2 94 52 152 22 114 72 132)(3 81 61 155 23 101 41 135)(4 108 70 158 24 88 50 138)(5 95 79 121 25 115 59 141)(6 82 48 124 26 102 68 144)(7 109 57 127 27 89 77 147)(8 96 66 130 28 116 46 150)(9 83 75 133 29 103 55 153)(10 110 44 136 30 90 64 156)(11 97 53 139 31 117 73 159)(12 84 62 142 32 104 42 122)(13 111 71 145 33 91 51 125)(14 98 80 148 34 118 60 128)(15 85 49 151 35 105 69 131)(16 112 58 154 36 92 78 134)(17 99 67 157 37 119 47 137)(18 86 76 160 38 106 56 140)(19 113 45 123 39 93 65 143)(20 100 54 126 40 120 74 146)
G:=sub<Sym(160)| (1,53)(2,54)(3,55)(4,56)(5,57)(6,58)(7,59)(8,60)(9,61)(10,62)(11,63)(12,64)(13,65)(14,66)(15,67)(16,68)(17,69)(18,70)(19,71)(20,72)(21,73)(22,74)(23,75)(24,76)(25,77)(26,78)(27,79)(28,80)(29,41)(30,42)(31,43)(32,44)(33,45)(34,46)(35,47)(36,48)(37,49)(38,50)(39,51)(40,52)(81,153)(82,154)(83,155)(84,156)(85,157)(86,158)(87,159)(88,160)(89,121)(90,122)(91,123)(92,124)(93,125)(94,126)(95,127)(96,128)(97,129)(98,130)(99,131)(100,132)(101,133)(102,134)(103,135)(104,136)(105,137)(106,138)(107,139)(108,140)(109,141)(110,142)(111,143)(112,144)(113,145)(114,146)(115,147)(116,148)(117,149)(118,150)(119,151)(120,152), (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), (1,107,43,149,21,87,63,129)(2,94,52,152,22,114,72,132)(3,81,61,155,23,101,41,135)(4,108,70,158,24,88,50,138)(5,95,79,121,25,115,59,141)(6,82,48,124,26,102,68,144)(7,109,57,127,27,89,77,147)(8,96,66,130,28,116,46,150)(9,83,75,133,29,103,55,153)(10,110,44,136,30,90,64,156)(11,97,53,139,31,117,73,159)(12,84,62,142,32,104,42,122)(13,111,71,145,33,91,51,125)(14,98,80,148,34,118,60,128)(15,85,49,151,35,105,69,131)(16,112,58,154,36,92,78,134)(17,99,67,157,37,119,47,137)(18,86,76,160,38,106,56,140)(19,113,45,123,39,93,65,143)(20,100,54,126,40,120,74,146)>;
G:=Group( (1,53)(2,54)(3,55)(4,56)(5,57)(6,58)(7,59)(8,60)(9,61)(10,62)(11,63)(12,64)(13,65)(14,66)(15,67)(16,68)(17,69)(18,70)(19,71)(20,72)(21,73)(22,74)(23,75)(24,76)(25,77)(26,78)(27,79)(28,80)(29,41)(30,42)(31,43)(32,44)(33,45)(34,46)(35,47)(36,48)(37,49)(38,50)(39,51)(40,52)(81,153)(82,154)(83,155)(84,156)(85,157)(86,158)(87,159)(88,160)(89,121)(90,122)(91,123)(92,124)(93,125)(94,126)(95,127)(96,128)(97,129)(98,130)(99,131)(100,132)(101,133)(102,134)(103,135)(104,136)(105,137)(106,138)(107,139)(108,140)(109,141)(110,142)(111,143)(112,144)(113,145)(114,146)(115,147)(116,148)(117,149)(118,150)(119,151)(120,152), (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), (1,107,43,149,21,87,63,129)(2,94,52,152,22,114,72,132)(3,81,61,155,23,101,41,135)(4,108,70,158,24,88,50,138)(5,95,79,121,25,115,59,141)(6,82,48,124,26,102,68,144)(7,109,57,127,27,89,77,147)(8,96,66,130,28,116,46,150)(9,83,75,133,29,103,55,153)(10,110,44,136,30,90,64,156)(11,97,53,139,31,117,73,159)(12,84,62,142,32,104,42,122)(13,111,71,145,33,91,51,125)(14,98,80,148,34,118,60,128)(15,85,49,151,35,105,69,131)(16,112,58,154,36,92,78,134)(17,99,67,157,37,119,47,137)(18,86,76,160,38,106,56,140)(19,113,45,123,39,93,65,143)(20,100,54,126,40,120,74,146) );
G=PermutationGroup([[(1,53),(2,54),(3,55),(4,56),(5,57),(6,58),(7,59),(8,60),(9,61),(10,62),(11,63),(12,64),(13,65),(14,66),(15,67),(16,68),(17,69),(18,70),(19,71),(20,72),(21,73),(22,74),(23,75),(24,76),(25,77),(26,78),(27,79),(28,80),(29,41),(30,42),(31,43),(32,44),(33,45),(34,46),(35,47),(36,48),(37,49),(38,50),(39,51),(40,52),(81,153),(82,154),(83,155),(84,156),(85,157),(86,158),(87,159),(88,160),(89,121),(90,122),(91,123),(92,124),(93,125),(94,126),(95,127),(96,128),(97,129),(98,130),(99,131),(100,132),(101,133),(102,134),(103,135),(104,136),(105,137),(106,138),(107,139),(108,140),(109,141),(110,142),(111,143),(112,144),(113,145),(114,146),(115,147),(116,148),(117,149),(118,150),(119,151),(120,152)], [(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)], [(1,107,43,149,21,87,63,129),(2,94,52,152,22,114,72,132),(3,81,61,155,23,101,41,135),(4,108,70,158,24,88,50,138),(5,95,79,121,25,115,59,141),(6,82,48,124,26,102,68,144),(7,109,57,127,27,89,77,147),(8,96,66,130,28,116,46,150),(9,83,75,133,29,103,55,153),(10,110,44,136,30,90,64,156),(11,97,53,139,31,117,73,159),(12,84,62,142,32,104,42,122),(13,111,71,145,33,91,51,125),(14,98,80,148,34,118,60,128),(15,85,49,151,35,105,69,131),(16,112,58,154,36,92,78,134),(17,99,67,157,37,119,47,137),(18,86,76,160,38,106,56,140),(19,113,45,123,39,93,65,143),(20,100,54,126,40,120,74,146)]])
44 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 5 | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 8I | ··· | 8P | 10A | 10B | 10C | 20A | 20B | 20C | 20D | 40A | ··· | 40H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | ··· | 8 | 10 | 10 | 10 | 20 | 20 | 20 | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 10 | 10 | 2 | 2 | 5 | 5 | 5 | 5 | 4 | 2 | 2 | 2 | 2 | 10 | 10 | 10 | 10 | 20 | ··· | 20 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | - | + | - | + | + | + | |||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C4 | D4 | Q8 | D4 | Q8 | C8.C4 | F5 | C2×F5 | C2×F5 | C4⋊F5 | C4⋊F5 | C40.C4 |
| kernel | C2×C40.C4 | C40.C4 | D5×C2×C8 | C2×C4.F5 | C8×D5 | C2×C5⋊2C8 | C2×C40 | C4×D5 | C4×D5 | C2×Dic5 | C22×D5 | C10 | C2×C8 | C8 | C2×C4 | C4 | C22 | C2 |
| # reps | 1 | 4 | 1 | 2 | 4 | 2 | 2 | 1 | 1 | 1 | 1 | 8 | 1 | 2 | 1 | 2 | 2 | 8 |
Matrix representation of C2×C40.C4 ►in GL6(𝔽41)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 14 | 2 | 0 | 0 | 0 | 0 |
| 0 | 38 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 27 | 7 | 34 |
| 0 | 0 | 14 | 27 | 34 | 0 |
| 0 | 0 | 7 | 0 | 34 | 27 |
| 0 | 0 | 14 | 34 | 7 | 27 |
| 38 | 0 | 0 | 0 | 0 | 0 |
| 5 | 3 | 0 | 0 | 0 | 0 |
| 0 | 0 | 5 | 0 | 36 | 8 |
| 0 | 0 | 0 | 8 | 28 | 13 |
| 0 | 0 | 33 | 13 | 28 | 8 |
| 0 | 0 | 33 | 8 | 36 | 0 |
G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[14,0,0,0,0,0,2,38,0,0,0,0,0,0,0,14,7,14,0,0,27,27,0,34,0,0,7,34,34,7,0,0,34,0,27,27],[38,5,0,0,0,0,0,3,0,0,0,0,0,0,5,0,33,33,0,0,0,8,13,8,0,0,36,28,28,36,0,0,8,13,8,0] >;
C2×C40.C4 in GAP, Magma, Sage, TeX
C_2\times C_{40}.C_4 % in TeX
G:=Group("C2xC40.C4"); // GroupNames label
G:=SmallGroup(320,1060);
// by ID
G=gap.SmallGroup(320,1060);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,422,100,136,1684,102,6278,1595]);
// Polycyclic
G:=Group<a,b,c|a^2=b^40=1,c^4=b^20,a*b=b*a,a*c=c*a,c*b*c^-1=b^3>;
// generators/relations