direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×C40.6C4, C23.14Dic10, (C2×C40).43C4, C4.84(C2×D20), C20.79(C4⋊C4), (C2×C20).62Q8, C40.115(C2×C4), C20.304(C2×D4), (C2×C4).170D20, (C2×C8).314D10, (C2×C20).402D4, C8.19(C2×Dic5), (C2×C8).11Dic5, C10⋊3(C8.C4), (C22×C8).12D5, (C22×C40).18C2, C4.24(C4⋊Dic5), (C2×C4).51Dic10, (C22×C10).23Q8, (C2×C40).386C22, (C2×C20).795C23, C20.231(C22×C4), (C22×C4).426D10, C4.26(C22×Dic5), C22.8(C2×Dic10), C22.14(C4⋊Dic5), C4.Dic5.35C22, (C22×C20).540C22, C5⋊5(C2×C8.C4), C10.70(C2×C4⋊C4), C2.13(C2×C4⋊Dic5), (C2×C10).40(C2×Q8), (C2×C10).80(C4⋊C4), (C2×C20).480(C2×C4), (C2×C4).84(C2×Dic5), (C2×C4.Dic5).5C2, (C2×C4).713(C22×D5), SmallGroup(320,734)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×C40.6C4
G = < a,b,c | a2=b40=1, c4=b20, ab=ba, ac=ca, cbc-1=b19 >
Subgroups: 238 in 106 conjugacy classes, 71 normal (41 characteristic)
C1, C2, C2 [×2], C2 [×2], C4 [×4], C22 [×3], C22 [×2], C5, C8 [×4], C8 [×4], C2×C4 [×6], C23, C10, C10 [×2], C10 [×2], C2×C8 [×2], C2×C8 [×4], C2×C8 [×2], M4(2) [×6], C22×C4, C20 [×4], C2×C10 [×3], C2×C10 [×2], C8.C4 [×4], C22×C8, C2×M4(2) [×2], C5⋊2C8 [×4], C40 [×4], C2×C20 [×6], C22×C10, C2×C8.C4, C2×C5⋊2C8 [×2], C4.Dic5 [×4], C4.Dic5 [×2], C2×C40 [×2], C2×C40 [×4], C22×C20, C40.6C4 [×4], C2×C4.Dic5 [×2], C22×C40, C2×C40.6C4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×2], Q8 [×2], C23, D5, C4⋊C4 [×4], C22×C4, C2×D4, C2×Q8, Dic5 [×4], D10 [×3], C8.C4 [×2], C2×C4⋊C4, Dic10 [×2], D20 [×2], C2×Dic5 [×6], C22×D5, C2×C8.C4, C4⋊Dic5 [×4], C2×Dic10, C2×D20, C22×Dic5, C40.6C4 [×2], C2×C4⋊Dic5, C2×C40.6C4
►On 160 points
Generators in S160
(1 97)(2 98)(3 99)(4 100)(5 101)(6 102)(7 103)(8 104)(9 105)(10 106)(11 107)(12 108)(13 109)(14 110)(15 111)(16 112)(17 113)(18 114)(19 115)(20 116)(21 117)(22 118)(23 119)(24 120)(25 81)(26 82)(27 83)(28 84)(29 85)(30 86)(31 87)(32 88)(33 89)(34 90)(35 91)(36 92)(37 93)(38 94)(39 95)(40 96)(41 133)(42 134)(43 135)(44 136)(45 137)(46 138)(47 139)(48 140)(49 141)(50 142)(51 143)(52 144)(53 145)(54 146)(55 147)(56 148)(57 149)(58 150)(59 151)(60 152)(61 153)(62 154)(63 155)(64 156)(65 157)(66 158)(67 159)(68 160)(69 121)(70 122)(71 123)(72 124)(73 125)(74 126)(75 127)(76 128)(77 129)(78 130)(79 131)(80 132)
(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 43 107 125 21 63 87 145)(2 62 108 144 22 42 88 124)(3 41 109 123 23 61 89 143)(4 60 110 142 24 80 90 122)(5 79 111 121 25 59 91 141)(6 58 112 140 26 78 92 160)(7 77 113 159 27 57 93 139)(8 56 114 138 28 76 94 158)(9 75 115 157 29 55 95 137)(10 54 116 136 30 74 96 156)(11 73 117 155 31 53 97 135)(12 52 118 134 32 72 98 154)(13 71 119 153 33 51 99 133)(14 50 120 132 34 70 100 152)(15 69 81 151 35 49 101 131)(16 48 82 130 36 68 102 150)(17 67 83 149 37 47 103 129)(18 46 84 128 38 66 104 148)(19 65 85 147 39 45 105 127)(20 44 86 126 40 64 106 146)
G:=sub<Sym(160)| (1,97)(2,98)(3,99)(4,100)(5,101)(6,102)(7,103)(8,104)(9,105)(10,106)(11,107)(12,108)(13,109)(14,110)(15,111)(16,112)(17,113)(18,114)(19,115)(20,116)(21,117)(22,118)(23,119)(24,120)(25,81)(26,82)(27,83)(28,84)(29,85)(30,86)(31,87)(32,88)(33,89)(34,90)(35,91)(36,92)(37,93)(38,94)(39,95)(40,96)(41,133)(42,134)(43,135)(44,136)(45,137)(46,138)(47,139)(48,140)(49,141)(50,142)(51,143)(52,144)(53,145)(54,146)(55,147)(56,148)(57,149)(58,150)(59,151)(60,152)(61,153)(62,154)(63,155)(64,156)(65,157)(66,158)(67,159)(68,160)(69,121)(70,122)(71,123)(72,124)(73,125)(74,126)(75,127)(76,128)(77,129)(78,130)(79,131)(80,132), (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), (1,43,107,125,21,63,87,145)(2,62,108,144,22,42,88,124)(3,41,109,123,23,61,89,143)(4,60,110,142,24,80,90,122)(5,79,111,121,25,59,91,141)(6,58,112,140,26,78,92,160)(7,77,113,159,27,57,93,139)(8,56,114,138,28,76,94,158)(9,75,115,157,29,55,95,137)(10,54,116,136,30,74,96,156)(11,73,117,155,31,53,97,135)(12,52,118,134,32,72,98,154)(13,71,119,153,33,51,99,133)(14,50,120,132,34,70,100,152)(15,69,81,151,35,49,101,131)(16,48,82,130,36,68,102,150)(17,67,83,149,37,47,103,129)(18,46,84,128,38,66,104,148)(19,65,85,147,39,45,105,127)(20,44,86,126,40,64,106,146)>;
G:=Group( (1,97)(2,98)(3,99)(4,100)(5,101)(6,102)(7,103)(8,104)(9,105)(10,106)(11,107)(12,108)(13,109)(14,110)(15,111)(16,112)(17,113)(18,114)(19,115)(20,116)(21,117)(22,118)(23,119)(24,120)(25,81)(26,82)(27,83)(28,84)(29,85)(30,86)(31,87)(32,88)(33,89)(34,90)(35,91)(36,92)(37,93)(38,94)(39,95)(40,96)(41,133)(42,134)(43,135)(44,136)(45,137)(46,138)(47,139)(48,140)(49,141)(50,142)(51,143)(52,144)(53,145)(54,146)(55,147)(56,148)(57,149)(58,150)(59,151)(60,152)(61,153)(62,154)(63,155)(64,156)(65,157)(66,158)(67,159)(68,160)(69,121)(70,122)(71,123)(72,124)(73,125)(74,126)(75,127)(76,128)(77,129)(78,130)(79,131)(80,132), (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), (1,43,107,125,21,63,87,145)(2,62,108,144,22,42,88,124)(3,41,109,123,23,61,89,143)(4,60,110,142,24,80,90,122)(5,79,111,121,25,59,91,141)(6,58,112,140,26,78,92,160)(7,77,113,159,27,57,93,139)(8,56,114,138,28,76,94,158)(9,75,115,157,29,55,95,137)(10,54,116,136,30,74,96,156)(11,73,117,155,31,53,97,135)(12,52,118,134,32,72,98,154)(13,71,119,153,33,51,99,133)(14,50,120,132,34,70,100,152)(15,69,81,151,35,49,101,131)(16,48,82,130,36,68,102,150)(17,67,83,149,37,47,103,129)(18,46,84,128,38,66,104,148)(19,65,85,147,39,45,105,127)(20,44,86,126,40,64,106,146) );
G=PermutationGroup([(1,97),(2,98),(3,99),(4,100),(5,101),(6,102),(7,103),(8,104),(9,105),(10,106),(11,107),(12,108),(13,109),(14,110),(15,111),(16,112),(17,113),(18,114),(19,115),(20,116),(21,117),(22,118),(23,119),(24,120),(25,81),(26,82),(27,83),(28,84),(29,85),(30,86),(31,87),(32,88),(33,89),(34,90),(35,91),(36,92),(37,93),(38,94),(39,95),(40,96),(41,133),(42,134),(43,135),(44,136),(45,137),(46,138),(47,139),(48,140),(49,141),(50,142),(51,143),(52,144),(53,145),(54,146),(55,147),(56,148),(57,149),(58,150),(59,151),(60,152),(61,153),(62,154),(63,155),(64,156),(65,157),(66,158),(67,159),(68,160),(69,121),(70,122),(71,123),(72,124),(73,125),(74,126),(75,127),(76,128),(77,129),(78,130),(79,131),(80,132)], [(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)], [(1,43,107,125,21,63,87,145),(2,62,108,144,22,42,88,124),(3,41,109,123,23,61,89,143),(4,60,110,142,24,80,90,122),(5,79,111,121,25,59,91,141),(6,58,112,140,26,78,92,160),(7,77,113,159,27,57,93,139),(8,56,114,138,28,76,94,158),(9,75,115,157,29,55,95,137),(10,54,116,136,30,74,96,156),(11,73,117,155,31,53,97,135),(12,52,118,134,32,72,98,154),(13,71,119,153,33,51,99,133),(14,50,120,132,34,70,100,152),(15,69,81,151,35,49,101,131),(16,48,82,130,36,68,102,150),(17,67,83,149,37,47,103,129),(18,46,84,128,38,66,104,148),(19,65,85,147,39,45,105,127),(20,44,86,126,40,64,106,146)])
92 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 5A | 5B | 8A | ··· | 8H | 8I | ··· | 8P | 10A | ··· | 10N | 20A | ··· | 20P | 40A | ··· | 40AF |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | ··· | 8 | 8 | ··· | 8 | 10 | ··· | 10 | 20 | ··· | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 20 | ··· | 20 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
92 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | - | - | + | - | + | + | - | + | - | |||
| image | C1 | C2 | C2 | C2 | C4 | D4 | Q8 | Q8 | D5 | Dic5 | D10 | D10 | C8.C4 | Dic10 | D20 | Dic10 | C40.6C4 |
| kernel | C2×C40.6C4 | C40.6C4 | C2×C4.Dic5 | C22×C40 | C2×C40 | C2×C20 | C2×C20 | C22×C10 | C22×C8 | C2×C8 | C2×C8 | C22×C4 | C10 | C2×C4 | C2×C4 | C23 | C2 |
| # reps | 1 | 4 | 2 | 1 | 8 | 2 | 1 | 1 | 2 | 8 | 4 | 2 | 8 | 4 | 8 | 4 | 32 |
Matrix representation of C2×C40.6C4 ►in GL3(𝔽41) generated by
| 40 | 0 | 0 |
| 0 | 40 | 0 |
| 0 | 0 | 40 |
| 1 | 0 | 0 |
| 0 | 19 | 0 |
| 0 | 0 | 28 |
| 40 | 0 | 0 |
| 0 | 0 | 27 |
| 0 | 14 | 0 |
G:=sub<GL(3,GF(41))| [40,0,0,0,40,0,0,0,40],[1,0,0,0,19,0,0,0,28],[40,0,0,0,0,14,0,27,0] >;
C2×C40.6C4 in GAP, Magma, Sage, TeX
C_2\times C_{40}._6C_4 % in TeX
G:=Group("C2xC40.6C4"); // GroupNames label
G:=SmallGroup(320,734);
// by ID
G=gap.SmallGroup(320,734);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,422,100,136,1684,102,12550]);
// 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^19>;
// generators/relations