metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C20⋊7D8, D4⋊1D20, C42.49D10, (C5×D4)⋊8D4, (C4×D4)⋊3D5, C4⋊3(D4⋊D5), (D4×C20)⋊3C2, C5⋊3(C4⋊D8), C20⋊4D4⋊8C2, C20⋊3C8⋊23C2, C10.52(C2×D8), C4.13(C2×D20), C20.17(C2×D4), (C2×C20).60D4, C4⋊C4.243D10, C4.9(C4○D20), D20⋊6C4⋊30C2, (C2×D4).190D10, C20.50(C4○D4), (C4×C20).86C22, C2.8(D4⋊D10), C2.12(C20⋊7D4), C10.64(C4⋊D4), (C2×C20).337C23, (C2×D20).97C22, C10.109(C8⋊C22), (D4×C10).232C22, (C2×D4⋊D5)⋊7C2, C2.7(C2×D4⋊D5), (C2×C10).468(C2×D4), (C2×C4).245(C5⋊D4), (C5×C4⋊C4).274C22, (C2×C5⋊2C8).93C22, (C2×C4).437(C22×D5), C22.149(C2×C5⋊D4), SmallGroup(320,642)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C20⋊7D8
G = < a,b,c | a20=b8=c2=1, bab-1=cac=a-1, cbc=b-1 >
Subgroups: 694 in 140 conjugacy classes, 47 normal (31 characteristic)
C1, C2 [×3], C2 [×4], C4 [×2], C4 [×2], C4 [×2], C22, C22 [×10], C5, C8 [×2], C2×C4 [×3], C2×C4 [×3], D4 [×2], D4 [×9], C23 [×3], D5 [×2], C10 [×3], C10 [×2], C42, C22⋊C4, C4⋊C4, C2×C8 [×2], D8 [×4], C22×C4, C2×D4, C2×D4 [×4], C20 [×2], C20 [×2], C20 [×2], D10 [×6], C2×C10, C2×C10 [×4], D4⋊C4 [×2], C4⋊C8, C4×D4, C4⋊1D4, C2×D8 [×2], C5⋊2C8 [×2], D20 [×8], C2×C20 [×3], C2×C20 [×3], C5×D4 [×2], C5×D4, C22×D5 [×2], C22×C10, C4⋊D8, C2×C5⋊2C8 [×2], D4⋊D5 [×4], C4×C20, C5×C22⋊C4, C5×C4⋊C4, C2×D20 [×2], C2×D20 [×2], C22×C20, D4×C10, C20⋊3C8, D20⋊6C4 [×2], C20⋊4D4, C2×D4⋊D5 [×2], D4×C20, C20⋊7D8
Quotients: C1, C2 [×7], C22 [×7], D4 [×4], C23, D5, D8 [×2], C2×D4 [×2], C4○D4, D10 [×3], C4⋊D4, C2×D8, C8⋊C22, D20 [×2], C5⋊D4 [×2], C22×D5, C4⋊D8, D4⋊D5 [×2], C2×D20, C4○D20, C2×C5⋊D4, C20⋊7D4, C2×D4⋊D5, D4⋊D10, C20⋊7D8
►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)
(1 157 130 41 24 74 109 99)(2 156 131 60 25 73 110 98)(3 155 132 59 26 72 111 97)(4 154 133 58 27 71 112 96)(5 153 134 57 28 70 113 95)(6 152 135 56 29 69 114 94)(7 151 136 55 30 68 115 93)(8 150 137 54 31 67 116 92)(9 149 138 53 32 66 117 91)(10 148 139 52 33 65 118 90)(11 147 140 51 34 64 119 89)(12 146 121 50 35 63 120 88)(13 145 122 49 36 62 101 87)(14 144 123 48 37 61 102 86)(15 143 124 47 38 80 103 85)(16 142 125 46 39 79 104 84)(17 141 126 45 40 78 105 83)(18 160 127 44 21 77 106 82)(19 159 128 43 22 76 107 81)(20 158 129 42 23 75 108 100)
(2 20)(3 19)(4 18)(5 17)(6 16)(7 15)(8 14)(9 13)(10 12)(21 27)(22 26)(23 25)(28 40)(29 39)(30 38)(31 37)(32 36)(33 35)(41 74)(42 73)(43 72)(44 71)(45 70)(46 69)(47 68)(48 67)(49 66)(50 65)(51 64)(52 63)(53 62)(54 61)(55 80)(56 79)(57 78)(58 77)(59 76)(60 75)(81 155)(82 154)(83 153)(84 152)(85 151)(86 150)(87 149)(88 148)(89 147)(90 146)(91 145)(92 144)(93 143)(94 142)(95 141)(96 160)(97 159)(98 158)(99 157)(100 156)(101 138)(102 137)(103 136)(104 135)(105 134)(106 133)(107 132)(108 131)(109 130)(110 129)(111 128)(112 127)(113 126)(114 125)(115 124)(116 123)(117 122)(118 121)(119 140)(120 139)
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), (1,157,130,41,24,74,109,99)(2,156,131,60,25,73,110,98)(3,155,132,59,26,72,111,97)(4,154,133,58,27,71,112,96)(5,153,134,57,28,70,113,95)(6,152,135,56,29,69,114,94)(7,151,136,55,30,68,115,93)(8,150,137,54,31,67,116,92)(9,149,138,53,32,66,117,91)(10,148,139,52,33,65,118,90)(11,147,140,51,34,64,119,89)(12,146,121,50,35,63,120,88)(13,145,122,49,36,62,101,87)(14,144,123,48,37,61,102,86)(15,143,124,47,38,80,103,85)(16,142,125,46,39,79,104,84)(17,141,126,45,40,78,105,83)(18,160,127,44,21,77,106,82)(19,159,128,43,22,76,107,81)(20,158,129,42,23,75,108,100), (2,20)(3,19)(4,18)(5,17)(6,16)(7,15)(8,14)(9,13)(10,12)(21,27)(22,26)(23,25)(28,40)(29,39)(30,38)(31,37)(32,36)(33,35)(41,74)(42,73)(43,72)(44,71)(45,70)(46,69)(47,68)(48,67)(49,66)(50,65)(51,64)(52,63)(53,62)(54,61)(55,80)(56,79)(57,78)(58,77)(59,76)(60,75)(81,155)(82,154)(83,153)(84,152)(85,151)(86,150)(87,149)(88,148)(89,147)(90,146)(91,145)(92,144)(93,143)(94,142)(95,141)(96,160)(97,159)(98,158)(99,157)(100,156)(101,138)(102,137)(103,136)(104,135)(105,134)(106,133)(107,132)(108,131)(109,130)(110,129)(111,128)(112,127)(113,126)(114,125)(115,124)(116,123)(117,122)(118,121)(119,140)(120,139)>;
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), (1,157,130,41,24,74,109,99)(2,156,131,60,25,73,110,98)(3,155,132,59,26,72,111,97)(4,154,133,58,27,71,112,96)(5,153,134,57,28,70,113,95)(6,152,135,56,29,69,114,94)(7,151,136,55,30,68,115,93)(8,150,137,54,31,67,116,92)(9,149,138,53,32,66,117,91)(10,148,139,52,33,65,118,90)(11,147,140,51,34,64,119,89)(12,146,121,50,35,63,120,88)(13,145,122,49,36,62,101,87)(14,144,123,48,37,61,102,86)(15,143,124,47,38,80,103,85)(16,142,125,46,39,79,104,84)(17,141,126,45,40,78,105,83)(18,160,127,44,21,77,106,82)(19,159,128,43,22,76,107,81)(20,158,129,42,23,75,108,100), (2,20)(3,19)(4,18)(5,17)(6,16)(7,15)(8,14)(9,13)(10,12)(21,27)(22,26)(23,25)(28,40)(29,39)(30,38)(31,37)(32,36)(33,35)(41,74)(42,73)(43,72)(44,71)(45,70)(46,69)(47,68)(48,67)(49,66)(50,65)(51,64)(52,63)(53,62)(54,61)(55,80)(56,79)(57,78)(58,77)(59,76)(60,75)(81,155)(82,154)(83,153)(84,152)(85,151)(86,150)(87,149)(88,148)(89,147)(90,146)(91,145)(92,144)(93,143)(94,142)(95,141)(96,160)(97,159)(98,158)(99,157)(100,156)(101,138)(102,137)(103,136)(104,135)(105,134)(106,133)(107,132)(108,131)(109,130)(110,129)(111,128)(112,127)(113,126)(114,125)(115,124)(116,123)(117,122)(118,121)(119,140)(120,139) );
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)], [(1,157,130,41,24,74,109,99),(2,156,131,60,25,73,110,98),(3,155,132,59,26,72,111,97),(4,154,133,58,27,71,112,96),(5,153,134,57,28,70,113,95),(6,152,135,56,29,69,114,94),(7,151,136,55,30,68,115,93),(8,150,137,54,31,67,116,92),(9,149,138,53,32,66,117,91),(10,148,139,52,33,65,118,90),(11,147,140,51,34,64,119,89),(12,146,121,50,35,63,120,88),(13,145,122,49,36,62,101,87),(14,144,123,48,37,61,102,86),(15,143,124,47,38,80,103,85),(16,142,125,46,39,79,104,84),(17,141,126,45,40,78,105,83),(18,160,127,44,21,77,106,82),(19,159,128,43,22,76,107,81),(20,158,129,42,23,75,108,100)], [(2,20),(3,19),(4,18),(5,17),(6,16),(7,15),(8,14),(9,13),(10,12),(21,27),(22,26),(23,25),(28,40),(29,39),(30,38),(31,37),(32,36),(33,35),(41,74),(42,73),(43,72),(44,71),(45,70),(46,69),(47,68),(48,67),(49,66),(50,65),(51,64),(52,63),(53,62),(54,61),(55,80),(56,79),(57,78),(58,77),(59,76),(60,75),(81,155),(82,154),(83,153),(84,152),(85,151),(86,150),(87,149),(88,148),(89,147),(90,146),(91,145),(92,144),(93,143),(94,142),(95,141),(96,160),(97,159),(98,158),(99,157),(100,156),(101,138),(102,137),(103,136),(104,135),(105,134),(106,133),(107,132),(108,131),(109,130),(110,129),(111,128),(112,127),(113,126),(114,125),(115,124),(116,123),(117,122),(118,121),(119,140),(120,139)])
59 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 5A | 5B | 8A | 8B | 8C | 8D | 10A | ··· | 10F | 10G | ··· | 10N | 20A | ··· | 20H | 20I | ··· | 20X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 40 | 40 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 2 | 2 | 20 | 20 | 20 | 20 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
59 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D5 | D8 | C4○D4 | D10 | D10 | D10 | C5⋊D4 | D20 | C4○D20 | C8⋊C22 | D4⋊D5 | D4⋊D10 |
| kernel | C20⋊7D8 | C20⋊3C8 | D20⋊6C4 | C20⋊4D4 | C2×D4⋊D5 | D4×C20 | C2×C20 | C5×D4 | C4×D4 | C20 | C20 | C42 | C4⋊C4 | C2×D4 | C2×C4 | D4 | C4 | C10 | C4 | C2 |
| # reps | 1 | 1 | 2 | 1 | 2 | 1 | 2 | 2 | 2 | 4 | 2 | 2 | 2 | 2 | 8 | 8 | 8 | 1 | 4 | 4 |
Matrix representation of C20⋊7D8 ►in GL6(𝔽41)
| 40 | 39 | 0 | 0 | 0 | 0 |
| 1 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 6 | 40 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 40 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 21 | 18 | 0 | 0 |
| 0 | 0 | 21 | 20 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 29 |
| 0 | 0 | 0 | 0 | 12 | 12 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 40 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 6 | 40 | 0 | 0 |
| 0 | 0 | 35 | 35 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
G:=sub<GL(6,GF(41))| [40,1,0,0,0,0,39,1,0,0,0,0,0,0,6,1,0,0,0,0,40,0,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[1,40,0,0,0,0,0,40,0,0,0,0,0,0,21,21,0,0,0,0,18,20,0,0,0,0,0,0,12,12,0,0,0,0,29,12],[1,40,0,0,0,0,0,40,0,0,0,0,0,0,6,35,0,0,0,0,40,35,0,0,0,0,0,0,1,0,0,0,0,0,0,40] >;
C20⋊7D8 in GAP, Magma, Sage, TeX
C_{20}\rtimes_7D_8 % in TeX
G:=Group("C20:7D8"); // GroupNames label
G:=SmallGroup(320,642);
// by ID
G=gap.SmallGroup(320,642);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,253,120,254,1123,297,136,12550]);
// Polycyclic
G:=Group<a,b,c|a^20=b^8=c^2=1,b*a*b^-1=c*a*c=a^-1,c*b*c=b^-1>;
// generators/relations