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, C2, C4, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, D4, C23, D5, C10, C10, C42, C22⋊C4, C4⋊C4, C2×C8, D8, C22×C4, C2×D4, C2×D4, C20, C20, C20, D10, C2×C10, C2×C10, D4⋊C4, C4⋊C8, C4×D4, C4⋊1D4, C2×D8, C5⋊2C8, D20, C2×C20, C2×C20, C5×D4, C5×D4, C22×D5, C22×C10, C4⋊D8, C2×C5⋊2C8, D4⋊D5, C4×C20, C5×C22⋊C4, C5×C4⋊C4, C2×D20, C2×D20, C22×C20, D4×C10, C20⋊3C8, D20⋊6C4, C20⋊4D4, C2×D4⋊D5, D4×C20, C20⋊7D8
Quotients: C1, C2, C22, D4, C23, D5, D8, C2×D4, C4○D4, D10, C4⋊D4, C2×D8, C8⋊C22, D20, C5⋊D4, C22×D5, C4⋊D8, D4⋊D5, 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 28 51 143 115 133 84 71)(2 27 52 142 116 132 85 70)(3 26 53 141 117 131 86 69)(4 25 54 160 118 130 87 68)(5 24 55 159 119 129 88 67)(6 23 56 158 120 128 89 66)(7 22 57 157 101 127 90 65)(8 21 58 156 102 126 91 64)(9 40 59 155 103 125 92 63)(10 39 60 154 104 124 93 62)(11 38 41 153 105 123 94 61)(12 37 42 152 106 122 95 80)(13 36 43 151 107 121 96 79)(14 35 44 150 108 140 97 78)(15 34 45 149 109 139 98 77)(16 33 46 148 110 138 99 76)(17 32 47 147 111 137 100 75)(18 31 48 146 112 136 81 74)(19 30 49 145 113 135 82 73)(20 29 50 144 114 134 83 72)
(2 20)(3 19)(4 18)(5 17)(6 16)(7 15)(8 14)(9 13)(10 12)(21 78)(22 77)(23 76)(24 75)(25 74)(26 73)(27 72)(28 71)(29 70)(30 69)(31 68)(32 67)(33 66)(34 65)(35 64)(36 63)(37 62)(38 61)(39 80)(40 79)(41 94)(42 93)(43 92)(44 91)(45 90)(46 89)(47 88)(48 87)(49 86)(50 85)(51 84)(52 83)(53 82)(54 81)(55 100)(56 99)(57 98)(58 97)(59 96)(60 95)(101 109)(102 108)(103 107)(104 106)(110 120)(111 119)(112 118)(113 117)(114 116)(121 155)(122 154)(123 153)(124 152)(125 151)(126 150)(127 149)(128 148)(129 147)(130 146)(131 145)(132 144)(133 143)(134 142)(135 141)(136 160)(137 159)(138 158)(139 157)(140 156)
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), (1,28,51,143,115,133,84,71)(2,27,52,142,116,132,85,70)(3,26,53,141,117,131,86,69)(4,25,54,160,118,130,87,68)(5,24,55,159,119,129,88,67)(6,23,56,158,120,128,89,66)(7,22,57,157,101,127,90,65)(8,21,58,156,102,126,91,64)(9,40,59,155,103,125,92,63)(10,39,60,154,104,124,93,62)(11,38,41,153,105,123,94,61)(12,37,42,152,106,122,95,80)(13,36,43,151,107,121,96,79)(14,35,44,150,108,140,97,78)(15,34,45,149,109,139,98,77)(16,33,46,148,110,138,99,76)(17,32,47,147,111,137,100,75)(18,31,48,146,112,136,81,74)(19,30,49,145,113,135,82,73)(20,29,50,144,114,134,83,72), (2,20)(3,19)(4,18)(5,17)(6,16)(7,15)(8,14)(9,13)(10,12)(21,78)(22,77)(23,76)(24,75)(25,74)(26,73)(27,72)(28,71)(29,70)(30,69)(31,68)(32,67)(33,66)(34,65)(35,64)(36,63)(37,62)(38,61)(39,80)(40,79)(41,94)(42,93)(43,92)(44,91)(45,90)(46,89)(47,88)(48,87)(49,86)(50,85)(51,84)(52,83)(53,82)(54,81)(55,100)(56,99)(57,98)(58,97)(59,96)(60,95)(101,109)(102,108)(103,107)(104,106)(110,120)(111,119)(112,118)(113,117)(114,116)(121,155)(122,154)(123,153)(124,152)(125,151)(126,150)(127,149)(128,148)(129,147)(130,146)(131,145)(132,144)(133,143)(134,142)(135,141)(136,160)(137,159)(138,158)(139,157)(140,156)>;
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), (1,28,51,143,115,133,84,71)(2,27,52,142,116,132,85,70)(3,26,53,141,117,131,86,69)(4,25,54,160,118,130,87,68)(5,24,55,159,119,129,88,67)(6,23,56,158,120,128,89,66)(7,22,57,157,101,127,90,65)(8,21,58,156,102,126,91,64)(9,40,59,155,103,125,92,63)(10,39,60,154,104,124,93,62)(11,38,41,153,105,123,94,61)(12,37,42,152,106,122,95,80)(13,36,43,151,107,121,96,79)(14,35,44,150,108,140,97,78)(15,34,45,149,109,139,98,77)(16,33,46,148,110,138,99,76)(17,32,47,147,111,137,100,75)(18,31,48,146,112,136,81,74)(19,30,49,145,113,135,82,73)(20,29,50,144,114,134,83,72), (2,20)(3,19)(4,18)(5,17)(6,16)(7,15)(8,14)(9,13)(10,12)(21,78)(22,77)(23,76)(24,75)(25,74)(26,73)(27,72)(28,71)(29,70)(30,69)(31,68)(32,67)(33,66)(34,65)(35,64)(36,63)(37,62)(38,61)(39,80)(40,79)(41,94)(42,93)(43,92)(44,91)(45,90)(46,89)(47,88)(48,87)(49,86)(50,85)(51,84)(52,83)(53,82)(54,81)(55,100)(56,99)(57,98)(58,97)(59,96)(60,95)(101,109)(102,108)(103,107)(104,106)(110,120)(111,119)(112,118)(113,117)(114,116)(121,155)(122,154)(123,153)(124,152)(125,151)(126,150)(127,149)(128,148)(129,147)(130,146)(131,145)(132,144)(133,143)(134,142)(135,141)(136,160)(137,159)(138,158)(139,157)(140,156) );
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)], [(1,28,51,143,115,133,84,71),(2,27,52,142,116,132,85,70),(3,26,53,141,117,131,86,69),(4,25,54,160,118,130,87,68),(5,24,55,159,119,129,88,67),(6,23,56,158,120,128,89,66),(7,22,57,157,101,127,90,65),(8,21,58,156,102,126,91,64),(9,40,59,155,103,125,92,63),(10,39,60,154,104,124,93,62),(11,38,41,153,105,123,94,61),(12,37,42,152,106,122,95,80),(13,36,43,151,107,121,96,79),(14,35,44,150,108,140,97,78),(15,34,45,149,109,139,98,77),(16,33,46,148,110,138,99,76),(17,32,47,147,111,137,100,75),(18,31,48,146,112,136,81,74),(19,30,49,145,113,135,82,73),(20,29,50,144,114,134,83,72)], [(2,20),(3,19),(4,18),(5,17),(6,16),(7,15),(8,14),(9,13),(10,12),(21,78),(22,77),(23,76),(24,75),(25,74),(26,73),(27,72),(28,71),(29,70),(30,69),(31,68),(32,67),(33,66),(34,65),(35,64),(36,63),(37,62),(38,61),(39,80),(40,79),(41,94),(42,93),(43,92),(44,91),(45,90),(46,89),(47,88),(48,87),(49,86),(50,85),(51,84),(52,83),(53,82),(54,81),(55,100),(56,99),(57,98),(58,97),(59,96),(60,95),(101,109),(102,108),(103,107),(104,106),(110,120),(111,119),(112,118),(113,117),(114,116),(121,155),(122,154),(123,153),(124,152),(125,151),(126,150),(127,149),(128,148),(129,147),(130,146),(131,145),(132,144),(133,143),(134,142),(135,141),(136,160),(137,159),(138,158),(139,157),(140,156)]])
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