metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C20.19D8, C20.19SD16, C42.224D10, C4⋊Q8⋊6D5, C4⋊C4.84D10, C4.7(D4⋊D5), C10.61(C2×D8), C5⋊4(C4.4D8), C4.5(Q8⋊D5), (C2×C20).158D4, D20⋊6C4⋊42C2, C20.83(C4○D4), C20⋊4D4.10C2, C10.78(C2×SD16), (C4×C20).135C22, (C2×C20).406C23, C4.16(Q8⋊2D5), C10.58(C4.4D4), (C2×D20).113C22, C2.11(C20.23D4), (C5×C4⋊Q8)⋊6C2, (C4×C5⋊2C8)⋊18C2, C2.16(C2×D4⋊D5), C2.16(C2×Q8⋊D5), (C2×C10).537(C2×D4), (C2×C4).137(C5⋊D4), (C5×C4⋊C4).131C22, (C2×C4).503(C22×D5), C22.209(C2×C5⋊D4), (C2×C5⋊2C8).271C22, SmallGroup(320,715)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C20.D8
G = < a,b,c | a20=b8=c2=1, bab-1=a9, cac=a-1, cbc=a10b-1 >
Subgroups: 606 in 118 conjugacy classes, 47 normal (23 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, Q8, C23, D5, C10, C42, C4⋊C4, C4⋊C4, C2×C8, C2×D4, C2×Q8, C20, C20, D10, C2×C10, C4×C8, D4⋊C4, C4⋊1D4, C4⋊Q8, C5⋊2C8, D20, C2×C20, C2×C20, C5×Q8, C22×D5, C4.4D8, C2×C5⋊2C8, C4×C20, C5×C4⋊C4, C5×C4⋊C4, C2×D20, C2×D20, Q8×C10, C4×C5⋊2C8, D20⋊6C4, C20⋊4D4, C5×C4⋊Q8, C20.D8
Quotients: C1, C2, C22, D4, C23, D5, D8, SD16, C2×D4, C4○D4, D10, C4.4D4, C2×D8, C2×SD16, C5⋊D4, C22×D5, C4.4D8, D4⋊D5, Q8⋊D5, Q8⋊2D5, C2×C5⋊D4, C2×D4⋊D5, C2×Q8⋊D5, C20.23D4, C20.D8
►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)
(1 155 49 89 33 68 127 107)(2 144 50 98 34 77 128 116)(3 153 51 87 35 66 129 105)(4 142 52 96 36 75 130 114)(5 151 53 85 37 64 131 103)(6 160 54 94 38 73 132 112)(7 149 55 83 39 62 133 101)(8 158 56 92 40 71 134 110)(9 147 57 81 21 80 135 119)(10 156 58 90 22 69 136 108)(11 145 59 99 23 78 137 117)(12 154 60 88 24 67 138 106)(13 143 41 97 25 76 139 115)(14 152 42 86 26 65 140 104)(15 141 43 95 27 74 121 113)(16 150 44 84 28 63 122 102)(17 159 45 93 29 72 123 111)(18 148 46 82 30 61 124 120)(19 157 47 91 31 70 125 109)(20 146 48 100 32 79 126 118)
(2 20)(3 19)(4 18)(5 17)(6 16)(7 15)(8 14)(9 13)(10 12)(21 25)(22 24)(26 40)(27 39)(28 38)(29 37)(30 36)(31 35)(32 34)(41 135)(42 134)(43 133)(44 132)(45 131)(46 130)(47 129)(48 128)(49 127)(50 126)(51 125)(52 124)(53 123)(54 122)(55 121)(56 140)(57 139)(58 138)(59 137)(60 136)(61 86)(62 85)(63 84)(64 83)(65 82)(66 81)(67 100)(68 99)(69 98)(70 97)(71 96)(72 95)(73 94)(74 93)(75 92)(76 91)(77 90)(78 89)(79 88)(80 87)(101 151)(102 150)(103 149)(104 148)(105 147)(106 146)(107 145)(108 144)(109 143)(110 142)(111 141)(112 160)(113 159)(114 158)(115 157)(116 156)(117 155)(118 154)(119 153)(120 152)
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), (1,155,49,89,33,68,127,107)(2,144,50,98,34,77,128,116)(3,153,51,87,35,66,129,105)(4,142,52,96,36,75,130,114)(5,151,53,85,37,64,131,103)(6,160,54,94,38,73,132,112)(7,149,55,83,39,62,133,101)(8,158,56,92,40,71,134,110)(9,147,57,81,21,80,135,119)(10,156,58,90,22,69,136,108)(11,145,59,99,23,78,137,117)(12,154,60,88,24,67,138,106)(13,143,41,97,25,76,139,115)(14,152,42,86,26,65,140,104)(15,141,43,95,27,74,121,113)(16,150,44,84,28,63,122,102)(17,159,45,93,29,72,123,111)(18,148,46,82,30,61,124,120)(19,157,47,91,31,70,125,109)(20,146,48,100,32,79,126,118), (2,20)(3,19)(4,18)(5,17)(6,16)(7,15)(8,14)(9,13)(10,12)(21,25)(22,24)(26,40)(27,39)(28,38)(29,37)(30,36)(31,35)(32,34)(41,135)(42,134)(43,133)(44,132)(45,131)(46,130)(47,129)(48,128)(49,127)(50,126)(51,125)(52,124)(53,123)(54,122)(55,121)(56,140)(57,139)(58,138)(59,137)(60,136)(61,86)(62,85)(63,84)(64,83)(65,82)(66,81)(67,100)(68,99)(69,98)(70,97)(71,96)(72,95)(73,94)(74,93)(75,92)(76,91)(77,90)(78,89)(79,88)(80,87)(101,151)(102,150)(103,149)(104,148)(105,147)(106,146)(107,145)(108,144)(109,143)(110,142)(111,141)(112,160)(113,159)(114,158)(115,157)(116,156)(117,155)(118,154)(119,153)(120,152)>;
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), (1,155,49,89,33,68,127,107)(2,144,50,98,34,77,128,116)(3,153,51,87,35,66,129,105)(4,142,52,96,36,75,130,114)(5,151,53,85,37,64,131,103)(6,160,54,94,38,73,132,112)(7,149,55,83,39,62,133,101)(8,158,56,92,40,71,134,110)(9,147,57,81,21,80,135,119)(10,156,58,90,22,69,136,108)(11,145,59,99,23,78,137,117)(12,154,60,88,24,67,138,106)(13,143,41,97,25,76,139,115)(14,152,42,86,26,65,140,104)(15,141,43,95,27,74,121,113)(16,150,44,84,28,63,122,102)(17,159,45,93,29,72,123,111)(18,148,46,82,30,61,124,120)(19,157,47,91,31,70,125,109)(20,146,48,100,32,79,126,118), (2,20)(3,19)(4,18)(5,17)(6,16)(7,15)(8,14)(9,13)(10,12)(21,25)(22,24)(26,40)(27,39)(28,38)(29,37)(30,36)(31,35)(32,34)(41,135)(42,134)(43,133)(44,132)(45,131)(46,130)(47,129)(48,128)(49,127)(50,126)(51,125)(52,124)(53,123)(54,122)(55,121)(56,140)(57,139)(58,138)(59,137)(60,136)(61,86)(62,85)(63,84)(64,83)(65,82)(66,81)(67,100)(68,99)(69,98)(70,97)(71,96)(72,95)(73,94)(74,93)(75,92)(76,91)(77,90)(78,89)(79,88)(80,87)(101,151)(102,150)(103,149)(104,148)(105,147)(106,146)(107,145)(108,144)(109,143)(110,142)(111,141)(112,160)(113,159)(114,158)(115,157)(116,156)(117,155)(118,154)(119,153)(120,152) );
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)], [(1,155,49,89,33,68,127,107),(2,144,50,98,34,77,128,116),(3,153,51,87,35,66,129,105),(4,142,52,96,36,75,130,114),(5,151,53,85,37,64,131,103),(6,160,54,94,38,73,132,112),(7,149,55,83,39,62,133,101),(8,158,56,92,40,71,134,110),(9,147,57,81,21,80,135,119),(10,156,58,90,22,69,136,108),(11,145,59,99,23,78,137,117),(12,154,60,88,24,67,138,106),(13,143,41,97,25,76,139,115),(14,152,42,86,26,65,140,104),(15,141,43,95,27,74,121,113),(16,150,44,84,28,63,122,102),(17,159,45,93,29,72,123,111),(18,148,46,82,30,61,124,120),(19,157,47,91,31,70,125,109),(20,146,48,100,32,79,126,118)], [(2,20),(3,19),(4,18),(5,17),(6,16),(7,15),(8,14),(9,13),(10,12),(21,25),(22,24),(26,40),(27,39),(28,38),(29,37),(30,36),(31,35),(32,34),(41,135),(42,134),(43,133),(44,132),(45,131),(46,130),(47,129),(48,128),(49,127),(50,126),(51,125),(52,124),(53,123),(54,122),(55,121),(56,140),(57,139),(58,138),(59,137),(60,136),(61,86),(62,85),(63,84),(64,83),(65,82),(66,81),(67,100),(68,99),(69,98),(70,97),(71,96),(72,95),(73,94),(74,93),(75,92),(76,91),(77,90),(78,89),(79,88),(80,87),(101,151),(102,150),(103,149),(104,148),(105,147),(106,146),(107,145),(108,144),(109,143),(110,142),(111,141),(112,160),(113,159),(114,158),(115,157),(116,156),(117,155),(118,154),(119,153),(120,152)]])
50 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | ··· | 4F | 4G | 4H | 5A | 5B | 8A | ··· | 8H | 10A | ··· | 10F | 20A | ··· | 20L | 20M | ··· | 20T |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 5 | 5 | 8 | ··· | 8 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 40 | 40 | 2 | ··· | 2 | 8 | 8 | 2 | 2 | 10 | ··· | 10 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | ··· | 8 |
50 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | D4 | D5 | D8 | SD16 | C4○D4 | D10 | D10 | C5⋊D4 | D4⋊D5 | Q8⋊D5 | Q8⋊2D5 |
| kernel | C20.D8 | C4×C5⋊2C8 | D20⋊6C4 | C20⋊4D4 | C5×C4⋊Q8 | C2×C20 | C4⋊Q8 | C20 | C20 | C20 | C42 | C4⋊C4 | C2×C4 | C4 | C4 | C4 |
| # reps | 1 | 1 | 4 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 2 | 4 | 8 | 4 | 4 | 4 |
Matrix representation of C20.D8 ►in GL6(𝔽41)
| 20 | 9 | 0 | 0 | 0 | 0 |
| 1 | 21 | 0 | 0 | 0 | 0 |
| 0 | 0 | 6 | 1 | 0 | 0 |
| 0 | 0 | 5 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 22 | 8 |
| 0 | 0 | 0 | 0 | 6 | 19 |
| 28 | 12 | 0 | 0 | 0 | 0 |
| 15 | 2 | 0 | 0 | 0 | 0 |
| 0 | 0 | 6 | 1 | 0 | 0 |
| 0 | 0 | 6 | 35 | 0 | 0 |
| 0 | 0 | 0 | 0 | 7 | 10 |
| 0 | 0 | 0 | 0 | 28 | 34 |
| 20 | 9 | 0 | 0 | 0 | 0 |
| 24 | 21 | 0 | 0 | 0 | 0 |
| 0 | 0 | 6 | 1 | 0 | 0 |
| 0 | 0 | 6 | 35 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 15 | 40 |
G:=sub<GL(6,GF(41))| [20,1,0,0,0,0,9,21,0,0,0,0,0,0,6,5,0,0,0,0,1,1,0,0,0,0,0,0,22,6,0,0,0,0,8,19],[28,15,0,0,0,0,12,2,0,0,0,0,0,0,6,6,0,0,0,0,1,35,0,0,0,0,0,0,7,28,0,0,0,0,10,34],[20,24,0,0,0,0,9,21,0,0,0,0,0,0,6,6,0,0,0,0,1,35,0,0,0,0,0,0,1,15,0,0,0,0,0,40] >;
C20.D8 in GAP, Magma, Sage, TeX
C_{20}.D_8 % in TeX
G:=Group("C20.D8"); // GroupNames label
G:=SmallGroup(320,715);
// by ID
G=gap.SmallGroup(320,715);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,253,120,254,219,100,1123,297,136,12550]);
// Polycyclic
G:=Group<a,b,c|a^20=b^8=c^2=1,b*a*b^-1=a^9,c*a*c=a^-1,c*b*c=a^10*b^-1>;
// generators/relations