metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D80⋊5C2, Q32⋊3D5, D10.7D8, C16.10D10, C80.8C22, Q16.5D10, C40.22C23, Dic5.26D8, D40.5C22, (D5×C16)⋊3C2, C5⋊4(C4○D16), (C5×Q32)⋊3C2, C4.10(D4×D5), C2.25(D5×D8), (C4×D5).62D4, C10.41(C2×D8), C20.16(C2×D4), C5⋊SD32⋊4C2, C5⋊2C8.28D4, Q8.D10⋊5C2, C8.28(C22×D5), C5⋊2C16.8C22, (C8×D5).43C22, (C5×Q16).6C22, SmallGroup(320,546)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D80⋊5C2
G = < a,b,c | a80=b2=c2=1, bab=a-1, cac=a49, cbc=a8b >
Subgroups: 454 in 84 conjugacy classes, 31 normal (21 characteristic)
C1, C2, C2, C4, C4, C22, C5, C8, C8, C2×C4, D4, Q8, D5, C10, C16, C16, C2×C8, D8, SD16, Q16, C4○D4, Dic5, C20, C20, D10, D10, C2×C16, D16, SD32, Q32, C4○D8, C5⋊2C8, C40, C4×D5, C4×D5, D20, C5×Q8, C4○D16, C5⋊2C16, C80, C8×D5, D40, Q8⋊D5, C5×Q16, Q8⋊2D5, D5×C16, D80, C5⋊SD32, C5×Q32, Q8.D10, D80⋊5C2
Quotients: C1, C2, C22, D4, C23, D5, D8, C2×D4, D10, C2×D8, C22×D5, C4○D16, D4×D5, D5×D8, D80⋊5C2
►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 80)(2 79)(3 78)(4 77)(5 76)(6 75)(7 74)(8 73)(9 72)(10 71)(11 70)(12 69)(13 68)(14 67)(15 66)(16 65)(17 64)(18 63)(19 62)(20 61)(21 60)(22 59)(23 58)(24 57)(25 56)(26 55)(27 54)(28 53)(29 52)(30 51)(31 50)(32 49)(33 48)(34 47)(35 46)(36 45)(37 44)(38 43)(39 42)(40 41)(81 96)(82 95)(83 94)(84 93)(85 92)(86 91)(87 90)(88 89)(97 160)(98 159)(99 158)(100 157)(101 156)(102 155)(103 154)(104 153)(105 152)(106 151)(107 150)(108 149)(109 148)(110 147)(111 146)(112 145)(113 144)(114 143)(115 142)(116 141)(117 140)(118 139)(119 138)(120 137)(121 136)(122 135)(123 134)(124 133)(125 132)(126 131)(127 130)(128 129)
(1 109)(2 158)(3 127)(4 96)(5 145)(6 114)(7 83)(8 132)(9 101)(10 150)(11 119)(12 88)(13 137)(14 106)(15 155)(16 124)(17 93)(18 142)(19 111)(20 160)(21 129)(22 98)(23 147)(24 116)(25 85)(26 134)(27 103)(28 152)(29 121)(30 90)(31 139)(32 108)(33 157)(34 126)(35 95)(36 144)(37 113)(38 82)(39 131)(40 100)(41 149)(42 118)(43 87)(44 136)(45 105)(46 154)(47 123)(48 92)(49 141)(50 110)(51 159)(52 128)(53 97)(54 146)(55 115)(56 84)(57 133)(58 102)(59 151)(60 120)(61 89)(62 138)(63 107)(64 156)(65 125)(66 94)(67 143)(68 112)(69 81)(70 130)(71 99)(72 148)(73 117)(74 86)(75 135)(76 104)(77 153)(78 122)(79 91)(80 140)
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), (1,80)(2,79)(3,78)(4,77)(5,76)(6,75)(7,74)(8,73)(9,72)(10,71)(11,70)(12,69)(13,68)(14,67)(15,66)(16,65)(17,64)(18,63)(19,62)(20,61)(21,60)(22,59)(23,58)(24,57)(25,56)(26,55)(27,54)(28,53)(29,52)(30,51)(31,50)(32,49)(33,48)(34,47)(35,46)(36,45)(37,44)(38,43)(39,42)(40,41)(81,96)(82,95)(83,94)(84,93)(85,92)(86,91)(87,90)(88,89)(97,160)(98,159)(99,158)(100,157)(101,156)(102,155)(103,154)(104,153)(105,152)(106,151)(107,150)(108,149)(109,148)(110,147)(111,146)(112,145)(113,144)(114,143)(115,142)(116,141)(117,140)(118,139)(119,138)(120,137)(121,136)(122,135)(123,134)(124,133)(125,132)(126,131)(127,130)(128,129), (1,109)(2,158)(3,127)(4,96)(5,145)(6,114)(7,83)(8,132)(9,101)(10,150)(11,119)(12,88)(13,137)(14,106)(15,155)(16,124)(17,93)(18,142)(19,111)(20,160)(21,129)(22,98)(23,147)(24,116)(25,85)(26,134)(27,103)(28,152)(29,121)(30,90)(31,139)(32,108)(33,157)(34,126)(35,95)(36,144)(37,113)(38,82)(39,131)(40,100)(41,149)(42,118)(43,87)(44,136)(45,105)(46,154)(47,123)(48,92)(49,141)(50,110)(51,159)(52,128)(53,97)(54,146)(55,115)(56,84)(57,133)(58,102)(59,151)(60,120)(61,89)(62,138)(63,107)(64,156)(65,125)(66,94)(67,143)(68,112)(69,81)(70,130)(71,99)(72,148)(73,117)(74,86)(75,135)(76,104)(77,153)(78,122)(79,91)(80,140)>;
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), (1,80)(2,79)(3,78)(4,77)(5,76)(6,75)(7,74)(8,73)(9,72)(10,71)(11,70)(12,69)(13,68)(14,67)(15,66)(16,65)(17,64)(18,63)(19,62)(20,61)(21,60)(22,59)(23,58)(24,57)(25,56)(26,55)(27,54)(28,53)(29,52)(30,51)(31,50)(32,49)(33,48)(34,47)(35,46)(36,45)(37,44)(38,43)(39,42)(40,41)(81,96)(82,95)(83,94)(84,93)(85,92)(86,91)(87,90)(88,89)(97,160)(98,159)(99,158)(100,157)(101,156)(102,155)(103,154)(104,153)(105,152)(106,151)(107,150)(108,149)(109,148)(110,147)(111,146)(112,145)(113,144)(114,143)(115,142)(116,141)(117,140)(118,139)(119,138)(120,137)(121,136)(122,135)(123,134)(124,133)(125,132)(126,131)(127,130)(128,129), (1,109)(2,158)(3,127)(4,96)(5,145)(6,114)(7,83)(8,132)(9,101)(10,150)(11,119)(12,88)(13,137)(14,106)(15,155)(16,124)(17,93)(18,142)(19,111)(20,160)(21,129)(22,98)(23,147)(24,116)(25,85)(26,134)(27,103)(28,152)(29,121)(30,90)(31,139)(32,108)(33,157)(34,126)(35,95)(36,144)(37,113)(38,82)(39,131)(40,100)(41,149)(42,118)(43,87)(44,136)(45,105)(46,154)(47,123)(48,92)(49,141)(50,110)(51,159)(52,128)(53,97)(54,146)(55,115)(56,84)(57,133)(58,102)(59,151)(60,120)(61,89)(62,138)(63,107)(64,156)(65,125)(66,94)(67,143)(68,112)(69,81)(70,130)(71,99)(72,148)(73,117)(74,86)(75,135)(76,104)(77,153)(78,122)(79,91)(80,140) );
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)], [(1,80),(2,79),(3,78),(4,77),(5,76),(6,75),(7,74),(8,73),(9,72),(10,71),(11,70),(12,69),(13,68),(14,67),(15,66),(16,65),(17,64),(18,63),(19,62),(20,61),(21,60),(22,59),(23,58),(24,57),(25,56),(26,55),(27,54),(28,53),(29,52),(30,51),(31,50),(32,49),(33,48),(34,47),(35,46),(36,45),(37,44),(38,43),(39,42),(40,41),(81,96),(82,95),(83,94),(84,93),(85,92),(86,91),(87,90),(88,89),(97,160),(98,159),(99,158),(100,157),(101,156),(102,155),(103,154),(104,153),(105,152),(106,151),(107,150),(108,149),(109,148),(110,147),(111,146),(112,145),(113,144),(114,143),(115,142),(116,141),(117,140),(118,139),(119,138),(120,137),(121,136),(122,135),(123,134),(124,133),(125,132),(126,131),(127,130),(128,129)], [(1,109),(2,158),(3,127),(4,96),(5,145),(6,114),(7,83),(8,132),(9,101),(10,150),(11,119),(12,88),(13,137),(14,106),(15,155),(16,124),(17,93),(18,142),(19,111),(20,160),(21,129),(22,98),(23,147),(24,116),(25,85),(26,134),(27,103),(28,152),(29,121),(30,90),(31,139),(32,108),(33,157),(34,126),(35,95),(36,144),(37,113),(38,82),(39,131),(40,100),(41,149),(42,118),(43,87),(44,136),(45,105),(46,154),(47,123),(48,92),(49,141),(50,110),(51,159),(52,128),(53,97),(54,146),(55,115),(56,84),(57,133),(58,102),(59,151),(60,120),(61,89),(62,138),(63,107),(64,156),(65,125),(66,94),(67,143),(68,112),(69,81),(70,130),(71,99),(72,148),(73,117),(74,86),(75,135),(76,104),(77,153),(78,122),(79,91),(80,140)]])
44 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | 4D | 4E | 5A | 5B | 8A | 8B | 8C | 8D | 10A | 10B | 16A | 16B | 16C | 16D | 16E | 16F | 16G | 16H | 20A | 20B | 20C | 20D | 20E | 20F | 40A | 40B | 40C | 40D | 80A | ··· | 80H |
| order | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 10 | 10 | 16 | 16 | 16 | 16 | 16 | 16 | 16 | 16 | 20 | 20 | 20 | 20 | 20 | 20 | 40 | 40 | 40 | 40 | 80 | ··· | 80 |
| size | 1 | 1 | 10 | 40 | 40 | 2 | 5 | 5 | 8 | 8 | 2 | 2 | 2 | 2 | 10 | 10 | 2 | 2 | 2 | 2 | 2 | 2 | 10 | 10 | 10 | 10 | 4 | 4 | 16 | 16 | 16 | 16 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D5 | D8 | D8 | D10 | D10 | C4○D16 | D4×D5 | D5×D8 | D80⋊5C2 |
| kernel | D80⋊5C2 | D5×C16 | D80 | C5⋊SD32 | C5×Q32 | Q8.D10 | C5⋊2C8 | C4×D5 | Q32 | Dic5 | D10 | C16 | Q16 | C5 | C4 | C2 | C1 |
| # reps | 1 | 1 | 1 | 2 | 1 | 2 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 8 | 2 | 4 | 8 |
Matrix representation of D80⋊5C2 ►in GL4(𝔽241) generated by
| 189 | 240 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 183 | 71 |
| 0 | 0 | 85 | 112 |
| 189 | 240 | 0 | 0 |
| 52 | 52 | 0 | 0 |
| 0 | 0 | 183 | 71 |
| 0 | 0 | 214 | 58 |
| 1 | 0 | 0 | 0 |
| 189 | 240 | 0 | 0 |
| 0 | 0 | 177 | 113 |
| 0 | 0 | 64 | 64 |
G:=sub<GL(4,GF(241))| [189,1,0,0,240,0,0,0,0,0,183,85,0,0,71,112],[189,52,0,0,240,52,0,0,0,0,183,214,0,0,71,58],[1,189,0,0,0,240,0,0,0,0,177,64,0,0,113,64] >;
D80⋊5C2 in GAP, Magma, Sage, TeX
D_{80}\rtimes_5C_2 % in TeX
G:=Group("D80:5C2"); // GroupNames label
G:=SmallGroup(320,546);
// by ID
G=gap.SmallGroup(320,546);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,477,232,758,135,184,346,185,192,851,438,102,12550]);
// Polycyclic
G:=Group<a,b,c|a^80=b^2=c^2=1,b*a*b=a^-1,c*a*c=a^49,c*b*c=a^8*b>;
// generators/relations