direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×C5⋊D16, D8⋊7D10, C10⋊2D16, C20.20D8, C40.20D4, D40⋊13C22, C40.23C23, C5⋊3(C2×D16), (C2×D8)⋊1D5, (C10×D8)⋊4C2, (C2×D40)⋊17C2, C4.8(D4⋊D5), (C2×C10).41D8, C10.62(C2×D8), (C5×D8)⋊7C22, C5⋊2C16⋊7C22, (C2×C20).179D4, C20.159(C2×D4), (C2×C8).233D10, C8.13(C5⋊D4), C8.29(C22×D5), (C2×C40).85C22, C22.21(D4⋊D5), (C2×C5⋊2C16)⋊6C2, C4.1(C2×C5⋊D4), C2.17(C2×D4⋊D5), (C2×C4).142(C5⋊D4), SmallGroup(320,773)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×C5⋊D16
G = < a,b,c,d | a2=b5=c16=d2=1, ab=ba, ac=ca, ad=da, cbc-1=dbd=b-1, dcd=c-1 >
Subgroups: 542 in 98 conjugacy classes, 39 normal (23 characteristic)
C1, C2, C2, C2, C4, C22, C22, C5, C8, C2×C4, D4, C23, D5, C10, C10, C10, C16, C2×C8, D8, D8, C2×D4, C20, D10, C2×C10, C2×C10, C2×C16, D16, C2×D8, C2×D8, C40, D20, C2×C20, C5×D4, C22×D5, C22×C10, C2×D16, C5⋊2C16, D40, D40, C2×C40, C5×D8, C5×D8, C2×D20, D4×C10, C2×C5⋊2C16, C5⋊D16, C2×D40, C10×D8, C2×C5⋊D16
Quotients: C1, C2, C22, D4, C23, D5, D8, C2×D4, D10, D16, C2×D8, C5⋊D4, C22×D5, C2×D16, D4⋊D5, C2×C5⋊D4, C5⋊D16, C2×D4⋊D5, C2×C5⋊D16
►On 160 points
Generators in S160
(1 31)(2 32)(3 17)(4 18)(5 19)(6 20)(7 21)(8 22)(9 23)(10 24)(11 25)(12 26)(13 27)(14 28)(15 29)(16 30)(33 88)(34 89)(35 90)(36 91)(37 92)(38 93)(39 94)(40 95)(41 96)(42 81)(43 82)(44 83)(45 84)(46 85)(47 86)(48 87)(49 70)(50 71)(51 72)(52 73)(53 74)(54 75)(55 76)(56 77)(57 78)(58 79)(59 80)(60 65)(61 66)(62 67)(63 68)(64 69)(97 137)(98 138)(99 139)(100 140)(101 141)(102 142)(103 143)(104 144)(105 129)(106 130)(107 131)(108 132)(109 133)(110 134)(111 135)(112 136)(113 153)(114 154)(115 155)(116 156)(117 157)(118 158)(119 159)(120 160)(121 145)(122 146)(123 147)(124 148)(125 149)(126 150)(127 151)(128 152)
(1 119 137 91 54)(2 55 92 138 120)(3 121 139 93 56)(4 57 94 140 122)(5 123 141 95 58)(6 59 96 142 124)(7 125 143 81 60)(8 61 82 144 126)(9 127 129 83 62)(10 63 84 130 128)(11 113 131 85 64)(12 49 86 132 114)(13 115 133 87 50)(14 51 88 134 116)(15 117 135 89 52)(16 53 90 136 118)(17 145 99 38 77)(18 78 39 100 146)(19 147 101 40 79)(20 80 41 102 148)(21 149 103 42 65)(22 66 43 104 150)(23 151 105 44 67)(24 68 45 106 152)(25 153 107 46 69)(26 70 47 108 154)(27 155 109 48 71)(28 72 33 110 156)(29 157 111 34 73)(30 74 35 112 158)(31 159 97 36 75)(32 76 37 98 160)
(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)
(2 16)(3 15)(4 14)(5 13)(6 12)(7 11)(8 10)(17 29)(18 28)(19 27)(20 26)(21 25)(22 24)(30 32)(33 100)(34 99)(35 98)(36 97)(37 112)(38 111)(39 110)(40 109)(41 108)(42 107)(43 106)(44 105)(45 104)(46 103)(47 102)(48 101)(49 124)(50 123)(51 122)(52 121)(53 120)(54 119)(55 118)(56 117)(57 116)(58 115)(59 114)(60 113)(61 128)(62 127)(63 126)(64 125)(65 153)(66 152)(67 151)(68 150)(69 149)(70 148)(71 147)(72 146)(73 145)(74 160)(75 159)(76 158)(77 157)(78 156)(79 155)(80 154)(81 131)(82 130)(83 129)(84 144)(85 143)(86 142)(87 141)(88 140)(89 139)(90 138)(91 137)(92 136)(93 135)(94 134)(95 133)(96 132)
G:=sub<Sym(160)| (1,31)(2,32)(3,17)(4,18)(5,19)(6,20)(7,21)(8,22)(9,23)(10,24)(11,25)(12,26)(13,27)(14,28)(15,29)(16,30)(33,88)(34,89)(35,90)(36,91)(37,92)(38,93)(39,94)(40,95)(41,96)(42,81)(43,82)(44,83)(45,84)(46,85)(47,86)(48,87)(49,70)(50,71)(51,72)(52,73)(53,74)(54,75)(55,76)(56,77)(57,78)(58,79)(59,80)(60,65)(61,66)(62,67)(63,68)(64,69)(97,137)(98,138)(99,139)(100,140)(101,141)(102,142)(103,143)(104,144)(105,129)(106,130)(107,131)(108,132)(109,133)(110,134)(111,135)(112,136)(113,153)(114,154)(115,155)(116,156)(117,157)(118,158)(119,159)(120,160)(121,145)(122,146)(123,147)(124,148)(125,149)(126,150)(127,151)(128,152), (1,119,137,91,54)(2,55,92,138,120)(3,121,139,93,56)(4,57,94,140,122)(5,123,141,95,58)(6,59,96,142,124)(7,125,143,81,60)(8,61,82,144,126)(9,127,129,83,62)(10,63,84,130,128)(11,113,131,85,64)(12,49,86,132,114)(13,115,133,87,50)(14,51,88,134,116)(15,117,135,89,52)(16,53,90,136,118)(17,145,99,38,77)(18,78,39,100,146)(19,147,101,40,79)(20,80,41,102,148)(21,149,103,42,65)(22,66,43,104,150)(23,151,105,44,67)(24,68,45,106,152)(25,153,107,46,69)(26,70,47,108,154)(27,155,109,48,71)(28,72,33,110,156)(29,157,111,34,73)(30,74,35,112,158)(31,159,97,36,75)(32,76,37,98,160), (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), (2,16)(3,15)(4,14)(5,13)(6,12)(7,11)(8,10)(17,29)(18,28)(19,27)(20,26)(21,25)(22,24)(30,32)(33,100)(34,99)(35,98)(36,97)(37,112)(38,111)(39,110)(40,109)(41,108)(42,107)(43,106)(44,105)(45,104)(46,103)(47,102)(48,101)(49,124)(50,123)(51,122)(52,121)(53,120)(54,119)(55,118)(56,117)(57,116)(58,115)(59,114)(60,113)(61,128)(62,127)(63,126)(64,125)(65,153)(66,152)(67,151)(68,150)(69,149)(70,148)(71,147)(72,146)(73,145)(74,160)(75,159)(76,158)(77,157)(78,156)(79,155)(80,154)(81,131)(82,130)(83,129)(84,144)(85,143)(86,142)(87,141)(88,140)(89,139)(90,138)(91,137)(92,136)(93,135)(94,134)(95,133)(96,132)>;
G:=Group( (1,31)(2,32)(3,17)(4,18)(5,19)(6,20)(7,21)(8,22)(9,23)(10,24)(11,25)(12,26)(13,27)(14,28)(15,29)(16,30)(33,88)(34,89)(35,90)(36,91)(37,92)(38,93)(39,94)(40,95)(41,96)(42,81)(43,82)(44,83)(45,84)(46,85)(47,86)(48,87)(49,70)(50,71)(51,72)(52,73)(53,74)(54,75)(55,76)(56,77)(57,78)(58,79)(59,80)(60,65)(61,66)(62,67)(63,68)(64,69)(97,137)(98,138)(99,139)(100,140)(101,141)(102,142)(103,143)(104,144)(105,129)(106,130)(107,131)(108,132)(109,133)(110,134)(111,135)(112,136)(113,153)(114,154)(115,155)(116,156)(117,157)(118,158)(119,159)(120,160)(121,145)(122,146)(123,147)(124,148)(125,149)(126,150)(127,151)(128,152), (1,119,137,91,54)(2,55,92,138,120)(3,121,139,93,56)(4,57,94,140,122)(5,123,141,95,58)(6,59,96,142,124)(7,125,143,81,60)(8,61,82,144,126)(9,127,129,83,62)(10,63,84,130,128)(11,113,131,85,64)(12,49,86,132,114)(13,115,133,87,50)(14,51,88,134,116)(15,117,135,89,52)(16,53,90,136,118)(17,145,99,38,77)(18,78,39,100,146)(19,147,101,40,79)(20,80,41,102,148)(21,149,103,42,65)(22,66,43,104,150)(23,151,105,44,67)(24,68,45,106,152)(25,153,107,46,69)(26,70,47,108,154)(27,155,109,48,71)(28,72,33,110,156)(29,157,111,34,73)(30,74,35,112,158)(31,159,97,36,75)(32,76,37,98,160), (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), (2,16)(3,15)(4,14)(5,13)(6,12)(7,11)(8,10)(17,29)(18,28)(19,27)(20,26)(21,25)(22,24)(30,32)(33,100)(34,99)(35,98)(36,97)(37,112)(38,111)(39,110)(40,109)(41,108)(42,107)(43,106)(44,105)(45,104)(46,103)(47,102)(48,101)(49,124)(50,123)(51,122)(52,121)(53,120)(54,119)(55,118)(56,117)(57,116)(58,115)(59,114)(60,113)(61,128)(62,127)(63,126)(64,125)(65,153)(66,152)(67,151)(68,150)(69,149)(70,148)(71,147)(72,146)(73,145)(74,160)(75,159)(76,158)(77,157)(78,156)(79,155)(80,154)(81,131)(82,130)(83,129)(84,144)(85,143)(86,142)(87,141)(88,140)(89,139)(90,138)(91,137)(92,136)(93,135)(94,134)(95,133)(96,132) );
G=PermutationGroup([[(1,31),(2,32),(3,17),(4,18),(5,19),(6,20),(7,21),(8,22),(9,23),(10,24),(11,25),(12,26),(13,27),(14,28),(15,29),(16,30),(33,88),(34,89),(35,90),(36,91),(37,92),(38,93),(39,94),(40,95),(41,96),(42,81),(43,82),(44,83),(45,84),(46,85),(47,86),(48,87),(49,70),(50,71),(51,72),(52,73),(53,74),(54,75),(55,76),(56,77),(57,78),(58,79),(59,80),(60,65),(61,66),(62,67),(63,68),(64,69),(97,137),(98,138),(99,139),(100,140),(101,141),(102,142),(103,143),(104,144),(105,129),(106,130),(107,131),(108,132),(109,133),(110,134),(111,135),(112,136),(113,153),(114,154),(115,155),(116,156),(117,157),(118,158),(119,159),(120,160),(121,145),(122,146),(123,147),(124,148),(125,149),(126,150),(127,151),(128,152)], [(1,119,137,91,54),(2,55,92,138,120),(3,121,139,93,56),(4,57,94,140,122),(5,123,141,95,58),(6,59,96,142,124),(7,125,143,81,60),(8,61,82,144,126),(9,127,129,83,62),(10,63,84,130,128),(11,113,131,85,64),(12,49,86,132,114),(13,115,133,87,50),(14,51,88,134,116),(15,117,135,89,52),(16,53,90,136,118),(17,145,99,38,77),(18,78,39,100,146),(19,147,101,40,79),(20,80,41,102,148),(21,149,103,42,65),(22,66,43,104,150),(23,151,105,44,67),(24,68,45,106,152),(25,153,107,46,69),(26,70,47,108,154),(27,155,109,48,71),(28,72,33,110,156),(29,157,111,34,73),(30,74,35,112,158),(31,159,97,36,75),(32,76,37,98,160)], [(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)], [(2,16),(3,15),(4,14),(5,13),(6,12),(7,11),(8,10),(17,29),(18,28),(19,27),(20,26),(21,25),(22,24),(30,32),(33,100),(34,99),(35,98),(36,97),(37,112),(38,111),(39,110),(40,109),(41,108),(42,107),(43,106),(44,105),(45,104),(46,103),(47,102),(48,101),(49,124),(50,123),(51,122),(52,121),(53,120),(54,119),(55,118),(56,117),(57,116),(58,115),(59,114),(60,113),(61,128),(62,127),(63,126),(64,125),(65,153),(66,152),(67,151),(68,150),(69,149),(70,148),(71,147),(72,146),(73,145),(74,160),(75,159),(76,158),(77,157),(78,156),(79,155),(80,154),(81,131),(82,130),(83,129),(84,144),(85,143),(86,142),(87,141),(88,140),(89,139),(90,138),(91,137),(92,136),(93,135),(94,134),(95,133),(96,132)]])
50 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 5A | 5B | 8A | 8B | 8C | 8D | 10A | ··· | 10F | 10G | ··· | 10N | 16A | ··· | 16H | 20A | 20B | 20C | 20D | 40A | ··· | 40H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 10 | ··· | 10 | 16 | ··· | 16 | 20 | 20 | 20 | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 8 | 8 | 40 | 40 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 8 | ··· | 8 | 10 | ··· | 10 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
50 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | D4 | D4 | D5 | D8 | D8 | D10 | D10 | D16 | C5⋊D4 | C5⋊D4 | D4⋊D5 | D4⋊D5 | C5⋊D16 |
| kernel | C2×C5⋊D16 | C2×C5⋊2C16 | C5⋊D16 | C2×D40 | C10×D8 | C40 | C2×C20 | C2×D8 | C20 | C2×C10 | C2×C8 | D8 | C10 | C8 | C2×C4 | C4 | C22 | C2 |
| # reps | 1 | 1 | 4 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 8 | 4 | 4 | 2 | 2 | 8 |
Matrix representation of C2×C5⋊D16 ►in GL5(𝔽241)
| 240 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 189 | 240 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 240 | 0 | 0 | 0 | 0 |
| 0 | 214 | 138 | 0 | 0 |
| 0 | 96 | 27 | 0 | 0 |
| 0 | 0 | 0 | 27 | 85 |
| 0 | 0 | 0 | 156 | 27 |
| 240 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 189 | 240 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 240 |
G:=sub<GL(5,GF(241))| [240,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,189,1,0,0,0,240,0,0,0,0,0,0,1,0,0,0,0,0,1],[240,0,0,0,0,0,214,96,0,0,0,138,27,0,0,0,0,0,27,156,0,0,0,85,27],[240,0,0,0,0,0,1,189,0,0,0,0,240,0,0,0,0,0,1,0,0,0,0,0,240] >;
C2×C5⋊D16 in GAP, Magma, Sage, TeX
C_2\times C_5\rtimes D_{16} % in TeX
G:=Group("C2xC5:D16"); // GroupNames label
G:=SmallGroup(320,773);
// by ID
G=gap.SmallGroup(320,773);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,254,675,185,192,1684,438,102,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^5=c^16=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations