metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D10⋊2SD16, C5⋊2C8⋊21D4, C5⋊2(C8⋊8D4), C4⋊C4.24D10, Q8⋊C4⋊2D5, C4.166(D4×D5), D10⋊3Q8⋊1C2, C4⋊D20.3C2, C20.121(C2×D4), (C2×C8).210D10, (C2×Q8).16D10, D20⋊5C4⋊24C2, C2.18(D5×SD16), C4.32(C4○D20), C20.19(C4○D4), C10.69(C4○D8), C20.Q8⋊11C2, C10.32(C2×SD16), (C22×D5).81D4, C22.197(D4×D5), C2.8(Q8.D10), C10.23(C4⋊D4), (C2×C20).247C23, (C2×C40).200C22, (C2×Dic5).139D4, (C2×D20).67C22, C4⋊Dic5.94C22, (Q8×C10).30C22, C2.26(D10⋊D4), (D5×C2×C8)⋊21C2, (C2×Q8⋊D5)⋊3C2, (C5×Q8⋊C4)⋊22C2, (C2×C10).260(C2×D4), (C5×C4⋊C4).48C22, (C2×C4×D5).298C22, (C2×C4).354(C22×D5), (C2×C5⋊2C8).228C22, SmallGroup(320,434)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D10⋊2SD16
G = < a,b,c,d | a10=b2=c8=d2=1, bab=cac-1=dad=a-1, cbc-1=a8b, dbd=a3b, dcd=c3 >
Subgroups: 582 in 124 conjugacy classes, 41 normal (37 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, Q8, C23, D5, C10, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, SD16, C22×C4, C2×D4, C2×Q8, Dic5, C20, C20, D10, D10, C2×C10, D4⋊C4, Q8⋊C4, C4.Q8, C4⋊D4, C22⋊Q8, C22×C8, C2×SD16, C5⋊2C8, C40, C4×D5, D20, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C5×Q8, C22×D5, C22×D5, C8⋊8D4, C8×D5, C2×C5⋊2C8, C10.D4, C4⋊Dic5, D10⋊C4, Q8⋊D5, C5×C4⋊C4, C2×C40, C2×C4×D5, C2×D20, C2×D20, Q8×C10, C20.Q8, D20⋊5C4, C5×Q8⋊C4, C4⋊D20, D5×C2×C8, C2×Q8⋊D5, D10⋊3Q8, D10⋊2SD16
Quotients: C1, C2, C22, D4, C23, D5, SD16, C2×D4, C4○D4, D10, C4⋊D4, C2×SD16, C4○D8, C22×D5, C8⋊8D4, C4○D20, D4×D5, D10⋊D4, D5×SD16, Q8.D10, D10⋊2SD16
►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)
(1 143)(2 142)(3 141)(4 150)(5 149)(6 148)(7 147)(8 146)(9 145)(10 144)(11 151)(12 160)(13 159)(14 158)(15 157)(16 156)(17 155)(18 154)(19 153)(20 152)(21 126)(22 125)(23 124)(24 123)(25 122)(26 121)(27 130)(28 129)(29 128)(30 127)(31 59)(32 58)(33 57)(34 56)(35 55)(36 54)(37 53)(38 52)(39 51)(40 60)(41 133)(42 132)(43 131)(44 140)(45 139)(46 138)(47 137)(48 136)(49 135)(50 134)(61 120)(62 119)(63 118)(64 117)(65 116)(66 115)(67 114)(68 113)(69 112)(70 111)(71 110)(72 109)(73 108)(74 107)(75 106)(76 105)(77 104)(78 103)(79 102)(80 101)(81 93)(82 92)(83 91)(84 100)(85 99)(86 98)(87 97)(88 96)(89 95)(90 94)
(1 50 15 64 55 78 28 95)(2 49 16 63 56 77 29 94)(3 48 17 62 57 76 30 93)(4 47 18 61 58 75 21 92)(5 46 19 70 59 74 22 91)(6 45 20 69 60 73 23 100)(7 44 11 68 51 72 24 99)(8 43 12 67 52 71 25 98)(9 42 13 66 53 80 26 97)(10 41 14 65 54 79 27 96)(31 109 125 85 149 140 153 113)(32 108 126 84 150 139 154 112)(33 107 127 83 141 138 155 111)(34 106 128 82 142 137 156 120)(35 105 129 81 143 136 157 119)(36 104 130 90 144 135 158 118)(37 103 121 89 145 134 159 117)(38 102 122 88 146 133 160 116)(39 101 123 87 147 132 151 115)(40 110 124 86 148 131 152 114)
(1 55)(2 54)(3 53)(4 52)(5 51)(6 60)(7 59)(8 58)(9 57)(10 56)(11 19)(12 18)(13 17)(14 16)(21 25)(22 24)(26 30)(27 29)(31 144)(32 143)(33 142)(34 141)(35 150)(36 149)(37 148)(38 147)(39 146)(40 145)(41 94)(42 93)(43 92)(44 91)(45 100)(46 99)(47 98)(48 97)(49 96)(50 95)(61 71)(62 80)(63 79)(64 78)(65 77)(66 76)(67 75)(68 74)(69 73)(70 72)(81 139)(82 138)(83 137)(84 136)(85 135)(86 134)(87 133)(88 132)(89 131)(90 140)(101 116)(102 115)(103 114)(104 113)(105 112)(106 111)(107 120)(108 119)(109 118)(110 117)(121 124)(122 123)(125 130)(126 129)(127 128)(151 160)(152 159)(153 158)(154 157)(155 156)
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), (1,143)(2,142)(3,141)(4,150)(5,149)(6,148)(7,147)(8,146)(9,145)(10,144)(11,151)(12,160)(13,159)(14,158)(15,157)(16,156)(17,155)(18,154)(19,153)(20,152)(21,126)(22,125)(23,124)(24,123)(25,122)(26,121)(27,130)(28,129)(29,128)(30,127)(31,59)(32,58)(33,57)(34,56)(35,55)(36,54)(37,53)(38,52)(39,51)(40,60)(41,133)(42,132)(43,131)(44,140)(45,139)(46,138)(47,137)(48,136)(49,135)(50,134)(61,120)(62,119)(63,118)(64,117)(65,116)(66,115)(67,114)(68,113)(69,112)(70,111)(71,110)(72,109)(73,108)(74,107)(75,106)(76,105)(77,104)(78,103)(79,102)(80,101)(81,93)(82,92)(83,91)(84,100)(85,99)(86,98)(87,97)(88,96)(89,95)(90,94), (1,50,15,64,55,78,28,95)(2,49,16,63,56,77,29,94)(3,48,17,62,57,76,30,93)(4,47,18,61,58,75,21,92)(5,46,19,70,59,74,22,91)(6,45,20,69,60,73,23,100)(7,44,11,68,51,72,24,99)(8,43,12,67,52,71,25,98)(9,42,13,66,53,80,26,97)(10,41,14,65,54,79,27,96)(31,109,125,85,149,140,153,113)(32,108,126,84,150,139,154,112)(33,107,127,83,141,138,155,111)(34,106,128,82,142,137,156,120)(35,105,129,81,143,136,157,119)(36,104,130,90,144,135,158,118)(37,103,121,89,145,134,159,117)(38,102,122,88,146,133,160,116)(39,101,123,87,147,132,151,115)(40,110,124,86,148,131,152,114), (1,55)(2,54)(3,53)(4,52)(5,51)(6,60)(7,59)(8,58)(9,57)(10,56)(11,19)(12,18)(13,17)(14,16)(21,25)(22,24)(26,30)(27,29)(31,144)(32,143)(33,142)(34,141)(35,150)(36,149)(37,148)(38,147)(39,146)(40,145)(41,94)(42,93)(43,92)(44,91)(45,100)(46,99)(47,98)(48,97)(49,96)(50,95)(61,71)(62,80)(63,79)(64,78)(65,77)(66,76)(67,75)(68,74)(69,73)(70,72)(81,139)(82,138)(83,137)(84,136)(85,135)(86,134)(87,133)(88,132)(89,131)(90,140)(101,116)(102,115)(103,114)(104,113)(105,112)(106,111)(107,120)(108,119)(109,118)(110,117)(121,124)(122,123)(125,130)(126,129)(127,128)(151,160)(152,159)(153,158)(154,157)(155,156)>;
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), (1,143)(2,142)(3,141)(4,150)(5,149)(6,148)(7,147)(8,146)(9,145)(10,144)(11,151)(12,160)(13,159)(14,158)(15,157)(16,156)(17,155)(18,154)(19,153)(20,152)(21,126)(22,125)(23,124)(24,123)(25,122)(26,121)(27,130)(28,129)(29,128)(30,127)(31,59)(32,58)(33,57)(34,56)(35,55)(36,54)(37,53)(38,52)(39,51)(40,60)(41,133)(42,132)(43,131)(44,140)(45,139)(46,138)(47,137)(48,136)(49,135)(50,134)(61,120)(62,119)(63,118)(64,117)(65,116)(66,115)(67,114)(68,113)(69,112)(70,111)(71,110)(72,109)(73,108)(74,107)(75,106)(76,105)(77,104)(78,103)(79,102)(80,101)(81,93)(82,92)(83,91)(84,100)(85,99)(86,98)(87,97)(88,96)(89,95)(90,94), (1,50,15,64,55,78,28,95)(2,49,16,63,56,77,29,94)(3,48,17,62,57,76,30,93)(4,47,18,61,58,75,21,92)(5,46,19,70,59,74,22,91)(6,45,20,69,60,73,23,100)(7,44,11,68,51,72,24,99)(8,43,12,67,52,71,25,98)(9,42,13,66,53,80,26,97)(10,41,14,65,54,79,27,96)(31,109,125,85,149,140,153,113)(32,108,126,84,150,139,154,112)(33,107,127,83,141,138,155,111)(34,106,128,82,142,137,156,120)(35,105,129,81,143,136,157,119)(36,104,130,90,144,135,158,118)(37,103,121,89,145,134,159,117)(38,102,122,88,146,133,160,116)(39,101,123,87,147,132,151,115)(40,110,124,86,148,131,152,114), (1,55)(2,54)(3,53)(4,52)(5,51)(6,60)(7,59)(8,58)(9,57)(10,56)(11,19)(12,18)(13,17)(14,16)(21,25)(22,24)(26,30)(27,29)(31,144)(32,143)(33,142)(34,141)(35,150)(36,149)(37,148)(38,147)(39,146)(40,145)(41,94)(42,93)(43,92)(44,91)(45,100)(46,99)(47,98)(48,97)(49,96)(50,95)(61,71)(62,80)(63,79)(64,78)(65,77)(66,76)(67,75)(68,74)(69,73)(70,72)(81,139)(82,138)(83,137)(84,136)(85,135)(86,134)(87,133)(88,132)(89,131)(90,140)(101,116)(102,115)(103,114)(104,113)(105,112)(106,111)(107,120)(108,119)(109,118)(110,117)(121,124)(122,123)(125,130)(126,129)(127,128)(151,160)(152,159)(153,158)(154,157)(155,156) );
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)], [(1,143),(2,142),(3,141),(4,150),(5,149),(6,148),(7,147),(8,146),(9,145),(10,144),(11,151),(12,160),(13,159),(14,158),(15,157),(16,156),(17,155),(18,154),(19,153),(20,152),(21,126),(22,125),(23,124),(24,123),(25,122),(26,121),(27,130),(28,129),(29,128),(30,127),(31,59),(32,58),(33,57),(34,56),(35,55),(36,54),(37,53),(38,52),(39,51),(40,60),(41,133),(42,132),(43,131),(44,140),(45,139),(46,138),(47,137),(48,136),(49,135),(50,134),(61,120),(62,119),(63,118),(64,117),(65,116),(66,115),(67,114),(68,113),(69,112),(70,111),(71,110),(72,109),(73,108),(74,107),(75,106),(76,105),(77,104),(78,103),(79,102),(80,101),(81,93),(82,92),(83,91),(84,100),(85,99),(86,98),(87,97),(88,96),(89,95),(90,94)], [(1,50,15,64,55,78,28,95),(2,49,16,63,56,77,29,94),(3,48,17,62,57,76,30,93),(4,47,18,61,58,75,21,92),(5,46,19,70,59,74,22,91),(6,45,20,69,60,73,23,100),(7,44,11,68,51,72,24,99),(8,43,12,67,52,71,25,98),(9,42,13,66,53,80,26,97),(10,41,14,65,54,79,27,96),(31,109,125,85,149,140,153,113),(32,108,126,84,150,139,154,112),(33,107,127,83,141,138,155,111),(34,106,128,82,142,137,156,120),(35,105,129,81,143,136,157,119),(36,104,130,90,144,135,158,118),(37,103,121,89,145,134,159,117),(38,102,122,88,146,133,160,116),(39,101,123,87,147,132,151,115),(40,110,124,86,148,131,152,114)], [(1,55),(2,54),(3,53),(4,52),(5,51),(6,60),(7,59),(8,58),(9,57),(10,56),(11,19),(12,18),(13,17),(14,16),(21,25),(22,24),(26,30),(27,29),(31,144),(32,143),(33,142),(34,141),(35,150),(36,149),(37,148),(38,147),(39,146),(40,145),(41,94),(42,93),(43,92),(44,91),(45,100),(46,99),(47,98),(48,97),(49,96),(50,95),(61,71),(62,80),(63,79),(64,78),(65,77),(66,76),(67,75),(68,74),(69,73),(70,72),(81,139),(82,138),(83,137),(84,136),(85,135),(86,134),(87,133),(88,132),(89,131),(90,140),(101,116),(102,115),(103,114),(104,113),(105,112),(106,111),(107,120),(108,119),(109,118),(110,117),(121,124),(122,123),(125,130),(126,129),(127,128),(151,160),(152,159),(153,158),(154,157),(155,156)]])
50 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 5A | 5B | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 10A | ··· | 10F | 20A | 20B | 20C | 20D | 20E | ··· | 20L | 40A | ··· | 40H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 20 | 20 | 20 | 20 | 20 | ··· | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 10 | 10 | 40 | 2 | 2 | 8 | 8 | 10 | 10 | 40 | 2 | 2 | 2 | 2 | 2 | 2 | 10 | 10 | 10 | 10 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 8 | ··· | 8 | 4 | ··· | 4 |
50 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | D5 | C4○D4 | SD16 | D10 | D10 | D10 | C4○D8 | C4○D20 | D4×D5 | D4×D5 | D5×SD16 | Q8.D10 |
| kernel | D10⋊2SD16 | C20.Q8 | D20⋊5C4 | C5×Q8⋊C4 | C4⋊D20 | D5×C2×C8 | C2×Q8⋊D5 | D10⋊3Q8 | C5⋊2C8 | C2×Dic5 | C22×D5 | Q8⋊C4 | C20 | D10 | C4⋊C4 | C2×C8 | C2×Q8 | C10 | C4 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 2 | 2 | 4 | 2 | 2 | 2 | 4 | 8 | 2 | 2 | 4 | 4 |
Matrix representation of D10⋊2SD16 ►in GL4(𝔽41) generated by
| 7 | 6 | 0 | 0 |
| 34 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 40 |
| 27 | 2 | 0 | 0 |
| 5 | 14 | 0 | 0 |
| 0 | 0 | 0 | 9 |
| 0 | 0 | 32 | 0 |
| 3 | 21 | 0 | 0 |
| 21 | 38 | 0 | 0 |
| 0 | 0 | 26 | 26 |
| 0 | 0 | 15 | 26 |
| 34 | 35 | 0 | 0 |
| 8 | 7 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 40 |
G:=sub<GL(4,GF(41))| [7,34,0,0,6,0,0,0,0,0,40,0,0,0,0,40],[27,5,0,0,2,14,0,0,0,0,0,32,0,0,9,0],[3,21,0,0,21,38,0,0,0,0,26,15,0,0,26,26],[34,8,0,0,35,7,0,0,0,0,1,0,0,0,0,40] >;
D10⋊2SD16 in GAP, Magma, Sage, TeX
D_{10}\rtimes_2{\rm SD}_{16} % in TeX
G:=Group("D10:2SD16"); // GroupNames label
G:=SmallGroup(320,434);
// by ID
G=gap.SmallGroup(320,434);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,64,590,219,184,297,136,438,102,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^10=b^2=c^8=d^2=1,b*a*b=c*a*c^-1=d*a*d=a^-1,c*b*c^-1=a^8*b,d*b*d=a^3*b,d*c*d=c^3>;
// generators/relations