metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D10⋊1D8, C5⋊2C8⋊19D4, C5⋊1(C8⋊7D4), C4⋊C4.9D10, C2.10(D5×D8), D4⋊C4⋊2D5, C20⋊2D4⋊1C2, C4⋊D20⋊2C2, C10.24(C2×D8), C4.158(D4×D5), (C2×D4).24D10, C20.7(C4○D4), C10.D8⋊6C2, (C2×C8).202D10, C20.108(C2×D4), D20⋊5C4⋊19C2, C4.24(C4○D20), C10.40(C4○D8), (C22×D5).78D4, C22.173(D4×D5), C10.15(C4⋊D4), (C2×C40).184C22, (C2×C20).215C23, (C2×Dic5).134D4, (C2×D20).53C22, (D4×C10).36C22, C4⋊Dic5.70C22, C2.18(D10⋊D4), C2.10(SD16⋊3D5), (D5×C2×C8)⋊18C2, (C2×D4⋊D5)⋊3C2, (C5×D4⋊C4)⋊22C2, (C2×C10).228(C2×D4), (C5×C4⋊C4).17C22, (C2×C4×D5).294C22, (C2×C4).322(C22×D5), (C2×C5⋊2C8).220C22, SmallGroup(320,402)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D10⋊D8
G = < a,b,c,d | a10=b2=c8=d2=1, bab=cac-1=dad=a-1, cbc-1=a8b, dbd=a3b, dcd=c-1 >
Subgroups: 662 in 134 conjugacy classes, 41 normal (37 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, C23, D5, C10, C10, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, D8, C22×C4, C2×D4, C2×D4, Dic5, C20, C20, D10, D10, C2×C10, C2×C10, D4⋊C4, D4⋊C4, C2.D8, C4⋊D4, C22×C8, C2×D8, C5⋊2C8, C40, C4×D5, D20, C2×Dic5, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C5×D4, C22×D5, C22×D5, C22×C10, C8⋊7D4, C8×D5, C2×C5⋊2C8, C4⋊Dic5, D10⋊C4, D4⋊D5, C23.D5, C5×C4⋊C4, C2×C40, C2×C4×D5, C2×D20, C2×D20, C2×C5⋊D4, D4×C10, C10.D8, D20⋊5C4, C5×D4⋊C4, C4⋊D20, D5×C2×C8, C2×D4⋊D5, C20⋊2D4, D10⋊D8
Quotients: C1, C2, C22, D4, C23, D5, D8, C2×D4, C4○D4, D10, C4⋊D4, C2×D8, C4○D8, C22×D5, C8⋊7D4, C4○D20, D4×D5, D10⋊D4, D5×D8, SD16⋊3D5, D10⋊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)
(1 147)(2 146)(3 145)(4 144)(5 143)(6 142)(7 141)(8 150)(9 149)(10 148)(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)(61 114)(62 113)(63 112)(64 111)(65 120)(66 119)(67 118)(68 117)(69 116)(70 115)(71 104)(72 103)(73 102)(74 101)(75 110)(76 109)(77 108)(78 107)(79 106)(80 105)(81 93)(82 92)(83 91)(84 100)(85 99)(86 98)(87 97)(88 96)(89 95)(90 94)
(1 68 28 50 55 95 15 78)(2 67 29 49 56 94 16 77)(3 66 30 48 57 93 17 76)(4 65 21 47 58 92 18 75)(5 64 22 46 59 91 19 74)(6 63 23 45 60 100 20 73)(7 62 24 44 51 99 11 72)(8 61 25 43 52 98 12 71)(9 70 26 42 53 97 13 80)(10 69 27 41 54 96 14 79)(31 85 153 103 143 113 125 140)(32 84 154 102 144 112 126 139)(33 83 155 101 145 111 127 138)(34 82 156 110 146 120 128 137)(35 81 157 109 147 119 129 136)(36 90 158 108 148 118 130 135)(37 89 159 107 149 117 121 134)(38 88 160 106 150 116 122 133)(39 87 151 105 141 115 123 132)(40 86 152 104 142 114 124 131)
(2 10)(3 9)(4 8)(5 7)(11 22)(12 21)(13 30)(14 29)(15 28)(16 27)(17 26)(18 25)(19 24)(20 23)(31 36)(32 35)(33 34)(37 40)(38 39)(41 94)(42 93)(43 92)(44 91)(45 100)(46 99)(47 98)(48 97)(49 96)(50 95)(51 59)(52 58)(53 57)(54 56)(61 75)(62 74)(63 73)(64 72)(65 71)(66 80)(67 79)(68 78)(69 77)(70 76)(81 139)(82 138)(83 137)(84 136)(85 135)(86 134)(87 133)(88 132)(89 131)(90 140)(101 120)(102 119)(103 118)(104 117)(105 116)(106 115)(107 114)(108 113)(109 112)(110 111)(121 152)(122 151)(123 160)(124 159)(125 158)(126 157)(127 156)(128 155)(129 154)(130 153)(141 150)(142 149)(143 148)(144 147)(145 146)
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), (1,147)(2,146)(3,145)(4,144)(5,143)(6,142)(7,141)(8,150)(9,149)(10,148)(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)(61,114)(62,113)(63,112)(64,111)(65,120)(66,119)(67,118)(68,117)(69,116)(70,115)(71,104)(72,103)(73,102)(74,101)(75,110)(76,109)(77,108)(78,107)(79,106)(80,105)(81,93)(82,92)(83,91)(84,100)(85,99)(86,98)(87,97)(88,96)(89,95)(90,94), (1,68,28,50,55,95,15,78)(2,67,29,49,56,94,16,77)(3,66,30,48,57,93,17,76)(4,65,21,47,58,92,18,75)(5,64,22,46,59,91,19,74)(6,63,23,45,60,100,20,73)(7,62,24,44,51,99,11,72)(8,61,25,43,52,98,12,71)(9,70,26,42,53,97,13,80)(10,69,27,41,54,96,14,79)(31,85,153,103,143,113,125,140)(32,84,154,102,144,112,126,139)(33,83,155,101,145,111,127,138)(34,82,156,110,146,120,128,137)(35,81,157,109,147,119,129,136)(36,90,158,108,148,118,130,135)(37,89,159,107,149,117,121,134)(38,88,160,106,150,116,122,133)(39,87,151,105,141,115,123,132)(40,86,152,104,142,114,124,131), (2,10)(3,9)(4,8)(5,7)(11,22)(12,21)(13,30)(14,29)(15,28)(16,27)(17,26)(18,25)(19,24)(20,23)(31,36)(32,35)(33,34)(37,40)(38,39)(41,94)(42,93)(43,92)(44,91)(45,100)(46,99)(47,98)(48,97)(49,96)(50,95)(51,59)(52,58)(53,57)(54,56)(61,75)(62,74)(63,73)(64,72)(65,71)(66,80)(67,79)(68,78)(69,77)(70,76)(81,139)(82,138)(83,137)(84,136)(85,135)(86,134)(87,133)(88,132)(89,131)(90,140)(101,120)(102,119)(103,118)(104,117)(105,116)(106,115)(107,114)(108,113)(109,112)(110,111)(121,152)(122,151)(123,160)(124,159)(125,158)(126,157)(127,156)(128,155)(129,154)(130,153)(141,150)(142,149)(143,148)(144,147)(145,146)>;
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), (1,147)(2,146)(3,145)(4,144)(5,143)(6,142)(7,141)(8,150)(9,149)(10,148)(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)(61,114)(62,113)(63,112)(64,111)(65,120)(66,119)(67,118)(68,117)(69,116)(70,115)(71,104)(72,103)(73,102)(74,101)(75,110)(76,109)(77,108)(78,107)(79,106)(80,105)(81,93)(82,92)(83,91)(84,100)(85,99)(86,98)(87,97)(88,96)(89,95)(90,94), (1,68,28,50,55,95,15,78)(2,67,29,49,56,94,16,77)(3,66,30,48,57,93,17,76)(4,65,21,47,58,92,18,75)(5,64,22,46,59,91,19,74)(6,63,23,45,60,100,20,73)(7,62,24,44,51,99,11,72)(8,61,25,43,52,98,12,71)(9,70,26,42,53,97,13,80)(10,69,27,41,54,96,14,79)(31,85,153,103,143,113,125,140)(32,84,154,102,144,112,126,139)(33,83,155,101,145,111,127,138)(34,82,156,110,146,120,128,137)(35,81,157,109,147,119,129,136)(36,90,158,108,148,118,130,135)(37,89,159,107,149,117,121,134)(38,88,160,106,150,116,122,133)(39,87,151,105,141,115,123,132)(40,86,152,104,142,114,124,131), (2,10)(3,9)(4,8)(5,7)(11,22)(12,21)(13,30)(14,29)(15,28)(16,27)(17,26)(18,25)(19,24)(20,23)(31,36)(32,35)(33,34)(37,40)(38,39)(41,94)(42,93)(43,92)(44,91)(45,100)(46,99)(47,98)(48,97)(49,96)(50,95)(51,59)(52,58)(53,57)(54,56)(61,75)(62,74)(63,73)(64,72)(65,71)(66,80)(67,79)(68,78)(69,77)(70,76)(81,139)(82,138)(83,137)(84,136)(85,135)(86,134)(87,133)(88,132)(89,131)(90,140)(101,120)(102,119)(103,118)(104,117)(105,116)(106,115)(107,114)(108,113)(109,112)(110,111)(121,152)(122,151)(123,160)(124,159)(125,158)(126,157)(127,156)(128,155)(129,154)(130,153)(141,150)(142,149)(143,148)(144,147)(145,146) );
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)], [(1,147),(2,146),(3,145),(4,144),(5,143),(6,142),(7,141),(8,150),(9,149),(10,148),(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),(61,114),(62,113),(63,112),(64,111),(65,120),(66,119),(67,118),(68,117),(69,116),(70,115),(71,104),(72,103),(73,102),(74,101),(75,110),(76,109),(77,108),(78,107),(79,106),(80,105),(81,93),(82,92),(83,91),(84,100),(85,99),(86,98),(87,97),(88,96),(89,95),(90,94)], [(1,68,28,50,55,95,15,78),(2,67,29,49,56,94,16,77),(3,66,30,48,57,93,17,76),(4,65,21,47,58,92,18,75),(5,64,22,46,59,91,19,74),(6,63,23,45,60,100,20,73),(7,62,24,44,51,99,11,72),(8,61,25,43,52,98,12,71),(9,70,26,42,53,97,13,80),(10,69,27,41,54,96,14,79),(31,85,153,103,143,113,125,140),(32,84,154,102,144,112,126,139),(33,83,155,101,145,111,127,138),(34,82,156,110,146,120,128,137),(35,81,157,109,147,119,129,136),(36,90,158,108,148,118,130,135),(37,89,159,107,149,117,121,134),(38,88,160,106,150,116,122,133),(39,87,151,105,141,115,123,132),(40,86,152,104,142,114,124,131)], [(2,10),(3,9),(4,8),(5,7),(11,22),(12,21),(13,30),(14,29),(15,28),(16,27),(17,26),(18,25),(19,24),(20,23),(31,36),(32,35),(33,34),(37,40),(38,39),(41,94),(42,93),(43,92),(44,91),(45,100),(46,99),(47,98),(48,97),(49,96),(50,95),(51,59),(52,58),(53,57),(54,56),(61,75),(62,74),(63,73),(64,72),(65,71),(66,80),(67,79),(68,78),(69,77),(70,76),(81,139),(82,138),(83,137),(84,136),(85,135),(86,134),(87,133),(88,132),(89,131),(90,140),(101,120),(102,119),(103,118),(104,117),(105,116),(106,115),(107,114),(108,113),(109,112),(110,111),(121,152),(122,151),(123,160),(124,159),(125,158),(126,157),(127,156),(128,155),(129,154),(130,153),(141,150),(142,149),(143,148),(144,147),(145,146)]])
50 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 5A | 5B | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 20A | 20B | 20C | 20D | 20E | 20F | 20G | 20H | 40A | ··· | 40H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 8 | 10 | 10 | 40 | 2 | 2 | 8 | 10 | 10 | 40 | 2 | 2 | 2 | 2 | 2 | 2 | 10 | 10 | 10 | 10 | 2 | ··· | 2 | 8 | 8 | 8 | 8 | 4 | 4 | 4 | 4 | 8 | 8 | 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 | D8 | D10 | D10 | D10 | C4○D8 | C4○D20 | D4×D5 | D4×D5 | D5×D8 | SD16⋊3D5 |
| kernel | D10⋊D8 | C10.D8 | D20⋊5C4 | C5×D4⋊C4 | C4⋊D20 | D5×C2×C8 | C2×D4⋊D5 | C20⋊2D4 | C5⋊2C8 | C2×Dic5 | C22×D5 | D4⋊C4 | C20 | D10 | C4⋊C4 | C2×C8 | C2×D4 | 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⋊D8 ►in GL4(𝔽41) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 34 |
| 0 | 0 | 7 | 34 |
| 40 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 30 | 9 |
| 0 | 0 | 14 | 11 |
| 29 | 12 | 0 | 0 |
| 29 | 29 | 0 | 0 |
| 0 | 0 | 38 | 38 |
| 0 | 0 | 17 | 3 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 34 | 7 |
| 0 | 0 | 40 | 7 |
G:=sub<GL(4,GF(41))| [1,0,0,0,0,1,0,0,0,0,1,7,0,0,34,34],[40,0,0,0,0,40,0,0,0,0,30,14,0,0,9,11],[29,29,0,0,12,29,0,0,0,0,38,17,0,0,38,3],[0,1,0,0,1,0,0,0,0,0,34,40,0,0,7,7] >;
D10⋊D8 in GAP, Magma, Sage, TeX
D_{10}\rtimes D_8 % in TeX
G:=Group("D10:D8"); // GroupNames label
G:=SmallGroup(320,402);
// by ID
G=gap.SmallGroup(320,402);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,64,590,555,297,136,438,102,12550]);
// 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^-1>;
// generators/relations