metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D10⋊8SD16, Dic10⋊7D4, (C5×Q8)⋊5D4, C4.63(D4×D5), Q8⋊2(C5⋊D4), C5⋊4(Q8⋊D4), C20.48(C2×D4), (C2×SD16)⋊11D5, (C2×D4).72D10, C20⋊2D4.8C2, (C2×C8).147D10, D10⋊1C8⋊33C2, C2.29(D5×SD16), C10.58C22≀C2, (C10×SD16)⋊21C2, Q8⋊Dic5⋊28C2, (C2×Q8).116D10, (C2×Dic5).80D4, C10.46(C2×SD16), C22.267(D4×D5), C20.44D4⋊35C2, (C2×C40).294C22, (C2×C20).447C23, (C22×D5).129D4, (D4×C10).96C22, (Q8×C10).76C22, C2.26(C23⋊D10), C2.29(SD16⋊D5), C10.49(C8.C22), C4⋊Dic5.174C22, (C2×Dic10).131C22, (C2×Q8×D5)⋊2C2, C4.43(C2×C5⋊D4), (C2×D4.D5)⋊20C2, (C2×C4×D5).52C22, (C2×C10).359(C2×D4), (C2×C4).536(C22×D5), (C2×C5⋊2C8).157C22, SmallGroup(320,797)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D10⋊8SD16
G = < a,b,c,d | a10=b2=c8=d2=1, bab=a-1, ac=ca, ad=da, cbc-1=dbd=a5b, dcd=c3 >
Subgroups: 654 in 158 conjugacy classes, 45 normal (37 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, Q8, Q8, C23, D5, C10, C10, C22⋊C4, C4⋊C4, C2×C8, C2×C8, SD16, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, Dic5, C20, C20, D10, D10, C2×C10, C2×C10, C22⋊C8, Q8⋊C4, C4⋊D4, C2×SD16, C2×SD16, C22×Q8, C5⋊2C8, C40, Dic10, Dic10, C4×D5, C2×Dic5, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C5×D4, C5×Q8, C5×Q8, C22×D5, C22×C10, Q8⋊D4, C2×C5⋊2C8, C4⋊Dic5, D4.D5, C23.D5, C2×C40, C5×SD16, C2×Dic10, C2×Dic10, C2×C4×D5, C2×C4×D5, Q8×D5, C2×C5⋊D4, D4×C10, Q8×C10, C20.44D4, D10⋊1C8, Q8⋊Dic5, C2×D4.D5, C20⋊2D4, C10×SD16, C2×Q8×D5, D10⋊8SD16
Quotients: C1, C2, C22, D4, C23, D5, SD16, C2×D4, D10, C22≀C2, C2×SD16, C8.C22, C5⋊D4, C22×D5, Q8⋊D4, D4×D5, C2×C5⋊D4, D5×SD16, SD16⋊D5, C23⋊D10, D10⋊8SD16
►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 32)(2 31)(3 40)(4 39)(5 38)(6 37)(7 36)(8 35)(9 34)(10 33)(11 48)(12 47)(13 46)(14 45)(15 44)(16 43)(17 42)(18 41)(19 50)(20 49)(21 145)(22 144)(23 143)(24 142)(25 141)(26 150)(27 149)(28 148)(29 147)(30 146)(51 74)(52 73)(53 72)(54 71)(55 80)(56 79)(57 78)(58 77)(59 76)(60 75)(61 89)(62 88)(63 87)(64 86)(65 85)(66 84)(67 83)(68 82)(69 81)(70 90)(91 119)(92 118)(93 117)(94 116)(95 115)(96 114)(97 113)(98 112)(99 111)(100 120)(101 124)(102 123)(103 122)(104 121)(105 130)(106 129)(107 128)(108 127)(109 126)(110 125)(131 154)(132 153)(133 152)(134 151)(135 160)(136 159)(137 158)(138 157)(139 156)(140 155)
(1 26 50 153 33 146 20 138)(2 27 41 154 34 147 11 139)(3 28 42 155 35 148 12 140)(4 29 43 156 36 149 13 131)(5 30 44 157 37 150 14 132)(6 21 45 158 38 141 15 133)(7 22 46 159 39 142 16 134)(8 23 47 160 40 143 17 135)(9 24 48 151 31 144 18 136)(10 25 49 152 32 145 19 137)(51 109 69 116 76 123 83 91)(52 110 70 117 77 124 84 92)(53 101 61 118 78 125 85 93)(54 102 62 119 79 126 86 94)(55 103 63 120 80 127 87 95)(56 104 64 111 71 128 88 96)(57 105 65 112 72 129 89 97)(58 106 66 113 73 130 90 98)(59 107 67 114 74 121 81 99)(60 108 68 115 75 122 82 100)
(1 58)(2 59)(3 60)(4 51)(5 52)(6 53)(7 54)(8 55)(9 56)(10 57)(11 67)(12 68)(13 69)(14 70)(15 61)(16 62)(17 63)(18 64)(19 65)(20 66)(21 118)(22 119)(23 120)(24 111)(25 112)(26 113)(27 114)(28 115)(29 116)(30 117)(31 71)(32 72)(33 73)(34 74)(35 75)(36 76)(37 77)(38 78)(39 79)(40 80)(41 81)(42 82)(43 83)(44 84)(45 85)(46 86)(47 87)(48 88)(49 89)(50 90)(91 149)(92 150)(93 141)(94 142)(95 143)(96 144)(97 145)(98 146)(99 147)(100 148)(101 158)(102 159)(103 160)(104 151)(105 152)(106 153)(107 154)(108 155)(109 156)(110 157)(121 139)(122 140)(123 131)(124 132)(125 133)(126 134)(127 135)(128 136)(129 137)(130 138)
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), (1,32)(2,31)(3,40)(4,39)(5,38)(6,37)(7,36)(8,35)(9,34)(10,33)(11,48)(12,47)(13,46)(14,45)(15,44)(16,43)(17,42)(18,41)(19,50)(20,49)(21,145)(22,144)(23,143)(24,142)(25,141)(26,150)(27,149)(28,148)(29,147)(30,146)(51,74)(52,73)(53,72)(54,71)(55,80)(56,79)(57,78)(58,77)(59,76)(60,75)(61,89)(62,88)(63,87)(64,86)(65,85)(66,84)(67,83)(68,82)(69,81)(70,90)(91,119)(92,118)(93,117)(94,116)(95,115)(96,114)(97,113)(98,112)(99,111)(100,120)(101,124)(102,123)(103,122)(104,121)(105,130)(106,129)(107,128)(108,127)(109,126)(110,125)(131,154)(132,153)(133,152)(134,151)(135,160)(136,159)(137,158)(138,157)(139,156)(140,155), (1,26,50,153,33,146,20,138)(2,27,41,154,34,147,11,139)(3,28,42,155,35,148,12,140)(4,29,43,156,36,149,13,131)(5,30,44,157,37,150,14,132)(6,21,45,158,38,141,15,133)(7,22,46,159,39,142,16,134)(8,23,47,160,40,143,17,135)(9,24,48,151,31,144,18,136)(10,25,49,152,32,145,19,137)(51,109,69,116,76,123,83,91)(52,110,70,117,77,124,84,92)(53,101,61,118,78,125,85,93)(54,102,62,119,79,126,86,94)(55,103,63,120,80,127,87,95)(56,104,64,111,71,128,88,96)(57,105,65,112,72,129,89,97)(58,106,66,113,73,130,90,98)(59,107,67,114,74,121,81,99)(60,108,68,115,75,122,82,100), (1,58)(2,59)(3,60)(4,51)(5,52)(6,53)(7,54)(8,55)(9,56)(10,57)(11,67)(12,68)(13,69)(14,70)(15,61)(16,62)(17,63)(18,64)(19,65)(20,66)(21,118)(22,119)(23,120)(24,111)(25,112)(26,113)(27,114)(28,115)(29,116)(30,117)(31,71)(32,72)(33,73)(34,74)(35,75)(36,76)(37,77)(38,78)(39,79)(40,80)(41,81)(42,82)(43,83)(44,84)(45,85)(46,86)(47,87)(48,88)(49,89)(50,90)(91,149)(92,150)(93,141)(94,142)(95,143)(96,144)(97,145)(98,146)(99,147)(100,148)(101,158)(102,159)(103,160)(104,151)(105,152)(106,153)(107,154)(108,155)(109,156)(110,157)(121,139)(122,140)(123,131)(124,132)(125,133)(126,134)(127,135)(128,136)(129,137)(130,138)>;
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), (1,32)(2,31)(3,40)(4,39)(5,38)(6,37)(7,36)(8,35)(9,34)(10,33)(11,48)(12,47)(13,46)(14,45)(15,44)(16,43)(17,42)(18,41)(19,50)(20,49)(21,145)(22,144)(23,143)(24,142)(25,141)(26,150)(27,149)(28,148)(29,147)(30,146)(51,74)(52,73)(53,72)(54,71)(55,80)(56,79)(57,78)(58,77)(59,76)(60,75)(61,89)(62,88)(63,87)(64,86)(65,85)(66,84)(67,83)(68,82)(69,81)(70,90)(91,119)(92,118)(93,117)(94,116)(95,115)(96,114)(97,113)(98,112)(99,111)(100,120)(101,124)(102,123)(103,122)(104,121)(105,130)(106,129)(107,128)(108,127)(109,126)(110,125)(131,154)(132,153)(133,152)(134,151)(135,160)(136,159)(137,158)(138,157)(139,156)(140,155), (1,26,50,153,33,146,20,138)(2,27,41,154,34,147,11,139)(3,28,42,155,35,148,12,140)(4,29,43,156,36,149,13,131)(5,30,44,157,37,150,14,132)(6,21,45,158,38,141,15,133)(7,22,46,159,39,142,16,134)(8,23,47,160,40,143,17,135)(9,24,48,151,31,144,18,136)(10,25,49,152,32,145,19,137)(51,109,69,116,76,123,83,91)(52,110,70,117,77,124,84,92)(53,101,61,118,78,125,85,93)(54,102,62,119,79,126,86,94)(55,103,63,120,80,127,87,95)(56,104,64,111,71,128,88,96)(57,105,65,112,72,129,89,97)(58,106,66,113,73,130,90,98)(59,107,67,114,74,121,81,99)(60,108,68,115,75,122,82,100), (1,58)(2,59)(3,60)(4,51)(5,52)(6,53)(7,54)(8,55)(9,56)(10,57)(11,67)(12,68)(13,69)(14,70)(15,61)(16,62)(17,63)(18,64)(19,65)(20,66)(21,118)(22,119)(23,120)(24,111)(25,112)(26,113)(27,114)(28,115)(29,116)(30,117)(31,71)(32,72)(33,73)(34,74)(35,75)(36,76)(37,77)(38,78)(39,79)(40,80)(41,81)(42,82)(43,83)(44,84)(45,85)(46,86)(47,87)(48,88)(49,89)(50,90)(91,149)(92,150)(93,141)(94,142)(95,143)(96,144)(97,145)(98,146)(99,147)(100,148)(101,158)(102,159)(103,160)(104,151)(105,152)(106,153)(107,154)(108,155)(109,156)(110,157)(121,139)(122,140)(123,131)(124,132)(125,133)(126,134)(127,135)(128,136)(129,137)(130,138) );
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)], [(1,32),(2,31),(3,40),(4,39),(5,38),(6,37),(7,36),(8,35),(9,34),(10,33),(11,48),(12,47),(13,46),(14,45),(15,44),(16,43),(17,42),(18,41),(19,50),(20,49),(21,145),(22,144),(23,143),(24,142),(25,141),(26,150),(27,149),(28,148),(29,147),(30,146),(51,74),(52,73),(53,72),(54,71),(55,80),(56,79),(57,78),(58,77),(59,76),(60,75),(61,89),(62,88),(63,87),(64,86),(65,85),(66,84),(67,83),(68,82),(69,81),(70,90),(91,119),(92,118),(93,117),(94,116),(95,115),(96,114),(97,113),(98,112),(99,111),(100,120),(101,124),(102,123),(103,122),(104,121),(105,130),(106,129),(107,128),(108,127),(109,126),(110,125),(131,154),(132,153),(133,152),(134,151),(135,160),(136,159),(137,158),(138,157),(139,156),(140,155)], [(1,26,50,153,33,146,20,138),(2,27,41,154,34,147,11,139),(3,28,42,155,35,148,12,140),(4,29,43,156,36,149,13,131),(5,30,44,157,37,150,14,132),(6,21,45,158,38,141,15,133),(7,22,46,159,39,142,16,134),(8,23,47,160,40,143,17,135),(9,24,48,151,31,144,18,136),(10,25,49,152,32,145,19,137),(51,109,69,116,76,123,83,91),(52,110,70,117,77,124,84,92),(53,101,61,118,78,125,85,93),(54,102,62,119,79,126,86,94),(55,103,63,120,80,127,87,95),(56,104,64,111,71,128,88,96),(57,105,65,112,72,129,89,97),(58,106,66,113,73,130,90,98),(59,107,67,114,74,121,81,99),(60,108,68,115,75,122,82,100)], [(1,58),(2,59),(3,60),(4,51),(5,52),(6,53),(7,54),(8,55),(9,56),(10,57),(11,67),(12,68),(13,69),(14,70),(15,61),(16,62),(17,63),(18,64),(19,65),(20,66),(21,118),(22,119),(23,120),(24,111),(25,112),(26,113),(27,114),(28,115),(29,116),(30,117),(31,71),(32,72),(33,73),(34,74),(35,75),(36,76),(37,77),(38,78),(39,79),(40,80),(41,81),(42,82),(43,83),(44,84),(45,85),(46,86),(47,87),(48,88),(49,89),(50,90),(91,149),(92,150),(93,141),(94,142),(95,143),(96,144),(97,145),(98,146),(99,147),(100,148),(101,158),(102,159),(103,160),(104,151),(105,152),(106,153),(107,154),(108,155),(109,156),(110,157),(121,139),(122,140),(123,131),(124,132),(125,133),(126,134),(127,135),(128,136),(129,137),(130,138)]])
47 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 5A | 5B | 8A | 8B | 8C | 8D | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 20A | 20B | 20C | 20D | 20E | 20F | 20G | 20H | 40A | ··· | 40H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 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 | 2 | 2 | 4 | 4 | 20 | 20 | 20 | 40 | 2 | 2 | 4 | 4 | 20 | 20 | 2 | ··· | 2 | 8 | 8 | 8 | 8 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 4 | ··· | 4 |
47 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | + | - | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | D4 | D5 | SD16 | D10 | D10 | D10 | C5⋊D4 | C8.C22 | D4×D5 | D4×D5 | D5×SD16 | SD16⋊D5 |
| kernel | D10⋊8SD16 | C20.44D4 | D10⋊1C8 | Q8⋊Dic5 | C2×D4.D5 | C20⋊2D4 | C10×SD16 | C2×Q8×D5 | Dic10 | C2×Dic5 | C5×Q8 | C22×D5 | C2×SD16 | D10 | C2×C8 | C2×D4 | C2×Q8 | Q8 | C10 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 2 | 1 | 2 | 4 | 2 | 2 | 2 | 8 | 1 | 2 | 2 | 4 | 4 |
Matrix representation of D10⋊8SD16 ►in GL4(𝔽41) generated by
| 6 | 6 | 0 | 0 |
| 35 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 6 | 6 | 0 | 0 |
| 1 | 35 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 40 |
| 23 | 6 | 0 | 0 |
| 35 | 18 | 0 | 0 |
| 0 | 0 | 26 | 15 |
| 0 | 0 | 26 | 26 |
| 23 | 6 | 0 | 0 |
| 35 | 18 | 0 | 0 |
| 0 | 0 | 19 | 38 |
| 0 | 0 | 38 | 22 |
G:=sub<GL(4,GF(41))| [6,35,0,0,6,1,0,0,0,0,1,0,0,0,0,1],[6,1,0,0,6,35,0,0,0,0,40,0,0,0,0,40],[23,35,0,0,6,18,0,0,0,0,26,26,0,0,15,26],[23,35,0,0,6,18,0,0,0,0,19,38,0,0,38,22] >;
D10⋊8SD16 in GAP, Magma, Sage, TeX
D_{10}\rtimes_8{\rm SD}_{16} % in TeX
G:=Group("D10:8SD16"); // GroupNames label
G:=SmallGroup(320,797);
// by ID
G=gap.SmallGroup(320,797);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,232,254,219,851,438,102,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^10=b^2=c^8=d^2=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d=a^5*b,d*c*d=c^3>;
// generators/relations