metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D10.7Q16, Q8⋊C4⋊5D5, C40⋊5C4⋊10C2, (C2×C8).17D10, C2.10(D5×Q16), C4⋊C4.152D10, C10.D8⋊9C2, Q8⋊Dic5⋊8C2, (C2×Q8).17D10, C10.18(C2×Q16), D10⋊1C8.3C2, C4.56(C4○D20), (C2×C40).17C22, D10⋊3Q8.3C2, (C2×Dic5).40D4, C22.199(D4×D5), C20.162(C4○D4), C4.87(D4⋊2D5), C2.16(D40⋊C2), C10.62(C8⋊C22), (C2×C20).249C23, (C22×D5).116D4, C5⋊2(C23.48D4), C4⋊Dic5.95C22, (Q8×C10).32C22, C2.17(D10.12D4), C10.25(C22.D4), (D5×C4⋊C4).3C2, (C5×Q8⋊C4)⋊5C2, (C2×C4×D5).26C22, (C2×C10).262(C2×D4), (C5×C4⋊C4).50C22, (C2×C5⋊2C8).40C22, (C2×C4).356(C22×D5), SmallGroup(320,436)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D10.7Q16
G = < a,b,c,d | a10=b2=c8=1, d2=c4, bab=a-1, ac=ca, ad=da, cbc-1=dbd-1=a5b, dcd-1=a5c-1 >
Subgroups: 414 in 104 conjugacy classes, 39 normal (37 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, Q8, C23, D5, C10, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, C22×C4, C2×Q8, Dic5, C20, C20, D10, D10, C2×C10, C22⋊C8, Q8⋊C4, Q8⋊C4, C2.D8, C2×C4⋊C4, C22⋊Q8, C5⋊2C8, C40, C4×D5, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C5×Q8, C22×D5, C23.48D4, C2×C5⋊2C8, C10.D4, C4⋊Dic5, D10⋊C4, C5×C4⋊C4, C2×C40, C2×C4×D5, C2×C4×D5, Q8×C10, C10.D8, C40⋊5C4, D10⋊1C8, Q8⋊Dic5, C5×Q8⋊C4, D5×C4⋊C4, D10⋊3Q8, D10.7Q16
Quotients: C1, C2, C22, D4, C23, D5, Q16, C2×D4, C4○D4, D10, C22.D4, C2×Q16, C8⋊C22, C22×D5, C23.48D4, C4○D20, D4×D5, D4⋊2D5, D10.12D4, D40⋊C2, D5×Q16, D10.7Q16
►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 40)(2 39)(3 38)(4 37)(5 36)(6 35)(7 34)(8 33)(9 32)(10 31)(11 43)(12 42)(13 41)(14 50)(15 49)(16 48)(17 47)(18 46)(19 45)(20 44)(21 134)(22 133)(23 132)(24 131)(25 140)(26 139)(27 138)(28 137)(29 136)(30 135)(51 88)(52 87)(53 86)(54 85)(55 84)(56 83)(57 82)(58 81)(59 90)(60 89)(61 74)(62 73)(63 72)(64 71)(65 80)(66 79)(67 78)(68 77)(69 76)(70 75)(91 123)(92 122)(93 121)(94 130)(95 129)(96 128)(97 127)(98 126)(99 125)(100 124)(101 119)(102 118)(103 117)(104 116)(105 115)(106 114)(107 113)(108 112)(109 111)(110 120)(141 154)(142 153)(143 152)(144 151)(145 160)(146 159)(147 158)(148 157)(149 156)(150 155)
(1 26 50 148 31 135 15 153)(2 27 41 149 32 136 16 154)(3 28 42 150 33 137 17 155)(4 29 43 141 34 138 18 156)(5 30 44 142 35 139 19 157)(6 21 45 143 36 140 20 158)(7 22 46 144 37 131 11 159)(8 23 47 145 38 132 12 160)(9 24 48 146 39 133 13 151)(10 25 49 147 40 134 14 152)(51 119 64 91 86 104 79 126)(52 120 65 92 87 105 80 127)(53 111 66 93 88 106 71 128)(54 112 67 94 89 107 72 129)(55 113 68 95 90 108 73 130)(56 114 69 96 81 109 74 121)(57 115 70 97 82 110 75 122)(58 116 61 98 83 101 76 123)(59 117 62 99 84 102 77 124)(60 118 63 100 85 103 78 125)
(1 68 31 73)(2 69 32 74)(3 70 33 75)(4 61 34 76)(5 62 35 77)(6 63 36 78)(7 64 37 79)(8 65 38 80)(9 66 39 71)(10 67 40 72)(11 86 46 51)(12 87 47 52)(13 88 48 53)(14 89 49 54)(15 90 50 55)(16 81 41 56)(17 82 42 57)(18 83 43 58)(19 84 44 59)(20 85 45 60)(21 113 140 108)(22 114 131 109)(23 115 132 110)(24 116 133 101)(25 117 134 102)(26 118 135 103)(27 119 136 104)(28 120 137 105)(29 111 138 106)(30 112 139 107)(91 149 126 154)(92 150 127 155)(93 141 128 156)(94 142 129 157)(95 143 130 158)(96 144 121 159)(97 145 122 160)(98 146 123 151)(99 147 124 152)(100 148 125 153)
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), (1,40)(2,39)(3,38)(4,37)(5,36)(6,35)(7,34)(8,33)(9,32)(10,31)(11,43)(12,42)(13,41)(14,50)(15,49)(16,48)(17,47)(18,46)(19,45)(20,44)(21,134)(22,133)(23,132)(24,131)(25,140)(26,139)(27,138)(28,137)(29,136)(30,135)(51,88)(52,87)(53,86)(54,85)(55,84)(56,83)(57,82)(58,81)(59,90)(60,89)(61,74)(62,73)(63,72)(64,71)(65,80)(66,79)(67,78)(68,77)(69,76)(70,75)(91,123)(92,122)(93,121)(94,130)(95,129)(96,128)(97,127)(98,126)(99,125)(100,124)(101,119)(102,118)(103,117)(104,116)(105,115)(106,114)(107,113)(108,112)(109,111)(110,120)(141,154)(142,153)(143,152)(144,151)(145,160)(146,159)(147,158)(148,157)(149,156)(150,155), (1,26,50,148,31,135,15,153)(2,27,41,149,32,136,16,154)(3,28,42,150,33,137,17,155)(4,29,43,141,34,138,18,156)(5,30,44,142,35,139,19,157)(6,21,45,143,36,140,20,158)(7,22,46,144,37,131,11,159)(8,23,47,145,38,132,12,160)(9,24,48,146,39,133,13,151)(10,25,49,147,40,134,14,152)(51,119,64,91,86,104,79,126)(52,120,65,92,87,105,80,127)(53,111,66,93,88,106,71,128)(54,112,67,94,89,107,72,129)(55,113,68,95,90,108,73,130)(56,114,69,96,81,109,74,121)(57,115,70,97,82,110,75,122)(58,116,61,98,83,101,76,123)(59,117,62,99,84,102,77,124)(60,118,63,100,85,103,78,125), (1,68,31,73)(2,69,32,74)(3,70,33,75)(4,61,34,76)(5,62,35,77)(6,63,36,78)(7,64,37,79)(8,65,38,80)(9,66,39,71)(10,67,40,72)(11,86,46,51)(12,87,47,52)(13,88,48,53)(14,89,49,54)(15,90,50,55)(16,81,41,56)(17,82,42,57)(18,83,43,58)(19,84,44,59)(20,85,45,60)(21,113,140,108)(22,114,131,109)(23,115,132,110)(24,116,133,101)(25,117,134,102)(26,118,135,103)(27,119,136,104)(28,120,137,105)(29,111,138,106)(30,112,139,107)(91,149,126,154)(92,150,127,155)(93,141,128,156)(94,142,129,157)(95,143,130,158)(96,144,121,159)(97,145,122,160)(98,146,123,151)(99,147,124,152)(100,148,125,153)>;
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), (1,40)(2,39)(3,38)(4,37)(5,36)(6,35)(7,34)(8,33)(9,32)(10,31)(11,43)(12,42)(13,41)(14,50)(15,49)(16,48)(17,47)(18,46)(19,45)(20,44)(21,134)(22,133)(23,132)(24,131)(25,140)(26,139)(27,138)(28,137)(29,136)(30,135)(51,88)(52,87)(53,86)(54,85)(55,84)(56,83)(57,82)(58,81)(59,90)(60,89)(61,74)(62,73)(63,72)(64,71)(65,80)(66,79)(67,78)(68,77)(69,76)(70,75)(91,123)(92,122)(93,121)(94,130)(95,129)(96,128)(97,127)(98,126)(99,125)(100,124)(101,119)(102,118)(103,117)(104,116)(105,115)(106,114)(107,113)(108,112)(109,111)(110,120)(141,154)(142,153)(143,152)(144,151)(145,160)(146,159)(147,158)(148,157)(149,156)(150,155), (1,26,50,148,31,135,15,153)(2,27,41,149,32,136,16,154)(3,28,42,150,33,137,17,155)(4,29,43,141,34,138,18,156)(5,30,44,142,35,139,19,157)(6,21,45,143,36,140,20,158)(7,22,46,144,37,131,11,159)(8,23,47,145,38,132,12,160)(9,24,48,146,39,133,13,151)(10,25,49,147,40,134,14,152)(51,119,64,91,86,104,79,126)(52,120,65,92,87,105,80,127)(53,111,66,93,88,106,71,128)(54,112,67,94,89,107,72,129)(55,113,68,95,90,108,73,130)(56,114,69,96,81,109,74,121)(57,115,70,97,82,110,75,122)(58,116,61,98,83,101,76,123)(59,117,62,99,84,102,77,124)(60,118,63,100,85,103,78,125), (1,68,31,73)(2,69,32,74)(3,70,33,75)(4,61,34,76)(5,62,35,77)(6,63,36,78)(7,64,37,79)(8,65,38,80)(9,66,39,71)(10,67,40,72)(11,86,46,51)(12,87,47,52)(13,88,48,53)(14,89,49,54)(15,90,50,55)(16,81,41,56)(17,82,42,57)(18,83,43,58)(19,84,44,59)(20,85,45,60)(21,113,140,108)(22,114,131,109)(23,115,132,110)(24,116,133,101)(25,117,134,102)(26,118,135,103)(27,119,136,104)(28,120,137,105)(29,111,138,106)(30,112,139,107)(91,149,126,154)(92,150,127,155)(93,141,128,156)(94,142,129,157)(95,143,130,158)(96,144,121,159)(97,145,122,160)(98,146,123,151)(99,147,124,152)(100,148,125,153) );
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)], [(1,40),(2,39),(3,38),(4,37),(5,36),(6,35),(7,34),(8,33),(9,32),(10,31),(11,43),(12,42),(13,41),(14,50),(15,49),(16,48),(17,47),(18,46),(19,45),(20,44),(21,134),(22,133),(23,132),(24,131),(25,140),(26,139),(27,138),(28,137),(29,136),(30,135),(51,88),(52,87),(53,86),(54,85),(55,84),(56,83),(57,82),(58,81),(59,90),(60,89),(61,74),(62,73),(63,72),(64,71),(65,80),(66,79),(67,78),(68,77),(69,76),(70,75),(91,123),(92,122),(93,121),(94,130),(95,129),(96,128),(97,127),(98,126),(99,125),(100,124),(101,119),(102,118),(103,117),(104,116),(105,115),(106,114),(107,113),(108,112),(109,111),(110,120),(141,154),(142,153),(143,152),(144,151),(145,160),(146,159),(147,158),(148,157),(149,156),(150,155)], [(1,26,50,148,31,135,15,153),(2,27,41,149,32,136,16,154),(3,28,42,150,33,137,17,155),(4,29,43,141,34,138,18,156),(5,30,44,142,35,139,19,157),(6,21,45,143,36,140,20,158),(7,22,46,144,37,131,11,159),(8,23,47,145,38,132,12,160),(9,24,48,146,39,133,13,151),(10,25,49,147,40,134,14,152),(51,119,64,91,86,104,79,126),(52,120,65,92,87,105,80,127),(53,111,66,93,88,106,71,128),(54,112,67,94,89,107,72,129),(55,113,68,95,90,108,73,130),(56,114,69,96,81,109,74,121),(57,115,70,97,82,110,75,122),(58,116,61,98,83,101,76,123),(59,117,62,99,84,102,77,124),(60,118,63,100,85,103,78,125)], [(1,68,31,73),(2,69,32,74),(3,70,33,75),(4,61,34,76),(5,62,35,77),(6,63,36,78),(7,64,37,79),(8,65,38,80),(9,66,39,71),(10,67,40,72),(11,86,46,51),(12,87,47,52),(13,88,48,53),(14,89,49,54),(15,90,50,55),(16,81,41,56),(17,82,42,57),(18,83,43,58),(19,84,44,59),(20,85,45,60),(21,113,140,108),(22,114,131,109),(23,115,132,110),(24,116,133,101),(25,117,134,102),(26,118,135,103),(27,119,136,104),(28,120,137,105),(29,111,138,106),(30,112,139,107),(91,149,126,154),(92,150,127,155),(93,141,128,156),(94,142,129,157),(95,143,130,158),(96,144,121,159),(97,145,122,160),(98,146,123,151),(99,147,124,152),(100,148,125,153)]])
47 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 5A | 5B | 8A | 8B | 8C | 8D | 10A | ··· | 10F | 20A | 20B | 20C | 20D | 20E | ··· | 20L | 40A | ··· | 40H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 20 | 20 | 20 | 20 | 20 | ··· | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 10 | 10 | 2 | 2 | 4 | 4 | 8 | 20 | 20 | 20 | 40 | 2 | 2 | 4 | 4 | 20 | 20 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 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 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | - | + | + | + | + | - | + | + | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D5 | C4○D4 | Q16 | D10 | D10 | D10 | C4○D20 | C8⋊C22 | D4⋊2D5 | D4×D5 | D40⋊C2 | D5×Q16 |
| kernel | D10.7Q16 | C10.D8 | C40⋊5C4 | D10⋊1C8 | Q8⋊Dic5 | C5×Q8⋊C4 | D5×C4⋊C4 | D10⋊3Q8 | C2×Dic5 | C22×D5 | Q8⋊C4 | C20 | D10 | C4⋊C4 | C2×C8 | C2×Q8 | C4 | C10 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 4 | 4 | 2 | 2 | 2 | 8 | 1 | 2 | 2 | 4 | 4 |
Matrix representation of D10.7Q16 ►in GL6(𝔽41)
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 35 | 0 | 0 |
| 0 | 0 | 6 | 35 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 13 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 6 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 12 | 5 | 0 | 0 | 0 | 0 |
| 37 | 29 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 29 |
| 0 | 0 | 0 | 0 | 12 | 12 |
| 15 | 37 | 0 | 0 | 0 | 0 |
| 15 | 26 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 21 | 38 |
| 0 | 0 | 0 | 0 | 38 | 20 |
G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,6,0,0,0,0,35,35,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[40,13,0,0,0,0,0,1,0,0,0,0,0,0,40,6,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[12,37,0,0,0,0,5,29,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,12,0,0,0,0,29,12],[15,15,0,0,0,0,37,26,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,21,38,0,0,0,0,38,20] >;
D10.7Q16 in GAP, Magma, Sage, TeX
D_{10}._7Q_{16} % in TeX
G:=Group("D10.7Q16"); // GroupNames label
G:=SmallGroup(320,436);
// by ID
G=gap.SmallGroup(320,436);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,64,926,219,184,851,438,102,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^10=b^2=c^8=1,d^2=c^4,b*a*b=a^-1,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d^-1=a^5*b,d*c*d^-1=a^5*c^-1>;
// generators/relations