metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: SD16:1Dic5, C40:19(C2xC4), C40:8C4:4C2, C8:3(C2xDic5), Q8:2(C2xDic5), (C5xSD16):5C4, (Q8xDic5):5C2, C40:5C4:26C2, (C2xC8).88D10, C10.126(C4xD4), (C2xSD16).1D5, (D4xDic5).9C2, D4.2(C2xDic5), C2.13(D4xDic5), (C2xD4).144D10, C20.98(C4oD4), Q8:Dic5:26C2, C5:8(SD16:C4), C2.7(D40:C2), (C2xQ8).114D10, (C10xSD16).1C2, C22.117(D4xD5), C4.31(D4:2D5), C4.4(C22xDic5), C10.76(C8:C22), C20.133(C22xC4), (C2xC40).113C22, (C2xC20).441C23, (C2xDic5).239D4, D4:Dic5.15C2, C2.7(SD16:D5), (D4xC10).90C22, (Q8xC10).71C22, C10.46(C8.C22), C4:Dic5.171C22, (C4xDic5).54C22, (C5xQ8):16(C2xC4), (C5xD4).23(C2xC4), (C2xC10).353(C2xD4), (C2xC4).530(C22xD5), (C2xC5:2C8).153C22, SmallGroup(320,791)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for SD16:Dic5
G = < a,b,c,d | a8=b2=c10=1, d2=c5, bab=a3, ac=ca, dad-1=a5, bc=cb, bd=db, dcd-1=c-1 >
Subgroups: 390 in 120 conjugacy classes, 57 normal (37 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C8, C2xC4, C2xC4, D4, D4, Q8, Q8, C23, C10, C10, C42, C22:C4, C4:C4, C2xC8, C2xC8, SD16, C22xC4, C2xD4, C2xQ8, Dic5, C20, C20, C2xC10, C2xC10, C8:C4, D4:C4, Q8:C4, C2.D8, C4xD4, C4xQ8, C2xSD16, C5:2C8, C40, C2xDic5, C2xDic5, C2xC20, C2xC20, C5xD4, C5xD4, C5xQ8, C5xQ8, C22xC10, SD16:C4, C2xC5:2C8, C4xDic5, C4xDic5, C4:Dic5, C4:Dic5, C23.D5, C2xC40, C5xSD16, C22xDic5, D4xC10, Q8xC10, C40:8C4, C40:5C4, D4:Dic5, Q8:Dic5, D4xDic5, Q8xDic5, C10xSD16, SD16:Dic5
Quotients: C1, C2, C4, C22, C2xC4, D4, C23, D5, C22xC4, C2xD4, C4oD4, Dic5, D10, C4xD4, C8:C22, C8.C22, C2xDic5, C22xD5, SD16:C4, D4xD5, D4:2D5, C22xDic5, D40:C2, SD16:D5, D4xDic5, SD16:Dic5
►On 160 points
Generators in S160
(1 85 18 79 39 64 48 58)(2 86 19 80 40 65 49 59)(3 87 20 71 31 66 50 60)(4 88 11 72 32 67 41 51)(5 89 12 73 33 68 42 52)(6 90 13 74 34 69 43 53)(7 81 14 75 35 70 44 54)(8 82 15 76 36 61 45 55)(9 83 16 77 37 62 46 56)(10 84 17 78 38 63 47 57)(21 122 156 92 141 101 132 116)(22 123 157 93 142 102 133 117)(23 124 158 94 143 103 134 118)(24 125 159 95 144 104 135 119)(25 126 160 96 145 105 136 120)(26 127 151 97 146 106 137 111)(27 128 152 98 147 107 138 112)(28 129 153 99 148 108 139 113)(29 130 154 100 149 109 140 114)(30 121 155 91 150 110 131 115)
(11 41)(12 42)(13 43)(14 44)(15 45)(16 46)(17 47)(18 48)(19 49)(20 50)(21 156)(22 157)(23 158)(24 159)(25 160)(26 151)(27 152)(28 153)(29 154)(30 155)(51 67)(52 68)(53 69)(54 70)(55 61)(56 62)(57 63)(58 64)(59 65)(60 66)(71 87)(72 88)(73 89)(74 90)(75 81)(76 82)(77 83)(78 84)(79 85)(80 86)(101 122)(102 123)(103 124)(104 125)(105 126)(106 127)(107 128)(108 129)(109 130)(110 121)(131 150)(132 141)(133 142)(134 143)(135 144)(136 145)(137 146)(138 147)(139 148)(140 149)
(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 92 6 97)(2 91 7 96)(3 100 8 95)(4 99 9 94)(5 98 10 93)(11 108 16 103)(12 107 17 102)(13 106 18 101)(14 105 19 110)(15 104 20 109)(21 90 26 85)(22 89 27 84)(23 88 28 83)(24 87 29 82)(25 86 30 81)(31 114 36 119)(32 113 37 118)(33 112 38 117)(34 111 39 116)(35 120 40 115)(41 129 46 124)(42 128 47 123)(43 127 48 122)(44 126 49 121)(45 125 50 130)(51 139 56 134)(52 138 57 133)(53 137 58 132)(54 136 59 131)(55 135 60 140)(61 144 66 149)(62 143 67 148)(63 142 68 147)(64 141 69 146)(65 150 70 145)(71 154 76 159)(72 153 77 158)(73 152 78 157)(74 151 79 156)(75 160 80 155)
G:=sub<Sym(160)| (1,85,18,79,39,64,48,58)(2,86,19,80,40,65,49,59)(3,87,20,71,31,66,50,60)(4,88,11,72,32,67,41,51)(5,89,12,73,33,68,42,52)(6,90,13,74,34,69,43,53)(7,81,14,75,35,70,44,54)(8,82,15,76,36,61,45,55)(9,83,16,77,37,62,46,56)(10,84,17,78,38,63,47,57)(21,122,156,92,141,101,132,116)(22,123,157,93,142,102,133,117)(23,124,158,94,143,103,134,118)(24,125,159,95,144,104,135,119)(25,126,160,96,145,105,136,120)(26,127,151,97,146,106,137,111)(27,128,152,98,147,107,138,112)(28,129,153,99,148,108,139,113)(29,130,154,100,149,109,140,114)(30,121,155,91,150,110,131,115), (11,41)(12,42)(13,43)(14,44)(15,45)(16,46)(17,47)(18,48)(19,49)(20,50)(21,156)(22,157)(23,158)(24,159)(25,160)(26,151)(27,152)(28,153)(29,154)(30,155)(51,67)(52,68)(53,69)(54,70)(55,61)(56,62)(57,63)(58,64)(59,65)(60,66)(71,87)(72,88)(73,89)(74,90)(75,81)(76,82)(77,83)(78,84)(79,85)(80,86)(101,122)(102,123)(103,124)(104,125)(105,126)(106,127)(107,128)(108,129)(109,130)(110,121)(131,150)(132,141)(133,142)(134,143)(135,144)(136,145)(137,146)(138,147)(139,148)(140,149), (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), (1,92,6,97)(2,91,7,96)(3,100,8,95)(4,99,9,94)(5,98,10,93)(11,108,16,103)(12,107,17,102)(13,106,18,101)(14,105,19,110)(15,104,20,109)(21,90,26,85)(22,89,27,84)(23,88,28,83)(24,87,29,82)(25,86,30,81)(31,114,36,119)(32,113,37,118)(33,112,38,117)(34,111,39,116)(35,120,40,115)(41,129,46,124)(42,128,47,123)(43,127,48,122)(44,126,49,121)(45,125,50,130)(51,139,56,134)(52,138,57,133)(53,137,58,132)(54,136,59,131)(55,135,60,140)(61,144,66,149)(62,143,67,148)(63,142,68,147)(64,141,69,146)(65,150,70,145)(71,154,76,159)(72,153,77,158)(73,152,78,157)(74,151,79,156)(75,160,80,155)>;
G:=Group( (1,85,18,79,39,64,48,58)(2,86,19,80,40,65,49,59)(3,87,20,71,31,66,50,60)(4,88,11,72,32,67,41,51)(5,89,12,73,33,68,42,52)(6,90,13,74,34,69,43,53)(7,81,14,75,35,70,44,54)(8,82,15,76,36,61,45,55)(9,83,16,77,37,62,46,56)(10,84,17,78,38,63,47,57)(21,122,156,92,141,101,132,116)(22,123,157,93,142,102,133,117)(23,124,158,94,143,103,134,118)(24,125,159,95,144,104,135,119)(25,126,160,96,145,105,136,120)(26,127,151,97,146,106,137,111)(27,128,152,98,147,107,138,112)(28,129,153,99,148,108,139,113)(29,130,154,100,149,109,140,114)(30,121,155,91,150,110,131,115), (11,41)(12,42)(13,43)(14,44)(15,45)(16,46)(17,47)(18,48)(19,49)(20,50)(21,156)(22,157)(23,158)(24,159)(25,160)(26,151)(27,152)(28,153)(29,154)(30,155)(51,67)(52,68)(53,69)(54,70)(55,61)(56,62)(57,63)(58,64)(59,65)(60,66)(71,87)(72,88)(73,89)(74,90)(75,81)(76,82)(77,83)(78,84)(79,85)(80,86)(101,122)(102,123)(103,124)(104,125)(105,126)(106,127)(107,128)(108,129)(109,130)(110,121)(131,150)(132,141)(133,142)(134,143)(135,144)(136,145)(137,146)(138,147)(139,148)(140,149), (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), (1,92,6,97)(2,91,7,96)(3,100,8,95)(4,99,9,94)(5,98,10,93)(11,108,16,103)(12,107,17,102)(13,106,18,101)(14,105,19,110)(15,104,20,109)(21,90,26,85)(22,89,27,84)(23,88,28,83)(24,87,29,82)(25,86,30,81)(31,114,36,119)(32,113,37,118)(33,112,38,117)(34,111,39,116)(35,120,40,115)(41,129,46,124)(42,128,47,123)(43,127,48,122)(44,126,49,121)(45,125,50,130)(51,139,56,134)(52,138,57,133)(53,137,58,132)(54,136,59,131)(55,135,60,140)(61,144,66,149)(62,143,67,148)(63,142,68,147)(64,141,69,146)(65,150,70,145)(71,154,76,159)(72,153,77,158)(73,152,78,157)(74,151,79,156)(75,160,80,155) );
G=PermutationGroup([[(1,85,18,79,39,64,48,58),(2,86,19,80,40,65,49,59),(3,87,20,71,31,66,50,60),(4,88,11,72,32,67,41,51),(5,89,12,73,33,68,42,52),(6,90,13,74,34,69,43,53),(7,81,14,75,35,70,44,54),(8,82,15,76,36,61,45,55),(9,83,16,77,37,62,46,56),(10,84,17,78,38,63,47,57),(21,122,156,92,141,101,132,116),(22,123,157,93,142,102,133,117),(23,124,158,94,143,103,134,118),(24,125,159,95,144,104,135,119),(25,126,160,96,145,105,136,120),(26,127,151,97,146,106,137,111),(27,128,152,98,147,107,138,112),(28,129,153,99,148,108,139,113),(29,130,154,100,149,109,140,114),(30,121,155,91,150,110,131,115)], [(11,41),(12,42),(13,43),(14,44),(15,45),(16,46),(17,47),(18,48),(19,49),(20,50),(21,156),(22,157),(23,158),(24,159),(25,160),(26,151),(27,152),(28,153),(29,154),(30,155),(51,67),(52,68),(53,69),(54,70),(55,61),(56,62),(57,63),(58,64),(59,65),(60,66),(71,87),(72,88),(73,89),(74,90),(75,81),(76,82),(77,83),(78,84),(79,85),(80,86),(101,122),(102,123),(103,124),(104,125),(105,126),(106,127),(107,128),(108,129),(109,130),(110,121),(131,150),(132,141),(133,142),(134,143),(135,144),(136,145),(137,146),(138,147),(139,148),(140,149)], [(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)], [(1,92,6,97),(2,91,7,96),(3,100,8,95),(4,99,9,94),(5,98,10,93),(11,108,16,103),(12,107,17,102),(13,106,18,101),(14,105,19,110),(15,104,20,109),(21,90,26,85),(22,89,27,84),(23,88,28,83),(24,87,29,82),(25,86,30,81),(31,114,36,119),(32,113,37,118),(33,112,38,117),(34,111,39,116),(35,120,40,115),(41,129,46,124),(42,128,47,123),(43,127,48,122),(44,126,49,121),(45,125,50,130),(51,139,56,134),(52,138,57,133),(53,137,58,132),(54,136,59,131),(55,135,60,140),(61,144,66,149),(62,143,67,148),(63,142,68,147),(64,141,69,146),(65,150,70,145),(71,154,76,159),(72,153,77,158),(73,152,78,157),(74,151,79,156),(75,160,80,155)]])
50 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 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 | 4 | 4 | 4 | 4 | 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 | 4 | 4 | 2 | 2 | 4 | 4 | 10 | 10 | 10 | 10 | 20 | 20 | 20 | 20 | 2 | 2 | 4 | 4 | 20 | 20 | 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 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | - | + | + | + | - | - | + | + | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | D4 | D5 | C4oD4 | D10 | Dic5 | D10 | D10 | C8:C22 | C8.C22 | D4:2D5 | D4xD5 | D40:C2 | SD16:D5 |
| kernel | SD16:Dic5 | C40:8C4 | C40:5C4 | D4:Dic5 | Q8:Dic5 | D4xDic5 | Q8xDic5 | C10xSD16 | C5xSD16 | C2xDic5 | C2xSD16 | C20 | C2xC8 | SD16 | C2xD4 | C2xQ8 | C10 | C10 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 8 | 2 | 2 | 2 | 2 | 8 | 2 | 2 | 1 | 1 | 2 | 2 | 4 | 4 |
Matrix representation of SD16:Dic5 ►in GL6(F41)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 11 | 6 | 6 |
| 0 | 0 | 3 | 34 | 0 | 11 |
| 0 | 0 | 7 | 26 | 0 | 0 |
| 0 | 0 | 1 | 26 | 6 | 6 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 38 | 23 | 40 | 0 |
| 0 | 0 | 38 | 23 | 0 | 40 |
| 35 | 40 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 40 | 0 | 0 |
| 0 | 0 | 36 | 6 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 40 |
| 0 | 0 | 35 | 40 | 1 | 7 |
| 14 | 39 | 0 | 0 | 0 | 0 |
| 37 | 27 | 0 | 0 | 0 | 0 |
| 0 | 0 | 13 | 19 | 0 | 0 |
| 0 | 0 | 4 | 28 | 0 | 0 |
| 0 | 0 | 22 | 0 | 19 | 19 |
| 0 | 0 | 32 | 19 | 9 | 22 |
G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,3,7,1,0,0,11,34,26,26,0,0,6,0,0,6,0,0,6,11,0,6],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,38,38,0,0,0,1,23,23,0,0,0,0,40,0,0,0,0,0,0,40],[35,1,0,0,0,0,40,0,0,0,0,0,0,0,1,36,1,35,0,0,40,6,0,40,0,0,0,0,0,1,0,0,0,0,40,7],[14,37,0,0,0,0,39,27,0,0,0,0,0,0,13,4,22,32,0,0,19,28,0,19,0,0,0,0,19,9,0,0,0,0,19,22] >;
SD16:Dic5 in GAP, Magma, Sage, TeX
{\rm SD}_{16}\rtimes {\rm Dic}_5 % in TeX
G:=Group("SD16:Dic5"); // GroupNames label
G:=SmallGroup(320,791);
// by ID
G=gap.SmallGroup(320,791);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,758,219,184,851,438,102,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^8=b^2=c^10=1,d^2=c^5,b*a*b=a^3,a*c=c*a,d*a*d^-1=a^5,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations