metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D20.11D4, Dic5:2SD16, C4.96(D4xD5), C4:C4.156D10, Q8:C4:14D5, C4.7(C4oD20), C20.128(C2xD4), (C2xC8).126D10, C5:1(D4.D4), (C2xQ8).23D10, C2.19(D5xSD16), Dic5:Q8:2C2, D20:8C4.3C2, C20.23(C4oD4), C10.Q16:13C2, C20.8Q8:14C2, (C2xDic5).43D4, C10.33(C2xSD16), C22.207(D4xD5), C10.27(C4:D4), (C2xC40).137C22, (C2xC20).258C23, (C2xD20).72C22, (Q8xC10).41C22, C2.30(D10:D4), C2.19(Q16:D5), C10.66(C8.C22), (C4xDic5).31C22, (C2xDic10).79C22, (C2xQ8:D5).3C2, (C2xC40:C2).4C2, (C5xQ8:C4):14C2, (C2xC10).271(C2xD4), (C5xC4:C4).59C22, (C2xC5:2C8).48C22, (C2xC4).365(C22xD5), SmallGroup(320,445)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic5:SD16
G = < a,b,c,d | a10=c8=d2=1, b2=a5, bab-1=cac-1=dad=a-1, cbc-1=a5b, bd=db, dcd=c3 >
Subgroups: 534 in 120 conjugacy classes, 41 normal (37 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2xC4, C2xC4, D4, Q8, C23, D5, C10, C42, C22:C4, C4:C4, C4:C4, C2xC8, C2xC8, SD16, C22xC4, C2xD4, C2xQ8, C2xQ8, Dic5, Dic5, C20, C20, D10, C2xC10, Q8:C4, Q8:C4, C4:C8, C4xD4, C4:Q8, C2xSD16, C5:2C8, C40, Dic10, C4xD5, D20, D20, C2xDic5, C2xDic5, C2xC20, C2xC20, C5xQ8, C22xD5, D4.D4, C40:C2, C2xC5:2C8, C4xDic5, C10.D4, D10:C4, Q8:D5, C5xC4:C4, C2xC40, C2xDic10, C2xC4xD5, C2xD20, Q8xC10, C10.Q16, C20.8Q8, C5xQ8:C4, D20:8C4, C2xC40:C2, C2xQ8:D5, Dic5:Q8, Dic5:SD16
Quotients: C1, C2, C22, D4, C23, D5, SD16, C2xD4, C4oD4, D10, C4:D4, C2xSD16, C8.C22, C22xD5, D4.D4, C4oD20, D4xD5, D10:D4, D5xSD16, Q16:D5, Dic5:SD16
►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 93 6 98)(2 92 7 97)(3 91 8 96)(4 100 9 95)(5 99 10 94)(11 124 16 129)(12 123 17 128)(13 122 18 127)(14 121 19 126)(15 130 20 125)(21 77 26 72)(22 76 27 71)(23 75 28 80)(24 74 29 79)(25 73 30 78)(31 112 36 117)(32 111 37 116)(33 120 38 115)(34 119 39 114)(35 118 40 113)(41 106 46 101)(42 105 47 110)(43 104 48 109)(44 103 49 108)(45 102 50 107)(51 141 56 146)(52 150 57 145)(53 149 58 144)(54 148 59 143)(55 147 60 142)(61 131 66 136)(62 140 67 135)(63 139 68 134)(64 138 69 133)(65 137 70 132)(81 156 86 151)(82 155 87 160)(83 154 88 159)(84 153 89 158)(85 152 90 157)
(1 83 17 79 39 69 43 58)(2 82 18 78 40 68 44 57)(3 81 19 77 31 67 45 56)(4 90 20 76 32 66 46 55)(5 89 11 75 33 65 47 54)(6 88 12 74 34 64 48 53)(7 87 13 73 35 63 49 52)(8 86 14 72 36 62 50 51)(9 85 15 71 37 61 41 60)(10 84 16 80 38 70 42 59)(21 112 140 102 141 91 151 126)(22 111 131 101 142 100 152 125)(23 120 132 110 143 99 153 124)(24 119 133 109 144 98 154 123)(25 118 134 108 145 97 155 122)(26 117 135 107 146 96 156 121)(27 116 136 106 147 95 157 130)(28 115 137 105 148 94 158 129)(29 114 138 104 149 93 159 128)(30 113 139 103 150 92 160 127)
(2 10)(3 9)(4 8)(5 7)(11 49)(12 48)(13 47)(14 46)(15 45)(16 44)(17 43)(18 42)(19 41)(20 50)(21 157)(22 156)(23 155)(24 154)(25 153)(26 152)(27 151)(28 160)(29 159)(30 158)(31 37)(32 36)(33 35)(38 40)(51 66)(52 65)(53 64)(54 63)(55 62)(56 61)(57 70)(58 69)(59 68)(60 67)(71 81)(72 90)(73 89)(74 88)(75 87)(76 86)(77 85)(78 84)(79 83)(80 82)(91 95)(92 94)(96 100)(97 99)(101 121)(102 130)(103 129)(104 128)(105 127)(106 126)(107 125)(108 124)(109 123)(110 122)(111 117)(112 116)(113 115)(118 120)(131 146)(132 145)(133 144)(134 143)(135 142)(136 141)(137 150)(138 149)(139 148)(140 147)
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), (1,93,6,98)(2,92,7,97)(3,91,8,96)(4,100,9,95)(5,99,10,94)(11,124,16,129)(12,123,17,128)(13,122,18,127)(14,121,19,126)(15,130,20,125)(21,77,26,72)(22,76,27,71)(23,75,28,80)(24,74,29,79)(25,73,30,78)(31,112,36,117)(32,111,37,116)(33,120,38,115)(34,119,39,114)(35,118,40,113)(41,106,46,101)(42,105,47,110)(43,104,48,109)(44,103,49,108)(45,102,50,107)(51,141,56,146)(52,150,57,145)(53,149,58,144)(54,148,59,143)(55,147,60,142)(61,131,66,136)(62,140,67,135)(63,139,68,134)(64,138,69,133)(65,137,70,132)(81,156,86,151)(82,155,87,160)(83,154,88,159)(84,153,89,158)(85,152,90,157), (1,83,17,79,39,69,43,58)(2,82,18,78,40,68,44,57)(3,81,19,77,31,67,45,56)(4,90,20,76,32,66,46,55)(5,89,11,75,33,65,47,54)(6,88,12,74,34,64,48,53)(7,87,13,73,35,63,49,52)(8,86,14,72,36,62,50,51)(9,85,15,71,37,61,41,60)(10,84,16,80,38,70,42,59)(21,112,140,102,141,91,151,126)(22,111,131,101,142,100,152,125)(23,120,132,110,143,99,153,124)(24,119,133,109,144,98,154,123)(25,118,134,108,145,97,155,122)(26,117,135,107,146,96,156,121)(27,116,136,106,147,95,157,130)(28,115,137,105,148,94,158,129)(29,114,138,104,149,93,159,128)(30,113,139,103,150,92,160,127), (2,10)(3,9)(4,8)(5,7)(11,49)(12,48)(13,47)(14,46)(15,45)(16,44)(17,43)(18,42)(19,41)(20,50)(21,157)(22,156)(23,155)(24,154)(25,153)(26,152)(27,151)(28,160)(29,159)(30,158)(31,37)(32,36)(33,35)(38,40)(51,66)(52,65)(53,64)(54,63)(55,62)(56,61)(57,70)(58,69)(59,68)(60,67)(71,81)(72,90)(73,89)(74,88)(75,87)(76,86)(77,85)(78,84)(79,83)(80,82)(91,95)(92,94)(96,100)(97,99)(101,121)(102,130)(103,129)(104,128)(105,127)(106,126)(107,125)(108,124)(109,123)(110,122)(111,117)(112,116)(113,115)(118,120)(131,146)(132,145)(133,144)(134,143)(135,142)(136,141)(137,150)(138,149)(139,148)(140,147)>;
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), (1,93,6,98)(2,92,7,97)(3,91,8,96)(4,100,9,95)(5,99,10,94)(11,124,16,129)(12,123,17,128)(13,122,18,127)(14,121,19,126)(15,130,20,125)(21,77,26,72)(22,76,27,71)(23,75,28,80)(24,74,29,79)(25,73,30,78)(31,112,36,117)(32,111,37,116)(33,120,38,115)(34,119,39,114)(35,118,40,113)(41,106,46,101)(42,105,47,110)(43,104,48,109)(44,103,49,108)(45,102,50,107)(51,141,56,146)(52,150,57,145)(53,149,58,144)(54,148,59,143)(55,147,60,142)(61,131,66,136)(62,140,67,135)(63,139,68,134)(64,138,69,133)(65,137,70,132)(81,156,86,151)(82,155,87,160)(83,154,88,159)(84,153,89,158)(85,152,90,157), (1,83,17,79,39,69,43,58)(2,82,18,78,40,68,44,57)(3,81,19,77,31,67,45,56)(4,90,20,76,32,66,46,55)(5,89,11,75,33,65,47,54)(6,88,12,74,34,64,48,53)(7,87,13,73,35,63,49,52)(8,86,14,72,36,62,50,51)(9,85,15,71,37,61,41,60)(10,84,16,80,38,70,42,59)(21,112,140,102,141,91,151,126)(22,111,131,101,142,100,152,125)(23,120,132,110,143,99,153,124)(24,119,133,109,144,98,154,123)(25,118,134,108,145,97,155,122)(26,117,135,107,146,96,156,121)(27,116,136,106,147,95,157,130)(28,115,137,105,148,94,158,129)(29,114,138,104,149,93,159,128)(30,113,139,103,150,92,160,127), (2,10)(3,9)(4,8)(5,7)(11,49)(12,48)(13,47)(14,46)(15,45)(16,44)(17,43)(18,42)(19,41)(20,50)(21,157)(22,156)(23,155)(24,154)(25,153)(26,152)(27,151)(28,160)(29,159)(30,158)(31,37)(32,36)(33,35)(38,40)(51,66)(52,65)(53,64)(54,63)(55,62)(56,61)(57,70)(58,69)(59,68)(60,67)(71,81)(72,90)(73,89)(74,88)(75,87)(76,86)(77,85)(78,84)(79,83)(80,82)(91,95)(92,94)(96,100)(97,99)(101,121)(102,130)(103,129)(104,128)(105,127)(106,126)(107,125)(108,124)(109,123)(110,122)(111,117)(112,116)(113,115)(118,120)(131,146)(132,145)(133,144)(134,143)(135,142)(136,141)(137,150)(138,149)(139,148)(140,147) );
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)], [(1,93,6,98),(2,92,7,97),(3,91,8,96),(4,100,9,95),(5,99,10,94),(11,124,16,129),(12,123,17,128),(13,122,18,127),(14,121,19,126),(15,130,20,125),(21,77,26,72),(22,76,27,71),(23,75,28,80),(24,74,29,79),(25,73,30,78),(31,112,36,117),(32,111,37,116),(33,120,38,115),(34,119,39,114),(35,118,40,113),(41,106,46,101),(42,105,47,110),(43,104,48,109),(44,103,49,108),(45,102,50,107),(51,141,56,146),(52,150,57,145),(53,149,58,144),(54,148,59,143),(55,147,60,142),(61,131,66,136),(62,140,67,135),(63,139,68,134),(64,138,69,133),(65,137,70,132),(81,156,86,151),(82,155,87,160),(83,154,88,159),(84,153,89,158),(85,152,90,157)], [(1,83,17,79,39,69,43,58),(2,82,18,78,40,68,44,57),(3,81,19,77,31,67,45,56),(4,90,20,76,32,66,46,55),(5,89,11,75,33,65,47,54),(6,88,12,74,34,64,48,53),(7,87,13,73,35,63,49,52),(8,86,14,72,36,62,50,51),(9,85,15,71,37,61,41,60),(10,84,16,80,38,70,42,59),(21,112,140,102,141,91,151,126),(22,111,131,101,142,100,152,125),(23,120,132,110,143,99,153,124),(24,119,133,109,144,98,154,123),(25,118,134,108,145,97,155,122),(26,117,135,107,146,96,156,121),(27,116,136,106,147,95,157,130),(28,115,137,105,148,94,158,129),(29,114,138,104,149,93,159,128),(30,113,139,103,150,92,160,127)], [(2,10),(3,9),(4,8),(5,7),(11,49),(12,48),(13,47),(14,46),(15,45),(16,44),(17,43),(18,42),(19,41),(20,50),(21,157),(22,156),(23,155),(24,154),(25,153),(26,152),(27,151),(28,160),(29,159),(30,158),(31,37),(32,36),(33,35),(38,40),(51,66),(52,65),(53,64),(54,63),(55,62),(56,61),(57,70),(58,69),(59,68),(60,67),(71,81),(72,90),(73,89),(74,88),(75,87),(76,86),(77,85),(78,84),(79,83),(80,82),(91,95),(92,94),(96,100),(97,99),(101,121),(102,130),(103,129),(104,128),(105,127),(106,126),(107,125),(108,124),(109,123),(110,122),(111,117),(112,116),(113,115),(118,120),(131,146),(132,145),(133,144),(134,143),(135,142),(136,141),(137,150),(138,149),(139,148),(140,147)]])
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 | 20 | 20 | 2 | 2 | 4 | 4 | 8 | 10 | 10 | 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 | SD16 | C4oD4 | D10 | D10 | D10 | C4oD20 | C8.C22 | D4xD5 | D4xD5 | D5xSD16 | Q16:D5 |
| kernel | Dic5:SD16 | C10.Q16 | C20.8Q8 | C5xQ8:C4 | D20:8C4 | C2xC40:C2 | C2xQ8:D5 | Dic5:Q8 | D20 | C2xDic5 | Q8:C4 | Dic5 | C20 | C4:C4 | C2xC8 | C2xQ8 | C4 | C10 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 2 | 2 | 2 | 2 | 8 | 1 | 2 | 2 | 4 | 4 |
Matrix representation of Dic5:SD16 ►in GL4(F41) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 35 | 40 |
| 0 | 0 | 1 | 0 |
| 40 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 13 | 28 |
| 0 | 0 | 32 | 28 |
| 0 | 17 | 0 | 0 |
| 29 | 30 | 0 | 0 |
| 0 | 0 | 21 | 21 |
| 0 | 0 | 18 | 20 |
| 1 | 18 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 6 | 35 |
| 0 | 0 | 40 | 35 |
G:=sub<GL(4,GF(41))| [1,0,0,0,0,1,0,0,0,0,35,1,0,0,40,0],[40,0,0,0,0,40,0,0,0,0,13,32,0,0,28,28],[0,29,0,0,17,30,0,0,0,0,21,18,0,0,21,20],[1,0,0,0,18,40,0,0,0,0,6,40,0,0,35,35] >;
Dic5:SD16 in GAP, Magma, Sage, TeX
{\rm Dic}_5\rtimes {\rm SD}_{16} % in TeX
G:=Group("Dic5:SD16"); // GroupNames label
G:=SmallGroup(320,445);
// by ID
G=gap.SmallGroup(320,445);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,64,590,219,184,1684,851,438,102,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^10=c^8=d^2=1,b^2=a^5,b*a*b^-1=c*a*c^-1=d*a*d=a^-1,c*b*c^-1=a^5*b,b*d=d*b,d*c*d=c^3>;
// generators/relations