direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: SD16×Dic5, C5⋊8(C4×SD16), C40⋊25(C2×C4), C8⋊5(C2×Dic5), Q8⋊1(C2×Dic5), (Q8×Dic5)⋊3C2, (C5×SD16)⋊7C4, (C8×Dic5)⋊9C2, C40⋊6C4⋊26C2, C2.7(D5×SD16), (C2×C8).261D10, C10.125(C4×D4), (D4×Dic5).7C2, D4.1(C2×Dic5), (C2×SD16).5D5, C2.12(D4×Dic5), (C2×D4).142D10, C20.95(C4○D4), C10.59(C4○D8), Q8⋊Dic5⋊24C2, (C2×Q8).112D10, (C10×SD16).3C2, C10.42(C2×SD16), C22.116(D4×D5), C4.30(D4⋊2D5), C4.3(C22×Dic5), (C2×C20).438C23, C20.132(C22×C4), (C2×C40).162C22, (C2×Dic5).281D4, D4⋊Dic5.14C2, (D4×C10).87C22, (Q8×C10).68C22, C2.7(SD16⋊3D5), C4⋊Dic5.168C22, (C4×Dic5).271C22, (C5×Q8)⋊15(C2×C4), (C5×D4).22(C2×C4), (C2×C10).350(C2×D4), (C2×C4).527(C22×D5), (C2×C5⋊2C8).280C22, SmallGroup(320,788)
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, ad=da, bc=cb, bd=db, dcd-1=c-1 >
Subgroups: 390 in 122 conjugacy classes, 59 normal (37 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C10, C10, C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, SD16, C22×C4, C2×D4, C2×Q8, Dic5, Dic5, C20, C20, C2×C10, C2×C10, C4×C8, D4⋊C4, Q8⋊C4, C4.Q8, C4×D4, C4×Q8, C2×SD16, C5⋊2C8, C40, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C5×D4, C5×D4, C5×Q8, C5×Q8, C22×C10, C4×SD16, C2×C5⋊2C8, C4×Dic5, C4×Dic5, C4⋊Dic5, C4⋊Dic5, C23.D5, C2×C40, C5×SD16, C22×Dic5, D4×C10, Q8×C10, C8×Dic5, C40⋊6C4, D4⋊Dic5, Q8⋊Dic5, D4×Dic5, Q8×Dic5, C10×SD16, SD16×Dic5
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D5, SD16, C22×C4, C2×D4, C4○D4, Dic5, D10, C4×D4, C2×SD16, C4○D8, C2×Dic5, C22×D5, C4×SD16, D4×D5, D4⋊2D5, C22×Dic5, D5×SD16, SD16⋊3D5, D4×Dic5, SD16×Dic5
►On 160 points
Generators in S160
(1 90 15 74 39 60 48 61)(2 81 16 75 40 51 49 62)(3 82 17 76 31 52 50 63)(4 83 18 77 32 53 41 64)(5 84 19 78 33 54 42 65)(6 85 20 79 34 55 43 66)(7 86 11 80 35 56 44 67)(8 87 12 71 36 57 45 68)(9 88 13 72 37 58 46 69)(10 89 14 73 38 59 47 70)(21 92 156 111 137 127 148 103)(22 93 157 112 138 128 149 104)(23 94 158 113 139 129 150 105)(24 95 159 114 140 130 141 106)(25 96 160 115 131 121 142 107)(26 97 151 116 132 122 143 108)(27 98 152 117 133 123 144 109)(28 99 153 118 134 124 145 110)(29 100 154 119 135 125 146 101)(30 91 155 120 136 126 147 102)
(11 44)(12 45)(13 46)(14 47)(15 48)(16 49)(17 50)(18 41)(19 42)(20 43)(21 156)(22 157)(23 158)(24 159)(25 160)(26 151)(27 152)(28 153)(29 154)(30 155)(51 62)(52 63)(53 64)(54 65)(55 66)(56 67)(57 68)(58 69)(59 70)(60 61)(71 87)(72 88)(73 89)(74 90)(75 81)(76 82)(77 83)(78 84)(79 85)(80 86)(91 126)(92 127)(93 128)(94 129)(95 130)(96 121)(97 122)(98 123)(99 124)(100 125)(131 142)(132 143)(133 144)(134 145)(135 146)(136 147)(137 148)(138 149)(139 150)(140 141)
(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)
(1 116 6 111)(2 115 7 120)(3 114 8 119)(4 113 9 118)(5 112 10 117)(11 126 16 121)(12 125 17 130)(13 124 18 129)(14 123 19 128)(15 122 20 127)(21 60 26 55)(22 59 27 54)(23 58 28 53)(24 57 29 52)(25 56 30 51)(31 106 36 101)(32 105 37 110)(33 104 38 109)(34 103 39 108)(35 102 40 107)(41 94 46 99)(42 93 47 98)(43 92 48 97)(44 91 49 96)(45 100 50 95)(61 151 66 156)(62 160 67 155)(63 159 68 154)(64 158 69 153)(65 157 70 152)(71 146 76 141)(72 145 77 150)(73 144 78 149)(74 143 79 148)(75 142 80 147)(81 131 86 136)(82 140 87 135)(83 139 88 134)(84 138 89 133)(85 137 90 132)
G:=sub<Sym(160)| (1,90,15,74,39,60,48,61)(2,81,16,75,40,51,49,62)(3,82,17,76,31,52,50,63)(4,83,18,77,32,53,41,64)(5,84,19,78,33,54,42,65)(6,85,20,79,34,55,43,66)(7,86,11,80,35,56,44,67)(8,87,12,71,36,57,45,68)(9,88,13,72,37,58,46,69)(10,89,14,73,38,59,47,70)(21,92,156,111,137,127,148,103)(22,93,157,112,138,128,149,104)(23,94,158,113,139,129,150,105)(24,95,159,114,140,130,141,106)(25,96,160,115,131,121,142,107)(26,97,151,116,132,122,143,108)(27,98,152,117,133,123,144,109)(28,99,153,118,134,124,145,110)(29,100,154,119,135,125,146,101)(30,91,155,120,136,126,147,102), (11,44)(12,45)(13,46)(14,47)(15,48)(16,49)(17,50)(18,41)(19,42)(20,43)(21,156)(22,157)(23,158)(24,159)(25,160)(26,151)(27,152)(28,153)(29,154)(30,155)(51,62)(52,63)(53,64)(54,65)(55,66)(56,67)(57,68)(58,69)(59,70)(60,61)(71,87)(72,88)(73,89)(74,90)(75,81)(76,82)(77,83)(78,84)(79,85)(80,86)(91,126)(92,127)(93,128)(94,129)(95,130)(96,121)(97,122)(98,123)(99,124)(100,125)(131,142)(132,143)(133,144)(134,145)(135,146)(136,147)(137,148)(138,149)(139,150)(140,141), (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), (1,116,6,111)(2,115,7,120)(3,114,8,119)(4,113,9,118)(5,112,10,117)(11,126,16,121)(12,125,17,130)(13,124,18,129)(14,123,19,128)(15,122,20,127)(21,60,26,55)(22,59,27,54)(23,58,28,53)(24,57,29,52)(25,56,30,51)(31,106,36,101)(32,105,37,110)(33,104,38,109)(34,103,39,108)(35,102,40,107)(41,94,46,99)(42,93,47,98)(43,92,48,97)(44,91,49,96)(45,100,50,95)(61,151,66,156)(62,160,67,155)(63,159,68,154)(64,158,69,153)(65,157,70,152)(71,146,76,141)(72,145,77,150)(73,144,78,149)(74,143,79,148)(75,142,80,147)(81,131,86,136)(82,140,87,135)(83,139,88,134)(84,138,89,133)(85,137,90,132)>;
G:=Group( (1,90,15,74,39,60,48,61)(2,81,16,75,40,51,49,62)(3,82,17,76,31,52,50,63)(4,83,18,77,32,53,41,64)(5,84,19,78,33,54,42,65)(6,85,20,79,34,55,43,66)(7,86,11,80,35,56,44,67)(8,87,12,71,36,57,45,68)(9,88,13,72,37,58,46,69)(10,89,14,73,38,59,47,70)(21,92,156,111,137,127,148,103)(22,93,157,112,138,128,149,104)(23,94,158,113,139,129,150,105)(24,95,159,114,140,130,141,106)(25,96,160,115,131,121,142,107)(26,97,151,116,132,122,143,108)(27,98,152,117,133,123,144,109)(28,99,153,118,134,124,145,110)(29,100,154,119,135,125,146,101)(30,91,155,120,136,126,147,102), (11,44)(12,45)(13,46)(14,47)(15,48)(16,49)(17,50)(18,41)(19,42)(20,43)(21,156)(22,157)(23,158)(24,159)(25,160)(26,151)(27,152)(28,153)(29,154)(30,155)(51,62)(52,63)(53,64)(54,65)(55,66)(56,67)(57,68)(58,69)(59,70)(60,61)(71,87)(72,88)(73,89)(74,90)(75,81)(76,82)(77,83)(78,84)(79,85)(80,86)(91,126)(92,127)(93,128)(94,129)(95,130)(96,121)(97,122)(98,123)(99,124)(100,125)(131,142)(132,143)(133,144)(134,145)(135,146)(136,147)(137,148)(138,149)(139,150)(140,141), (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), (1,116,6,111)(2,115,7,120)(3,114,8,119)(4,113,9,118)(5,112,10,117)(11,126,16,121)(12,125,17,130)(13,124,18,129)(14,123,19,128)(15,122,20,127)(21,60,26,55)(22,59,27,54)(23,58,28,53)(24,57,29,52)(25,56,30,51)(31,106,36,101)(32,105,37,110)(33,104,38,109)(34,103,39,108)(35,102,40,107)(41,94,46,99)(42,93,47,98)(43,92,48,97)(44,91,49,96)(45,100,50,95)(61,151,66,156)(62,160,67,155)(63,159,68,154)(64,158,69,153)(65,157,70,152)(71,146,76,141)(72,145,77,150)(73,144,78,149)(74,143,79,148)(75,142,80,147)(81,131,86,136)(82,140,87,135)(83,139,88,134)(84,138,89,133)(85,137,90,132) );
G=PermutationGroup([[(1,90,15,74,39,60,48,61),(2,81,16,75,40,51,49,62),(3,82,17,76,31,52,50,63),(4,83,18,77,32,53,41,64),(5,84,19,78,33,54,42,65),(6,85,20,79,34,55,43,66),(7,86,11,80,35,56,44,67),(8,87,12,71,36,57,45,68),(9,88,13,72,37,58,46,69),(10,89,14,73,38,59,47,70),(21,92,156,111,137,127,148,103),(22,93,157,112,138,128,149,104),(23,94,158,113,139,129,150,105),(24,95,159,114,140,130,141,106),(25,96,160,115,131,121,142,107),(26,97,151,116,132,122,143,108),(27,98,152,117,133,123,144,109),(28,99,153,118,134,124,145,110),(29,100,154,119,135,125,146,101),(30,91,155,120,136,126,147,102)], [(11,44),(12,45),(13,46),(14,47),(15,48),(16,49),(17,50),(18,41),(19,42),(20,43),(21,156),(22,157),(23,158),(24,159),(25,160),(26,151),(27,152),(28,153),(29,154),(30,155),(51,62),(52,63),(53,64),(54,65),(55,66),(56,67),(57,68),(58,69),(59,70),(60,61),(71,87),(72,88),(73,89),(74,90),(75,81),(76,82),(77,83),(78,84),(79,85),(80,86),(91,126),(92,127),(93,128),(94,129),(95,130),(96,121),(97,122),(98,123),(99,124),(100,125),(131,142),(132,143),(133,144),(134,145),(135,146),(136,147),(137,148),(138,149),(139,150),(140,141)], [(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)], [(1,116,6,111),(2,115,7,120),(3,114,8,119),(4,113,9,118),(5,112,10,117),(11,126,16,121),(12,125,17,130),(13,124,18,129),(14,123,19,128),(15,122,20,127),(21,60,26,55),(22,59,27,54),(23,58,28,53),(24,57,29,52),(25,56,30,51),(31,106,36,101),(32,105,37,110),(33,104,38,109),(34,103,39,108),(35,102,40,107),(41,94,46,99),(42,93,47,98),(43,92,48,97),(44,91,49,96),(45,100,50,95),(61,151,66,156),(62,160,67,155),(63,159,68,154),(64,158,69,153),(65,157,70,152),(71,146,76,141),(72,145,77,150),(73,144,78,149),(74,143,79,148),(75,142,80,147),(81,131,86,136),(82,140,87,135),(83,139,88,134),(84,138,89,133),(85,137,90,132)]])
56 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 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 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 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 | 4 | 4 | 2 | 2 | 4 | 4 | 5 | 5 | 5 | 5 | 10 | 10 | 20 | 20 | 20 | 20 | 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 |
56 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | - | + | + | - | + | ||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | D4 | D5 | SD16 | C4○D4 | D10 | Dic5 | D10 | D10 | C4○D8 | D4⋊2D5 | D4×D5 | D5×SD16 | SD16⋊3D5 |
| kernel | SD16×Dic5 | C8×Dic5 | C40⋊6C4 | D4⋊Dic5 | Q8⋊Dic5 | D4×Dic5 | Q8×Dic5 | C10×SD16 | C5×SD16 | C2×Dic5 | C2×SD16 | Dic5 | C20 | C2×C8 | SD16 | C2×D4 | C2×Q8 | C10 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 8 | 2 | 2 | 4 | 2 | 2 | 8 | 2 | 2 | 4 | 2 | 2 | 4 | 4 |
Matrix representation of SD16×Dic5 ►in GL4(𝔽41) generated by
| 0 | 26 | 0 | 0 |
| 30 | 30 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 40 |
| 1 | 1 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 40 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 1 | 40 |
| 0 | 0 | 36 | 6 |
| 32 | 0 | 0 | 0 |
| 0 | 32 | 0 | 0 |
| 0 | 0 | 9 | 32 |
| 0 | 0 | 0 | 32 |
G:=sub<GL(4,GF(41))| [0,30,0,0,26,30,0,0,0,0,40,0,0,0,0,40],[1,0,0,0,1,40,0,0,0,0,1,0,0,0,0,1],[40,0,0,0,0,40,0,0,0,0,1,36,0,0,40,6],[32,0,0,0,0,32,0,0,0,0,9,0,0,0,32,32] >;
SD16×Dic5 in GAP, Magma, Sage, TeX
{\rm SD}_{16}\times {\rm Dic}_5 % in TeX
G:=Group("SD16xDic5"); // GroupNames label
G:=SmallGroup(320,788);
// by ID
G=gap.SmallGroup(320,788);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,219,184,851,438,102,12550]);
// 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,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations