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 [×3], C2 [×2], C4 [×2], C4 [×7], C22, C22 [×4], C5, C8 [×2], C8, C2×C4, C2×C4 [×7], D4 [×2], D4, Q8 [×2], Q8, C23, C10 [×3], C10 [×2], C42 [×2], C22⋊C4, C4⋊C4 [×3], C2×C8, C2×C8, SD16 [×4], C22×C4, C2×D4, C2×Q8, Dic5 [×2], Dic5 [×3], C20 [×2], C20 [×2], C2×C10, C2×C10 [×4], C4×C8, D4⋊C4, Q8⋊C4, C4.Q8, C4×D4, C4×Q8, C2×SD16, C5⋊2C8, C40 [×2], C2×Dic5 [×2], C2×Dic5 [×4], C2×C20, C2×C20, C5×D4 [×2], C5×D4, C5×Q8 [×2], C5×Q8, C22×C10, C4×SD16, C2×C5⋊2C8, C4×Dic5, C4×Dic5, C4⋊Dic5 [×2], C4⋊Dic5, C23.D5, C2×C40, C5×SD16 [×4], 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 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×2], C23, D5, SD16 [×2], C22×C4, C2×D4, C4○D4, Dic5 [×4], D10 [×3], C4×D4, C2×SD16, C4○D8, C2×Dic5 [×6], 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 19 74 39 63 48 57)(2 81 20 75 40 64 49 58)(3 82 11 76 31 65 50 59)(4 83 12 77 32 66 41 60)(5 84 13 78 33 67 42 51)(6 85 14 79 34 68 43 52)(7 86 15 80 35 69 44 53)(8 87 16 71 36 70 45 54)(9 88 17 72 37 61 46 55)(10 89 18 73 38 62 47 56)(21 102 156 111 147 127 140 95)(22 103 157 112 148 128 131 96)(23 104 158 113 149 129 132 97)(24 105 159 114 150 130 133 98)(25 106 160 115 141 121 134 99)(26 107 151 116 142 122 135 100)(27 108 152 117 143 123 136 91)(28 109 153 118 144 124 137 92)(29 110 154 119 145 125 138 93)(30 101 155 120 146 126 139 94)
(11 50)(12 41)(13 42)(14 43)(15 44)(16 45)(17 46)(18 47)(19 48)(20 49)(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)(101 126)(102 127)(103 128)(104 129)(105 130)(106 121)(107 122)(108 123)(109 124)(110 125)(131 148)(132 149)(133 150)(134 141)(135 142)(136 143)(137 144)(138 145)(139 146)(140 147)
(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)(11 130 16 125)(12 129 17 124)(13 128 18 123)(14 127 19 122)(15 126 20 121)(21 63 26 68)(22 62 27 67)(23 61 28 66)(24 70 29 65)(25 69 30 64)(31 98 36 93)(32 97 37 92)(33 96 38 91)(34 95 39 100)(35 94 40 99)(41 104 46 109)(42 103 47 108)(43 102 48 107)(44 101 49 106)(45 110 50 105)(51 157 56 152)(52 156 57 151)(53 155 58 160)(54 154 59 159)(55 153 60 158)(71 138 76 133)(72 137 77 132)(73 136 78 131)(74 135 79 140)(75 134 80 139)(81 141 86 146)(82 150 87 145)(83 149 88 144)(84 148 89 143)(85 147 90 142)
G:=sub<Sym(160)| (1,90,19,74,39,63,48,57)(2,81,20,75,40,64,49,58)(3,82,11,76,31,65,50,59)(4,83,12,77,32,66,41,60)(5,84,13,78,33,67,42,51)(6,85,14,79,34,68,43,52)(7,86,15,80,35,69,44,53)(8,87,16,71,36,70,45,54)(9,88,17,72,37,61,46,55)(10,89,18,73,38,62,47,56)(21,102,156,111,147,127,140,95)(22,103,157,112,148,128,131,96)(23,104,158,113,149,129,132,97)(24,105,159,114,150,130,133,98)(25,106,160,115,141,121,134,99)(26,107,151,116,142,122,135,100)(27,108,152,117,143,123,136,91)(28,109,153,118,144,124,137,92)(29,110,154,119,145,125,138,93)(30,101,155,120,146,126,139,94), (11,50)(12,41)(13,42)(14,43)(15,44)(16,45)(17,46)(18,47)(19,48)(20,49)(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)(101,126)(102,127)(103,128)(104,129)(105,130)(106,121)(107,122)(108,123)(109,124)(110,125)(131,148)(132,149)(133,150)(134,141)(135,142)(136,143)(137,144)(138,145)(139,146)(140,147), (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)(11,130,16,125)(12,129,17,124)(13,128,18,123)(14,127,19,122)(15,126,20,121)(21,63,26,68)(22,62,27,67)(23,61,28,66)(24,70,29,65)(25,69,30,64)(31,98,36,93)(32,97,37,92)(33,96,38,91)(34,95,39,100)(35,94,40,99)(41,104,46,109)(42,103,47,108)(43,102,48,107)(44,101,49,106)(45,110,50,105)(51,157,56,152)(52,156,57,151)(53,155,58,160)(54,154,59,159)(55,153,60,158)(71,138,76,133)(72,137,77,132)(73,136,78,131)(74,135,79,140)(75,134,80,139)(81,141,86,146)(82,150,87,145)(83,149,88,144)(84,148,89,143)(85,147,90,142)>;
G:=Group( (1,90,19,74,39,63,48,57)(2,81,20,75,40,64,49,58)(3,82,11,76,31,65,50,59)(4,83,12,77,32,66,41,60)(5,84,13,78,33,67,42,51)(6,85,14,79,34,68,43,52)(7,86,15,80,35,69,44,53)(8,87,16,71,36,70,45,54)(9,88,17,72,37,61,46,55)(10,89,18,73,38,62,47,56)(21,102,156,111,147,127,140,95)(22,103,157,112,148,128,131,96)(23,104,158,113,149,129,132,97)(24,105,159,114,150,130,133,98)(25,106,160,115,141,121,134,99)(26,107,151,116,142,122,135,100)(27,108,152,117,143,123,136,91)(28,109,153,118,144,124,137,92)(29,110,154,119,145,125,138,93)(30,101,155,120,146,126,139,94), (11,50)(12,41)(13,42)(14,43)(15,44)(16,45)(17,46)(18,47)(19,48)(20,49)(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)(101,126)(102,127)(103,128)(104,129)(105,130)(106,121)(107,122)(108,123)(109,124)(110,125)(131,148)(132,149)(133,150)(134,141)(135,142)(136,143)(137,144)(138,145)(139,146)(140,147), (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)(11,130,16,125)(12,129,17,124)(13,128,18,123)(14,127,19,122)(15,126,20,121)(21,63,26,68)(22,62,27,67)(23,61,28,66)(24,70,29,65)(25,69,30,64)(31,98,36,93)(32,97,37,92)(33,96,38,91)(34,95,39,100)(35,94,40,99)(41,104,46,109)(42,103,47,108)(43,102,48,107)(44,101,49,106)(45,110,50,105)(51,157,56,152)(52,156,57,151)(53,155,58,160)(54,154,59,159)(55,153,60,158)(71,138,76,133)(72,137,77,132)(73,136,78,131)(74,135,79,140)(75,134,80,139)(81,141,86,146)(82,150,87,145)(83,149,88,144)(84,148,89,143)(85,147,90,142) );
G=PermutationGroup([(1,90,19,74,39,63,48,57),(2,81,20,75,40,64,49,58),(3,82,11,76,31,65,50,59),(4,83,12,77,32,66,41,60),(5,84,13,78,33,67,42,51),(6,85,14,79,34,68,43,52),(7,86,15,80,35,69,44,53),(8,87,16,71,36,70,45,54),(9,88,17,72,37,61,46,55),(10,89,18,73,38,62,47,56),(21,102,156,111,147,127,140,95),(22,103,157,112,148,128,131,96),(23,104,158,113,149,129,132,97),(24,105,159,114,150,130,133,98),(25,106,160,115,141,121,134,99),(26,107,151,116,142,122,135,100),(27,108,152,117,143,123,136,91),(28,109,153,118,144,124,137,92),(29,110,154,119,145,125,138,93),(30,101,155,120,146,126,139,94)], [(11,50),(12,41),(13,42),(14,43),(15,44),(16,45),(17,46),(18,47),(19,48),(20,49),(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),(101,126),(102,127),(103,128),(104,129),(105,130),(106,121),(107,122),(108,123),(109,124),(110,125),(131,148),(132,149),(133,150),(134,141),(135,142),(136,143),(137,144),(138,145),(139,146),(140,147)], [(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),(11,130,16,125),(12,129,17,124),(13,128,18,123),(14,127,19,122),(15,126,20,121),(21,63,26,68),(22,62,27,67),(23,61,28,66),(24,70,29,65),(25,69,30,64),(31,98,36,93),(32,97,37,92),(33,96,38,91),(34,95,39,100),(35,94,40,99),(41,104,46,109),(42,103,47,108),(43,102,48,107),(44,101,49,106),(45,110,50,105),(51,157,56,152),(52,156,57,151),(53,155,58,160),(54,154,59,159),(55,153,60,158),(71,138,76,133),(72,137,77,132),(73,136,78,131),(74,135,79,140),(75,134,80,139),(81,141,86,146),(82,150,87,145),(83,149,88,144),(84,148,89,143),(85,147,90,142)])
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