metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D20.11D4, Dic5⋊2SD16, C4.96(D4×D5), C4⋊C4.156D10, Q8⋊C4⋊14D5, C4.7(C4○D20), C20.128(C2×D4), (C2×C8).126D10, C5⋊1(D4.D4), (C2×Q8).23D10, C2.19(D5×SD16), Dic5⋊Q8⋊2C2, D20⋊8C4.3C2, C20.23(C4○D4), C10.Q16⋊13C2, C20.8Q8⋊14C2, (C2×Dic5).43D4, C10.33(C2×SD16), C22.207(D4×D5), C10.27(C4⋊D4), (C2×C40).137C22, (C2×C20).258C23, (C2×D20).72C22, (Q8×C10).41C22, C2.30(D10⋊D4), C2.19(Q16⋊D5), C10.66(C8.C22), (C4×Dic5).31C22, (C2×Dic10).79C22, (C2×Q8⋊D5).3C2, (C2×C40⋊C2).4C2, (C5×Q8⋊C4)⋊14C2, (C2×C10).271(C2×D4), (C5×C4⋊C4).59C22, (C2×C5⋊2C8).48C22, (C2×C4).365(C22×D5), 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 [×3], C2 [×2], C4 [×2], C4 [×6], C22, C22 [×4], C5, C8 [×2], C2×C4, C2×C4 [×7], D4 [×3], Q8 [×4], C23, D5 [×2], C10 [×3], C42, C22⋊C4, C4⋊C4, C4⋊C4 [×2], C2×C8, C2×C8, SD16 [×4], C22×C4, C2×D4, C2×Q8, C2×Q8, Dic5 [×2], Dic5 [×2], C20 [×2], C20 [×2], D10 [×4], C2×C10, Q8⋊C4, Q8⋊C4, C4⋊C8, C4×D4, C4⋊Q8, C2×SD16 [×2], C5⋊2C8, C40, Dic10 [×2], C4×D5 [×2], D20 [×2], D20, C2×Dic5 [×2], C2×Dic5, C2×C20, C2×C20 [×2], C5×Q8 [×2], C22×D5, D4.D4, C40⋊C2 [×2], C2×C5⋊2C8, C4×Dic5, C10.D4 [×2], D10⋊C4, Q8⋊D5 [×2], C5×C4⋊C4, C2×C40, C2×Dic10, C2×C4×D5, C2×D20, Q8×C10, C10.Q16, C20.8Q8, C5×Q8⋊C4, D20⋊8C4, C2×C40⋊C2, C2×Q8⋊D5, Dic5⋊Q8, Dic5⋊SD16
Quotients: C1, C2 [×7], C22 [×7], D4 [×4], C23, D5, SD16 [×2], C2×D4 [×2], C4○D4, D10 [×3], C4⋊D4, C2×SD16, C8.C22, C22×D5, D4.D4, C4○D20, D4×D5 [×2], D10⋊D4, D5×SD16, 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)(11 127 16 122)(12 126 17 121)(13 125 18 130)(14 124 19 129)(15 123 20 128)(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)(41 102 46 107)(42 101 47 106)(43 110 48 105)(44 109 49 104)(45 108 50 103)(51 147 56 142)(52 146 57 141)(53 145 58 150)(54 144 59 149)(55 143 60 148)(61 137 66 132)(62 136 67 131)(63 135 68 140)(64 134 69 139)(65 133 70 138)(81 156 86 151)(82 155 87 160)(83 154 88 159)(84 153 89 158)(85 152 90 157)
(1 83 20 79 39 65 43 58)(2 82 11 78 40 64 44 57)(3 81 12 77 31 63 45 56)(4 90 13 76 32 62 46 55)(5 89 14 75 33 61 47 54)(6 88 15 74 34 70 48 53)(7 87 16 73 35 69 49 52)(8 86 17 72 36 68 50 51)(9 85 18 71 37 67 41 60)(10 84 19 80 38 66 42 59)(21 112 140 108 147 91 151 126)(22 111 131 107 148 100 152 125)(23 120 132 106 149 99 153 124)(24 119 133 105 150 98 154 123)(25 118 134 104 141 97 155 122)(26 117 135 103 142 96 156 121)(27 116 136 102 143 95 157 130)(28 115 137 101 144 94 158 129)(29 114 138 110 145 93 159 128)(30 113 139 109 146 92 160 127)
(2 10)(3 9)(4 8)(5 7)(11 42)(12 41)(13 50)(14 49)(15 48)(16 47)(17 46)(18 45)(19 44)(20 43)(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 62)(52 61)(53 70)(54 69)(55 68)(56 67)(57 66)(58 65)(59 64)(60 63)(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)(101 127)(102 126)(103 125)(104 124)(105 123)(106 122)(107 121)(108 130)(109 129)(110 128)(111 117)(112 116)(113 115)(118 120)(131 142)(132 141)(133 150)(134 149)(135 148)(136 147)(137 146)(138 145)(139 144)(140 143)
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)(11,127,16,122)(12,126,17,121)(13,125,18,130)(14,124,19,129)(15,123,20,128)(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)(41,102,46,107)(42,101,47,106)(43,110,48,105)(44,109,49,104)(45,108,50,103)(51,147,56,142)(52,146,57,141)(53,145,58,150)(54,144,59,149)(55,143,60,148)(61,137,66,132)(62,136,67,131)(63,135,68,140)(64,134,69,139)(65,133,70,138)(81,156,86,151)(82,155,87,160)(83,154,88,159)(84,153,89,158)(85,152,90,157), (1,83,20,79,39,65,43,58)(2,82,11,78,40,64,44,57)(3,81,12,77,31,63,45,56)(4,90,13,76,32,62,46,55)(5,89,14,75,33,61,47,54)(6,88,15,74,34,70,48,53)(7,87,16,73,35,69,49,52)(8,86,17,72,36,68,50,51)(9,85,18,71,37,67,41,60)(10,84,19,80,38,66,42,59)(21,112,140,108,147,91,151,126)(22,111,131,107,148,100,152,125)(23,120,132,106,149,99,153,124)(24,119,133,105,150,98,154,123)(25,118,134,104,141,97,155,122)(26,117,135,103,142,96,156,121)(27,116,136,102,143,95,157,130)(28,115,137,101,144,94,158,129)(29,114,138,110,145,93,159,128)(30,113,139,109,146,92,160,127), (2,10)(3,9)(4,8)(5,7)(11,42)(12,41)(13,50)(14,49)(15,48)(16,47)(17,46)(18,45)(19,44)(20,43)(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)(51,62)(52,61)(53,70)(54,69)(55,68)(56,67)(57,66)(58,65)(59,64)(60,63)(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)(101,127)(102,126)(103,125)(104,124)(105,123)(106,122)(107,121)(108,130)(109,129)(110,128)(111,117)(112,116)(113,115)(118,120)(131,142)(132,141)(133,150)(134,149)(135,148)(136,147)(137,146)(138,145)(139,144)(140,143)>;
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)(11,127,16,122)(12,126,17,121)(13,125,18,130)(14,124,19,129)(15,123,20,128)(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)(41,102,46,107)(42,101,47,106)(43,110,48,105)(44,109,49,104)(45,108,50,103)(51,147,56,142)(52,146,57,141)(53,145,58,150)(54,144,59,149)(55,143,60,148)(61,137,66,132)(62,136,67,131)(63,135,68,140)(64,134,69,139)(65,133,70,138)(81,156,86,151)(82,155,87,160)(83,154,88,159)(84,153,89,158)(85,152,90,157), (1,83,20,79,39,65,43,58)(2,82,11,78,40,64,44,57)(3,81,12,77,31,63,45,56)(4,90,13,76,32,62,46,55)(5,89,14,75,33,61,47,54)(6,88,15,74,34,70,48,53)(7,87,16,73,35,69,49,52)(8,86,17,72,36,68,50,51)(9,85,18,71,37,67,41,60)(10,84,19,80,38,66,42,59)(21,112,140,108,147,91,151,126)(22,111,131,107,148,100,152,125)(23,120,132,106,149,99,153,124)(24,119,133,105,150,98,154,123)(25,118,134,104,141,97,155,122)(26,117,135,103,142,96,156,121)(27,116,136,102,143,95,157,130)(28,115,137,101,144,94,158,129)(29,114,138,110,145,93,159,128)(30,113,139,109,146,92,160,127), (2,10)(3,9)(4,8)(5,7)(11,42)(12,41)(13,50)(14,49)(15,48)(16,47)(17,46)(18,45)(19,44)(20,43)(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)(51,62)(52,61)(53,70)(54,69)(55,68)(56,67)(57,66)(58,65)(59,64)(60,63)(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)(101,127)(102,126)(103,125)(104,124)(105,123)(106,122)(107,121)(108,130)(109,129)(110,128)(111,117)(112,116)(113,115)(118,120)(131,142)(132,141)(133,150)(134,149)(135,148)(136,147)(137,146)(138,145)(139,144)(140,143) );
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),(11,127,16,122),(12,126,17,121),(13,125,18,130),(14,124,19,129),(15,123,20,128),(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),(41,102,46,107),(42,101,47,106),(43,110,48,105),(44,109,49,104),(45,108,50,103),(51,147,56,142),(52,146,57,141),(53,145,58,150),(54,144,59,149),(55,143,60,148),(61,137,66,132),(62,136,67,131),(63,135,68,140),(64,134,69,139),(65,133,70,138),(81,156,86,151),(82,155,87,160),(83,154,88,159),(84,153,89,158),(85,152,90,157)], [(1,83,20,79,39,65,43,58),(2,82,11,78,40,64,44,57),(3,81,12,77,31,63,45,56),(4,90,13,76,32,62,46,55),(5,89,14,75,33,61,47,54),(6,88,15,74,34,70,48,53),(7,87,16,73,35,69,49,52),(8,86,17,72,36,68,50,51),(9,85,18,71,37,67,41,60),(10,84,19,80,38,66,42,59),(21,112,140,108,147,91,151,126),(22,111,131,107,148,100,152,125),(23,120,132,106,149,99,153,124),(24,119,133,105,150,98,154,123),(25,118,134,104,141,97,155,122),(26,117,135,103,142,96,156,121),(27,116,136,102,143,95,157,130),(28,115,137,101,144,94,158,129),(29,114,138,110,145,93,159,128),(30,113,139,109,146,92,160,127)], [(2,10),(3,9),(4,8),(5,7),(11,42),(12,41),(13,50),(14,49),(15,48),(16,47),(17,46),(18,45),(19,44),(20,43),(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),(51,62),(52,61),(53,70),(54,69),(55,68),(56,67),(57,66),(58,65),(59,64),(60,63),(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),(101,127),(102,126),(103,125),(104,124),(105,123),(106,122),(107,121),(108,130),(109,129),(110,128),(111,117),(112,116),(113,115),(118,120),(131,142),(132,141),(133,150),(134,149),(135,148),(136,147),(137,146),(138,145),(139,144),(140,143)])
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 | C4○D4 | D10 | D10 | D10 | C4○D20 | C8.C22 | D4×D5 | D4×D5 | D5×SD16 | Q16⋊D5 |
| kernel | Dic5⋊SD16 | C10.Q16 | C20.8Q8 | C5×Q8⋊C4 | D20⋊8C4 | C2×C40⋊C2 | C2×Q8⋊D5 | Dic5⋊Q8 | D20 | C2×Dic5 | Q8⋊C4 | Dic5 | C20 | C4⋊C4 | C2×C8 | C2×Q8 | 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