metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Dic5⋊3SD16, (C5×D4).7D4, C20.170(C2×D4), (C2×C8).143D10, D4.2(C5⋊D4), C5⋊5(D4.D4), (C2×Q8).50D10, (D4×Dic5).8C2, (C2×SD16).3D5, C2.26(D5×SD16), Dic5⋊Q8⋊3C2, (C2×D4).143D10, C20.96(C4○D4), C4.9(D4⋊2D5), Q8⋊Dic5⋊25C2, C20.8Q8⋊32C2, (C10×SD16).6C2, C10.43(C2×SD16), C22.260(D4×D5), C20.44D4⋊33C2, (C2×C40).290C22, (C2×C20).439C23, (C2×Dic5).238D4, (D4×C10).88C22, (Q8×C10).69C22, C10.112(C4⋊D4), C2.26(SD16⋊D5), C10.45(C8.C22), C4⋊Dic5.169C22, (C4×Dic5).52C22, C2.24(Dic5⋊D4), (C2×Dic10).127C22, C4.38(C2×C5⋊D4), (C2×D4.D5).8C2, (C2×C10).351(C2×D4), (C2×C4).528(C22×D5), (C2×C5⋊2C8).151C22, SmallGroup(320,789)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic5⋊3SD16
G = < a,b,c,d | a10=c8=d2=1, b2=a5, bab-1=a-1, ac=ca, ad=da, cbc-1=a5b, bd=db, dcd=c3 >
Subgroups: 438 in 120 conjugacy classes, 43 normal (37 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, D4, Q8, C23, C10, C10, C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, SD16, C22×C4, C2×D4, C2×Q8, C2×Q8, Dic5, Dic5, C20, C20, C2×C10, C2×C10, Q8⋊C4, C4⋊C8, C4×D4, C4⋊Q8, C2×SD16, C2×SD16, C5⋊2C8, C40, Dic10, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C5×D4, C5×D4, C5×Q8, C22×C10, D4.D4, C2×C5⋊2C8, C4×Dic5, C10.D4, C4⋊Dic5, D4.D5, C23.D5, C2×C40, C5×SD16, C2×Dic10, C22×Dic5, D4×C10, Q8×C10, C20.8Q8, C20.44D4, Q8⋊Dic5, C2×D4.D5, D4×Dic5, Dic5⋊Q8, C10×SD16, Dic5⋊3SD16
Quotients: C1, C2, C22, D4, C23, D5, SD16, C2×D4, C4○D4, D10, C4⋊D4, C2×SD16, C8.C22, C5⋊D4, C22×D5, D4.D4, D4×D5, D4⋊2D5, C2×C5⋊D4, D5×SD16, SD16⋊D5, Dic5⋊D4, Dic5⋊3SD16
►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 99 6 94)(2 98 7 93)(3 97 8 92)(4 96 9 91)(5 95 10 100)(11 90 16 85)(12 89 17 84)(13 88 18 83)(14 87 19 82)(15 86 20 81)(21 109 26 104)(22 108 27 103)(23 107 28 102)(24 106 29 101)(25 105 30 110)(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)(51 134 56 139)(52 133 57 138)(53 132 58 137)(54 131 59 136)(55 140 60 135)(61 149 66 144)(62 148 67 143)(63 147 68 142)(64 146 69 141)(65 145 70 150)(71 154 76 159)(72 153 77 158)(73 152 78 157)(74 151 79 156)(75 160 80 155)
(1 85 36 68 28 76 48 56)(2 86 37 69 29 77 49 57)(3 87 38 70 30 78 50 58)(4 88 39 61 21 79 41 59)(5 89 40 62 22 80 42 60)(6 90 31 63 23 71 43 51)(7 81 32 64 24 72 44 52)(8 82 33 65 25 73 45 53)(9 83 34 66 26 74 46 54)(10 84 35 67 27 75 47 55)(11 114 142 107 159 127 139 94)(12 115 143 108 160 128 140 95)(13 116 144 109 151 129 131 96)(14 117 145 110 152 130 132 97)(15 118 146 101 153 121 133 98)(16 119 147 102 154 122 134 99)(17 120 148 103 155 123 135 100)(18 111 149 104 156 124 136 91)(19 112 150 105 157 125 137 92)(20 113 141 106 158 126 138 93)
(11 142)(12 143)(13 144)(14 145)(15 146)(16 147)(17 148)(18 149)(19 150)(20 141)(31 43)(32 44)(33 45)(34 46)(35 47)(36 48)(37 49)(38 50)(39 41)(40 42)(51 71)(52 72)(53 73)(54 74)(55 75)(56 76)(57 77)(58 78)(59 79)(60 80)(61 88)(62 89)(63 90)(64 81)(65 82)(66 83)(67 84)(68 85)(69 86)(70 87)(111 124)(112 125)(113 126)(114 127)(115 128)(116 129)(117 130)(118 121)(119 122)(120 123)(131 151)(132 152)(133 153)(134 154)(135 155)(136 156)(137 157)(138 158)(139 159)(140 160)
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), (1,99,6,94)(2,98,7,93)(3,97,8,92)(4,96,9,91)(5,95,10,100)(11,90,16,85)(12,89,17,84)(13,88,18,83)(14,87,19,82)(15,86,20,81)(21,109,26,104)(22,108,27,103)(23,107,28,102)(24,106,29,101)(25,105,30,110)(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)(51,134,56,139)(52,133,57,138)(53,132,58,137)(54,131,59,136)(55,140,60,135)(61,149,66,144)(62,148,67,143)(63,147,68,142)(64,146,69,141)(65,145,70,150)(71,154,76,159)(72,153,77,158)(73,152,78,157)(74,151,79,156)(75,160,80,155), (1,85,36,68,28,76,48,56)(2,86,37,69,29,77,49,57)(3,87,38,70,30,78,50,58)(4,88,39,61,21,79,41,59)(5,89,40,62,22,80,42,60)(6,90,31,63,23,71,43,51)(7,81,32,64,24,72,44,52)(8,82,33,65,25,73,45,53)(9,83,34,66,26,74,46,54)(10,84,35,67,27,75,47,55)(11,114,142,107,159,127,139,94)(12,115,143,108,160,128,140,95)(13,116,144,109,151,129,131,96)(14,117,145,110,152,130,132,97)(15,118,146,101,153,121,133,98)(16,119,147,102,154,122,134,99)(17,120,148,103,155,123,135,100)(18,111,149,104,156,124,136,91)(19,112,150,105,157,125,137,92)(20,113,141,106,158,126,138,93), (11,142)(12,143)(13,144)(14,145)(15,146)(16,147)(17,148)(18,149)(19,150)(20,141)(31,43)(32,44)(33,45)(34,46)(35,47)(36,48)(37,49)(38,50)(39,41)(40,42)(51,71)(52,72)(53,73)(54,74)(55,75)(56,76)(57,77)(58,78)(59,79)(60,80)(61,88)(62,89)(63,90)(64,81)(65,82)(66,83)(67,84)(68,85)(69,86)(70,87)(111,124)(112,125)(113,126)(114,127)(115,128)(116,129)(117,130)(118,121)(119,122)(120,123)(131,151)(132,152)(133,153)(134,154)(135,155)(136,156)(137,157)(138,158)(139,159)(140,160)>;
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), (1,99,6,94)(2,98,7,93)(3,97,8,92)(4,96,9,91)(5,95,10,100)(11,90,16,85)(12,89,17,84)(13,88,18,83)(14,87,19,82)(15,86,20,81)(21,109,26,104)(22,108,27,103)(23,107,28,102)(24,106,29,101)(25,105,30,110)(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)(51,134,56,139)(52,133,57,138)(53,132,58,137)(54,131,59,136)(55,140,60,135)(61,149,66,144)(62,148,67,143)(63,147,68,142)(64,146,69,141)(65,145,70,150)(71,154,76,159)(72,153,77,158)(73,152,78,157)(74,151,79,156)(75,160,80,155), (1,85,36,68,28,76,48,56)(2,86,37,69,29,77,49,57)(3,87,38,70,30,78,50,58)(4,88,39,61,21,79,41,59)(5,89,40,62,22,80,42,60)(6,90,31,63,23,71,43,51)(7,81,32,64,24,72,44,52)(8,82,33,65,25,73,45,53)(9,83,34,66,26,74,46,54)(10,84,35,67,27,75,47,55)(11,114,142,107,159,127,139,94)(12,115,143,108,160,128,140,95)(13,116,144,109,151,129,131,96)(14,117,145,110,152,130,132,97)(15,118,146,101,153,121,133,98)(16,119,147,102,154,122,134,99)(17,120,148,103,155,123,135,100)(18,111,149,104,156,124,136,91)(19,112,150,105,157,125,137,92)(20,113,141,106,158,126,138,93), (11,142)(12,143)(13,144)(14,145)(15,146)(16,147)(17,148)(18,149)(19,150)(20,141)(31,43)(32,44)(33,45)(34,46)(35,47)(36,48)(37,49)(38,50)(39,41)(40,42)(51,71)(52,72)(53,73)(54,74)(55,75)(56,76)(57,77)(58,78)(59,79)(60,80)(61,88)(62,89)(63,90)(64,81)(65,82)(66,83)(67,84)(68,85)(69,86)(70,87)(111,124)(112,125)(113,126)(114,127)(115,128)(116,129)(117,130)(118,121)(119,122)(120,123)(131,151)(132,152)(133,153)(134,154)(135,155)(136,156)(137,157)(138,158)(139,159)(140,160) );
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)], [(1,99,6,94),(2,98,7,93),(3,97,8,92),(4,96,9,91),(5,95,10,100),(11,90,16,85),(12,89,17,84),(13,88,18,83),(14,87,19,82),(15,86,20,81),(21,109,26,104),(22,108,27,103),(23,107,28,102),(24,106,29,101),(25,105,30,110),(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),(51,134,56,139),(52,133,57,138),(53,132,58,137),(54,131,59,136),(55,140,60,135),(61,149,66,144),(62,148,67,143),(63,147,68,142),(64,146,69,141),(65,145,70,150),(71,154,76,159),(72,153,77,158),(73,152,78,157),(74,151,79,156),(75,160,80,155)], [(1,85,36,68,28,76,48,56),(2,86,37,69,29,77,49,57),(3,87,38,70,30,78,50,58),(4,88,39,61,21,79,41,59),(5,89,40,62,22,80,42,60),(6,90,31,63,23,71,43,51),(7,81,32,64,24,72,44,52),(8,82,33,65,25,73,45,53),(9,83,34,66,26,74,46,54),(10,84,35,67,27,75,47,55),(11,114,142,107,159,127,139,94),(12,115,143,108,160,128,140,95),(13,116,144,109,151,129,131,96),(14,117,145,110,152,130,132,97),(15,118,146,101,153,121,133,98),(16,119,147,102,154,122,134,99),(17,120,148,103,155,123,135,100),(18,111,149,104,156,124,136,91),(19,112,150,105,157,125,137,92),(20,113,141,106,158,126,138,93)], [(11,142),(12,143),(13,144),(14,145),(15,146),(16,147),(17,148),(18,149),(19,150),(20,141),(31,43),(32,44),(33,45),(34,46),(35,47),(36,48),(37,49),(38,50),(39,41),(40,42),(51,71),(52,72),(53,73),(54,74),(55,75),(56,76),(57,77),(58,78),(59,79),(60,80),(61,88),(62,89),(63,90),(64,81),(65,82),(66,83),(67,84),(68,85),(69,86),(70,87),(111,124),(112,125),(113,126),(114,127),(115,128),(116,129),(117,130),(118,121),(119,122),(120,123),(131,151),(132,152),(133,153),(134,154),(135,155),(136,156),(137,157),(138,158),(139,159),(140,160)]])
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 | 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 | 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 | 8 | 10 | 10 | 20 | 20 | 20 | 40 | 2 | 2 | 4 | 4 | 20 | 20 | 2 | ··· | 2 | 8 | 8 | 8 | 8 | 4 | 4 | 4 | 4 | 8 | 8 | 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 | C5⋊D4 | C8.C22 | D4⋊2D5 | D4×D5 | D5×SD16 | SD16⋊D5 |
| kernel | Dic5⋊3SD16 | C20.8Q8 | C20.44D4 | Q8⋊Dic5 | C2×D4.D5 | D4×Dic5 | Dic5⋊Q8 | C10×SD16 | C2×Dic5 | C5×D4 | C2×SD16 | Dic5 | C20 | C2×C8 | C2×D4 | C2×Q8 | D4 | 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⋊3SD16 ►in GL6(𝔽41)
| 40 | 1 | 0 | 0 | 0 | 0 |
| 5 | 35 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 18 | 21 | 0 | 0 | 0 | 0 |
| 10 | 23 | 0 | 0 | 0 | 0 |
| 0 | 0 | 32 | 0 | 0 | 0 |
| 0 | 0 | 0 | 9 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 10 | 2 |
| 0 | 0 | 0 | 0 | 26 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 16 | 40 |
G:=sub<GL(6,GF(41))| [40,5,0,0,0,0,1,35,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[18,10,0,0,0,0,21,23,0,0,0,0,0,0,32,0,0,0,0,0,0,9,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,40,0,0,0,0,1,0,0,0,0,0,0,0,10,26,0,0,0,0,2,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,1,16,0,0,0,0,0,40] >;
Dic5⋊3SD16 in GAP, Magma, Sage, TeX
{\rm Dic}_5\rtimes_3{\rm SD}_{16} % in TeX
G:=Group("Dic5:3SD16"); // GroupNames label
G:=SmallGroup(320,789);
// by ID
G=gap.SmallGroup(320,789);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,422,135,184,570,297,136,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^10=c^8=d^2=1,b^2=a^5,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,c*b*c^-1=a^5*b,b*d=d*b,d*c*d=c^3>;
// generators/relations