metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Dic5⋊8SD16, C8⋊8(C4×D5), C5⋊5(C4×SD16), C40⋊23(C2×C4), C40⋊C2⋊9C4, C4.Q8⋊13D5, (C8×Dic5)⋊8C2, C10.79(C4×D4), C2.5(D5×SD16), C4⋊C4.158D10, D20.22(C2×C4), (C2×C8).255D10, C22.81(D4×D5), Dic10⋊15(C2×C4), Dic5⋊3Q8⋊6C2, D20⋊8C4.4C2, D20⋊6C4.4C2, C10.51(C4○D8), C20.25(C4○D4), C10.Q16⋊14C2, C20.99(C22×C4), C4.1(Q8⋊2D5), C10.34(C2×SD16), C2.9(D20⋊8C4), (C2×C40).156C22, (C2×C20).268C23, (C2×Dic5).273D4, (C2×D20).76C22, C2.5(SD16⋊3D5), (C2×Dic10).82C22, (C4×Dic5).259C22, C4.40(C2×C4×D5), (C5×C4.Q8)⋊6C2, (C2×C40⋊C2).9C2, (C2×C10).273(C2×D4), (C5×C4⋊C4).61C22, (C2×C4).371(C22×D5), (C2×C5⋊2C8).231C22, SmallGroup(320,479)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic5⋊8SD16
G = < a,b,c,d | a10=c8=d2=1, b2=a5, bab-1=dad=a-1, ac=ca, bc=cb, bd=db, dcd=c3 >
Subgroups: 502 in 122 conjugacy classes, 51 normal (37 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C8, C2×C4, C2×C4, D4, Q8, C23, D5, C10, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, SD16, C22×C4, C2×D4, C2×Q8, Dic5, Dic5, C20, C20, D10, C2×C10, C4×C8, D4⋊C4, Q8⋊C4, C4.Q8, C4×D4, C4×Q8, C2×SD16, C5⋊2C8, C40, Dic10, Dic10, C4×D5, D20, D20, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C22×D5, C4×SD16, C40⋊C2, C2×C5⋊2C8, C4×Dic5, C4×Dic5, C10.D4, D10⋊C4, C5×C4⋊C4, C2×C40, C2×Dic10, C2×C4×D5, C2×D20, D20⋊6C4, C10.Q16, C8×Dic5, C5×C4.Q8, Dic5⋊3Q8, D20⋊8C4, C2×C40⋊C2, Dic5⋊8SD16
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D5, SD16, C22×C4, C2×D4, C4○D4, D10, C4×D4, C2×SD16, C4○D8, C4×D5, C22×D5, C4×SD16, C2×C4×D5, D4×D5, Q8⋊2D5, D20⋊8C4, D5×SD16, SD16⋊3D5, Dic5⋊8SD16
►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)
(1 103 6 108)(2 102 7 107)(3 101 8 106)(4 110 9 105)(5 109 10 104)(11 57 16 52)(12 56 17 51)(13 55 18 60)(14 54 19 59)(15 53 20 58)(21 96 26 91)(22 95 27 100)(23 94 28 99)(24 93 29 98)(25 92 30 97)(31 112 36 117)(32 111 37 116)(33 120 38 115)(34 119 39 114)(35 118 40 113)(41 130 46 125)(42 129 47 124)(43 128 48 123)(44 127 49 122)(45 126 50 121)(61 156 66 151)(62 155 67 160)(63 154 68 159)(64 153 69 158)(65 152 70 157)(71 146 76 141)(72 145 77 150)(73 144 78 149)(74 143 79 148)(75 142 80 147)(81 136 86 131)(82 135 87 140)(83 134 88 139)(84 133 89 138)(85 132 90 137)
(1 148 34 14 28 134 48 154)(2 149 35 15 29 135 49 155)(3 150 36 16 30 136 50 156)(4 141 37 17 21 137 41 157)(5 142 38 18 22 138 42 158)(6 143 39 19 23 139 43 159)(7 144 40 20 24 140 44 160)(8 145 31 11 25 131 45 151)(9 146 32 12 26 132 46 152)(10 147 33 13 27 133 47 153)(51 96 85 130 65 110 71 116)(52 97 86 121 66 101 72 117)(53 98 87 122 67 102 73 118)(54 99 88 123 68 103 74 119)(55 100 89 124 69 104 75 120)(56 91 90 125 70 105 76 111)(57 92 81 126 61 106 77 112)(58 93 82 127 62 107 78 113)(59 94 83 128 63 108 79 114)(60 95 84 129 64 109 80 115)
(1 23)(2 22)(3 21)(4 30)(5 29)(6 28)(7 27)(8 26)(9 25)(10 24)(11 132)(12 131)(13 140)(14 139)(15 138)(16 137)(17 136)(18 135)(19 134)(20 133)(31 32)(33 40)(34 39)(35 38)(36 37)(41 50)(42 49)(43 48)(44 47)(45 46)(51 86)(52 85)(53 84)(54 83)(55 82)(56 81)(57 90)(58 89)(59 88)(60 87)(61 76)(62 75)(63 74)(64 73)(65 72)(66 71)(67 80)(68 79)(69 78)(70 77)(91 106)(92 105)(93 104)(94 103)(95 102)(96 101)(97 110)(98 109)(99 108)(100 107)(111 112)(113 120)(114 119)(115 118)(116 117)(121 130)(122 129)(123 128)(124 127)(125 126)(141 156)(142 155)(143 154)(144 153)(145 152)(146 151)(147 160)(148 159)(149 158)(150 157)
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), (1,103,6,108)(2,102,7,107)(3,101,8,106)(4,110,9,105)(5,109,10,104)(11,57,16,52)(12,56,17,51)(13,55,18,60)(14,54,19,59)(15,53,20,58)(21,96,26,91)(22,95,27,100)(23,94,28,99)(24,93,29,98)(25,92,30,97)(31,112,36,117)(32,111,37,116)(33,120,38,115)(34,119,39,114)(35,118,40,113)(41,130,46,125)(42,129,47,124)(43,128,48,123)(44,127,49,122)(45,126,50,121)(61,156,66,151)(62,155,67,160)(63,154,68,159)(64,153,69,158)(65,152,70,157)(71,146,76,141)(72,145,77,150)(73,144,78,149)(74,143,79,148)(75,142,80,147)(81,136,86,131)(82,135,87,140)(83,134,88,139)(84,133,89,138)(85,132,90,137), (1,148,34,14,28,134,48,154)(2,149,35,15,29,135,49,155)(3,150,36,16,30,136,50,156)(4,141,37,17,21,137,41,157)(5,142,38,18,22,138,42,158)(6,143,39,19,23,139,43,159)(7,144,40,20,24,140,44,160)(8,145,31,11,25,131,45,151)(9,146,32,12,26,132,46,152)(10,147,33,13,27,133,47,153)(51,96,85,130,65,110,71,116)(52,97,86,121,66,101,72,117)(53,98,87,122,67,102,73,118)(54,99,88,123,68,103,74,119)(55,100,89,124,69,104,75,120)(56,91,90,125,70,105,76,111)(57,92,81,126,61,106,77,112)(58,93,82,127,62,107,78,113)(59,94,83,128,63,108,79,114)(60,95,84,129,64,109,80,115), (1,23)(2,22)(3,21)(4,30)(5,29)(6,28)(7,27)(8,26)(9,25)(10,24)(11,132)(12,131)(13,140)(14,139)(15,138)(16,137)(17,136)(18,135)(19,134)(20,133)(31,32)(33,40)(34,39)(35,38)(36,37)(41,50)(42,49)(43,48)(44,47)(45,46)(51,86)(52,85)(53,84)(54,83)(55,82)(56,81)(57,90)(58,89)(59,88)(60,87)(61,76)(62,75)(63,74)(64,73)(65,72)(66,71)(67,80)(68,79)(69,78)(70,77)(91,106)(92,105)(93,104)(94,103)(95,102)(96,101)(97,110)(98,109)(99,108)(100,107)(111,112)(113,120)(114,119)(115,118)(116,117)(121,130)(122,129)(123,128)(124,127)(125,126)(141,156)(142,155)(143,154)(144,153)(145,152)(146,151)(147,160)(148,159)(149,158)(150,157)>;
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), (1,103,6,108)(2,102,7,107)(3,101,8,106)(4,110,9,105)(5,109,10,104)(11,57,16,52)(12,56,17,51)(13,55,18,60)(14,54,19,59)(15,53,20,58)(21,96,26,91)(22,95,27,100)(23,94,28,99)(24,93,29,98)(25,92,30,97)(31,112,36,117)(32,111,37,116)(33,120,38,115)(34,119,39,114)(35,118,40,113)(41,130,46,125)(42,129,47,124)(43,128,48,123)(44,127,49,122)(45,126,50,121)(61,156,66,151)(62,155,67,160)(63,154,68,159)(64,153,69,158)(65,152,70,157)(71,146,76,141)(72,145,77,150)(73,144,78,149)(74,143,79,148)(75,142,80,147)(81,136,86,131)(82,135,87,140)(83,134,88,139)(84,133,89,138)(85,132,90,137), (1,148,34,14,28,134,48,154)(2,149,35,15,29,135,49,155)(3,150,36,16,30,136,50,156)(4,141,37,17,21,137,41,157)(5,142,38,18,22,138,42,158)(6,143,39,19,23,139,43,159)(7,144,40,20,24,140,44,160)(8,145,31,11,25,131,45,151)(9,146,32,12,26,132,46,152)(10,147,33,13,27,133,47,153)(51,96,85,130,65,110,71,116)(52,97,86,121,66,101,72,117)(53,98,87,122,67,102,73,118)(54,99,88,123,68,103,74,119)(55,100,89,124,69,104,75,120)(56,91,90,125,70,105,76,111)(57,92,81,126,61,106,77,112)(58,93,82,127,62,107,78,113)(59,94,83,128,63,108,79,114)(60,95,84,129,64,109,80,115), (1,23)(2,22)(3,21)(4,30)(5,29)(6,28)(7,27)(8,26)(9,25)(10,24)(11,132)(12,131)(13,140)(14,139)(15,138)(16,137)(17,136)(18,135)(19,134)(20,133)(31,32)(33,40)(34,39)(35,38)(36,37)(41,50)(42,49)(43,48)(44,47)(45,46)(51,86)(52,85)(53,84)(54,83)(55,82)(56,81)(57,90)(58,89)(59,88)(60,87)(61,76)(62,75)(63,74)(64,73)(65,72)(66,71)(67,80)(68,79)(69,78)(70,77)(91,106)(92,105)(93,104)(94,103)(95,102)(96,101)(97,110)(98,109)(99,108)(100,107)(111,112)(113,120)(114,119)(115,118)(116,117)(121,130)(122,129)(123,128)(124,127)(125,126)(141,156)(142,155)(143,154)(144,153)(145,152)(146,151)(147,160)(148,159)(149,158)(150,157) );
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)], [(1,103,6,108),(2,102,7,107),(3,101,8,106),(4,110,9,105),(5,109,10,104),(11,57,16,52),(12,56,17,51),(13,55,18,60),(14,54,19,59),(15,53,20,58),(21,96,26,91),(22,95,27,100),(23,94,28,99),(24,93,29,98),(25,92,30,97),(31,112,36,117),(32,111,37,116),(33,120,38,115),(34,119,39,114),(35,118,40,113),(41,130,46,125),(42,129,47,124),(43,128,48,123),(44,127,49,122),(45,126,50,121),(61,156,66,151),(62,155,67,160),(63,154,68,159),(64,153,69,158),(65,152,70,157),(71,146,76,141),(72,145,77,150),(73,144,78,149),(74,143,79,148),(75,142,80,147),(81,136,86,131),(82,135,87,140),(83,134,88,139),(84,133,89,138),(85,132,90,137)], [(1,148,34,14,28,134,48,154),(2,149,35,15,29,135,49,155),(3,150,36,16,30,136,50,156),(4,141,37,17,21,137,41,157),(5,142,38,18,22,138,42,158),(6,143,39,19,23,139,43,159),(7,144,40,20,24,140,44,160),(8,145,31,11,25,131,45,151),(9,146,32,12,26,132,46,152),(10,147,33,13,27,133,47,153),(51,96,85,130,65,110,71,116),(52,97,86,121,66,101,72,117),(53,98,87,122,67,102,73,118),(54,99,88,123,68,103,74,119),(55,100,89,124,69,104,75,120),(56,91,90,125,70,105,76,111),(57,92,81,126,61,106,77,112),(58,93,82,127,62,107,78,113),(59,94,83,128,63,108,79,114),(60,95,84,129,64,109,80,115)], [(1,23),(2,22),(3,21),(4,30),(5,29),(6,28),(7,27),(8,26),(9,25),(10,24),(11,132),(12,131),(13,140),(14,139),(15,138),(16,137),(17,136),(18,135),(19,134),(20,133),(31,32),(33,40),(34,39),(35,38),(36,37),(41,50),(42,49),(43,48),(44,47),(45,46),(51,86),(52,85),(53,84),(54,83),(55,82),(56,81),(57,90),(58,89),(59,88),(60,87),(61,76),(62,75),(63,74),(64,73),(65,72),(66,71),(67,80),(68,79),(69,78),(70,77),(91,106),(92,105),(93,104),(94,103),(95,102),(96,101),(97,110),(98,109),(99,108),(100,107),(111,112),(113,120),(114,119),(115,118),(116,117),(121,130),(122,129),(123,128),(124,127),(125,126),(141,156),(142,155),(143,154),(144,153),(145,152),(146,151),(147,160),(148,159),(149,158),(150,157)]])
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 | 20A | 20B | 20C | 20D | 20E | ··· | 20L | 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 | 20 | 20 | 20 | 20 | 20 | ··· | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 20 | 20 | 2 | 2 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 10 | 10 | 20 | 20 | 2 | 2 | 2 | 2 | 2 | 2 | 10 | 10 | 10 | 10 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 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 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | D4 | D5 | SD16 | C4○D4 | D10 | D10 | C4○D8 | C4×D5 | Q8⋊2D5 | D4×D5 | D5×SD16 | SD16⋊3D5 |
| kernel | Dic5⋊8SD16 | D20⋊6C4 | C10.Q16 | C8×Dic5 | C5×C4.Q8 | Dic5⋊3Q8 | D20⋊8C4 | C2×C40⋊C2 | C40⋊C2 | C2×Dic5 | C4.Q8 | Dic5 | C20 | C4⋊C4 | C2×C8 | C10 | C8 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 8 | 2 | 2 | 4 | 2 | 4 | 2 | 4 | 8 | 2 | 2 | 4 | 4 |
Matrix representation of Dic5⋊8SD16 ►in GL5(𝔽41)
| 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 40 | 34 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 40 |
| 9 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 34 | 40 | 0 | 0 |
| 0 | 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 0 | 9 |
| 40 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 15 | 15 |
| 0 | 0 | 0 | 26 | 15 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 34 | 40 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 40 |
G:=sub<GL(5,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,1,34,0,0,0,0,0,40,0,0,0,0,0,40],[9,0,0,0,0,0,1,34,0,0,0,0,40,0,0,0,0,0,9,0,0,0,0,0,9],[40,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,15,26,0,0,0,15,15],[1,0,0,0,0,0,1,34,0,0,0,0,40,0,0,0,0,0,1,0,0,0,0,0,40] >;
Dic5⋊8SD16 in GAP, Magma, Sage, TeX
{\rm Dic}_5\rtimes_8{\rm SD}_{16} % in TeX
G:=Group("Dic5:8SD16"); // GroupNames label
G:=SmallGroup(320,479);
// by ID
G=gap.SmallGroup(320,479);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,120,135,268,570,297,136,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^10=c^8=d^2=1,b^2=a^5,b*a*b^-1=d*a*d=a^-1,a*c=c*a,b*c=c*b,b*d=d*b,d*c*d=c^3>;
// generators/relations