metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D8.10D10, D20.47D4, C20.18C24, C40.39C23, Q16.12D10, SD16.2D10, Dic10.47D4, Dic10.12C23, Dic20.16C22, C4○D8⋊6D5, C5⋊3(Q8○D8), (D5×Q16)⋊7C2, C5⋊D4.3D4, D8⋊3D5⋊7C2, C4.145(D4×D5), D4.D5.C22, C4○D4.13D10, D10.54(C2×D4), C20.351(C2×D4), (C2×C8).106D10, SD16⋊D5⋊6C2, C5⋊2C8.9C23, (C8×D5).8C22, C4.18(C23×D5), C8.18(C22×D5), C22.10(D4×D5), (C2×Dic20)⋊23C2, D20.3C4⋊8C2, D4.9D10⋊8C2, (Q8×D5).2C22, Dic5.60(C2×D4), (C4×D5).11C23, (C5×D4).12C23, D4.12(C22×D5), (C5×D8).10C22, C8⋊D5.2C22, D4.10D10⋊6C2, (C5×Q8).12C23, Q8.12(C22×D5), C5⋊Q16.2C22, (C2×C40).106C22, (C2×C20).535C23, C4○D20.56C22, D4⋊2D5.2C22, C10.119(C22×D4), (C5×Q16).12C22, (C5×SD16).2C22, C4.Dic5.49C22, (C2×Dic10).206C22, C2.92(C2×D4×D5), (C5×C4○D8)⋊6C2, (C2×C10).15(C2×D4), (C5×C4○D4).23C22, (C2×C4).234(C22×D5), SmallGroup(320,1443)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 854 in 248 conjugacy classes, 99 normal (31 characteristic)
C1, C2, C2 [×5], C4 [×2], C4 [×8], C22, C22 [×4], C5, C8 [×2], C8 [×2], C2×C4, C2×C4 [×14], D4 [×2], D4 [×9], Q8 [×2], Q8 [×11], D5 [×2], C10, C10 [×3], C2×C8, C2×C8 [×2], M4(2) [×3], D8, SD16 [×2], SD16 [×4], Q16, Q16 [×8], C2×Q8 [×8], C4○D4 [×2], C4○D4 [×11], Dic5 [×2], Dic5 [×4], C20 [×2], C20 [×2], D10 [×2], C2×C10, C2×C10 [×2], C8○D4, C2×Q16 [×3], C4○D8, C4○D8 [×2], C8.C22 [×6], 2- (1+4) [×2], C5⋊2C8 [×2], C40 [×2], Dic10, Dic10 [×4], Dic10 [×6], C4×D5 [×2], C4×D5 [×4], D20, C2×Dic5 [×6], C5⋊D4 [×2], C5⋊D4 [×4], C2×C20, C2×C20 [×2], C5×D4 [×2], C5×D4 [×2], C5×Q8 [×2], Q8○D8, C8×D5 [×2], C8⋊D5 [×2], Dic20 [×4], C4.Dic5, D4.D5 [×4], C5⋊Q16 [×4], C2×C40, C5×D8, C5×SD16 [×2], C5×Q16, C2×Dic10 [×2], C2×Dic10 [×2], C4○D20, C4○D20 [×2], D4⋊2D5 [×4], D4⋊2D5 [×4], Q8×D5 [×4], C5×C4○D4 [×2], D20.3C4, C2×Dic20, D8⋊3D5 [×2], SD16⋊D5 [×4], D5×Q16 [×2], D4.9D10 [×2], C5×C4○D8, D4.10D10 [×2], D8.10D10
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D5, C2×D4 [×6], C24, D10 [×7], C22×D4, C22×D5 [×7], Q8○D8, D4×D5 [×2], C23×D5, C2×D4×D5, D8.10D10
Generators and relations
G = < a,b,c,d | a8=b2=c10=1, d2=a4, bab=a-1, ac=ca, ad=da, cbc-1=a4b, bd=db, dcd-1=a4c-1 >
►On 160 points
Generators in S160
(1 95 30 153 123 86 60 13)(2 96 21 154 124 87 51 14)(3 97 22 155 125 88 52 15)(4 98 23 156 126 89 53 16)(5 99 24 157 127 90 54 17)(6 100 25 158 128 81 55 18)(7 91 26 159 129 82 56 19)(8 92 27 160 130 83 57 20)(9 93 28 151 121 84 58 11)(10 94 29 152 122 85 59 12)(31 144 104 41 61 114 134 71)(32 145 105 42 62 115 135 72)(33 146 106 43 63 116 136 73)(34 147 107 44 64 117 137 74)(35 148 108 45 65 118 138 75)(36 149 109 46 66 119 139 76)(37 150 110 47 67 120 140 77)(38 141 101 48 68 111 131 78)(39 142 102 49 69 112 132 79)(40 143 103 50 70 113 133 80)
(1 148)(2 119)(3 150)(4 111)(5 142)(6 113)(7 144)(8 115)(9 146)(10 117)(11 106)(12 137)(13 108)(14 139)(15 110)(16 131)(17 102)(18 133)(19 104)(20 135)(21 46)(22 77)(23 48)(24 79)(25 50)(26 71)(27 42)(28 73)(29 44)(30 75)(31 91)(32 83)(33 93)(34 85)(35 95)(36 87)(37 97)(38 89)(39 99)(40 81)(41 56)(43 58)(45 60)(47 52)(49 54)(51 76)(53 78)(55 80)(57 72)(59 74)(61 82)(62 92)(63 84)(64 94)(65 86)(66 96)(67 88)(68 98)(69 90)(70 100)(101 156)(103 158)(105 160)(107 152)(109 154)(112 127)(114 129)(116 121)(118 123)(120 125)(122 147)(124 149)(126 141)(128 143)(130 145)(132 157)(134 159)(136 151)(138 153)(140 155)
(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 10 123 122)(2 121 124 9)(3 8 125 130)(4 129 126 7)(5 6 127 128)(11 14 151 154)(12 153 152 13)(15 20 155 160)(16 159 156 19)(17 18 157 158)(21 58 51 28)(22 27 52 57)(23 56 53 26)(24 25 54 55)(29 60 59 30)(31 68 61 38)(32 37 62 67)(33 66 63 36)(34 35 64 65)(39 70 69 40)(41 78 71 48)(42 47 72 77)(43 76 73 46)(44 45 74 75)(49 80 79 50)(81 99 100 90)(82 89 91 98)(83 97 92 88)(84 87 93 96)(85 95 94 86)(101 104 131 134)(102 133 132 103)(105 110 135 140)(106 139 136 109)(107 108 137 138)(111 114 141 144)(112 143 142 113)(115 120 145 150)(116 149 146 119)(117 118 147 148)
G:=sub<Sym(160)| (1,95,30,153,123,86,60,13)(2,96,21,154,124,87,51,14)(3,97,22,155,125,88,52,15)(4,98,23,156,126,89,53,16)(5,99,24,157,127,90,54,17)(6,100,25,158,128,81,55,18)(7,91,26,159,129,82,56,19)(8,92,27,160,130,83,57,20)(9,93,28,151,121,84,58,11)(10,94,29,152,122,85,59,12)(31,144,104,41,61,114,134,71)(32,145,105,42,62,115,135,72)(33,146,106,43,63,116,136,73)(34,147,107,44,64,117,137,74)(35,148,108,45,65,118,138,75)(36,149,109,46,66,119,139,76)(37,150,110,47,67,120,140,77)(38,141,101,48,68,111,131,78)(39,142,102,49,69,112,132,79)(40,143,103,50,70,113,133,80), (1,148)(2,119)(3,150)(4,111)(5,142)(6,113)(7,144)(8,115)(9,146)(10,117)(11,106)(12,137)(13,108)(14,139)(15,110)(16,131)(17,102)(18,133)(19,104)(20,135)(21,46)(22,77)(23,48)(24,79)(25,50)(26,71)(27,42)(28,73)(29,44)(30,75)(31,91)(32,83)(33,93)(34,85)(35,95)(36,87)(37,97)(38,89)(39,99)(40,81)(41,56)(43,58)(45,60)(47,52)(49,54)(51,76)(53,78)(55,80)(57,72)(59,74)(61,82)(62,92)(63,84)(64,94)(65,86)(66,96)(67,88)(68,98)(69,90)(70,100)(101,156)(103,158)(105,160)(107,152)(109,154)(112,127)(114,129)(116,121)(118,123)(120,125)(122,147)(124,149)(126,141)(128,143)(130,145)(132,157)(134,159)(136,151)(138,153)(140,155), (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), (1,10,123,122)(2,121,124,9)(3,8,125,130)(4,129,126,7)(5,6,127,128)(11,14,151,154)(12,153,152,13)(15,20,155,160)(16,159,156,19)(17,18,157,158)(21,58,51,28)(22,27,52,57)(23,56,53,26)(24,25,54,55)(29,60,59,30)(31,68,61,38)(32,37,62,67)(33,66,63,36)(34,35,64,65)(39,70,69,40)(41,78,71,48)(42,47,72,77)(43,76,73,46)(44,45,74,75)(49,80,79,50)(81,99,100,90)(82,89,91,98)(83,97,92,88)(84,87,93,96)(85,95,94,86)(101,104,131,134)(102,133,132,103)(105,110,135,140)(106,139,136,109)(107,108,137,138)(111,114,141,144)(112,143,142,113)(115,120,145,150)(116,149,146,119)(117,118,147,148)>;
G:=Group( (1,95,30,153,123,86,60,13)(2,96,21,154,124,87,51,14)(3,97,22,155,125,88,52,15)(4,98,23,156,126,89,53,16)(5,99,24,157,127,90,54,17)(6,100,25,158,128,81,55,18)(7,91,26,159,129,82,56,19)(8,92,27,160,130,83,57,20)(9,93,28,151,121,84,58,11)(10,94,29,152,122,85,59,12)(31,144,104,41,61,114,134,71)(32,145,105,42,62,115,135,72)(33,146,106,43,63,116,136,73)(34,147,107,44,64,117,137,74)(35,148,108,45,65,118,138,75)(36,149,109,46,66,119,139,76)(37,150,110,47,67,120,140,77)(38,141,101,48,68,111,131,78)(39,142,102,49,69,112,132,79)(40,143,103,50,70,113,133,80), (1,148)(2,119)(3,150)(4,111)(5,142)(6,113)(7,144)(8,115)(9,146)(10,117)(11,106)(12,137)(13,108)(14,139)(15,110)(16,131)(17,102)(18,133)(19,104)(20,135)(21,46)(22,77)(23,48)(24,79)(25,50)(26,71)(27,42)(28,73)(29,44)(30,75)(31,91)(32,83)(33,93)(34,85)(35,95)(36,87)(37,97)(38,89)(39,99)(40,81)(41,56)(43,58)(45,60)(47,52)(49,54)(51,76)(53,78)(55,80)(57,72)(59,74)(61,82)(62,92)(63,84)(64,94)(65,86)(66,96)(67,88)(68,98)(69,90)(70,100)(101,156)(103,158)(105,160)(107,152)(109,154)(112,127)(114,129)(116,121)(118,123)(120,125)(122,147)(124,149)(126,141)(128,143)(130,145)(132,157)(134,159)(136,151)(138,153)(140,155), (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), (1,10,123,122)(2,121,124,9)(3,8,125,130)(4,129,126,7)(5,6,127,128)(11,14,151,154)(12,153,152,13)(15,20,155,160)(16,159,156,19)(17,18,157,158)(21,58,51,28)(22,27,52,57)(23,56,53,26)(24,25,54,55)(29,60,59,30)(31,68,61,38)(32,37,62,67)(33,66,63,36)(34,35,64,65)(39,70,69,40)(41,78,71,48)(42,47,72,77)(43,76,73,46)(44,45,74,75)(49,80,79,50)(81,99,100,90)(82,89,91,98)(83,97,92,88)(84,87,93,96)(85,95,94,86)(101,104,131,134)(102,133,132,103)(105,110,135,140)(106,139,136,109)(107,108,137,138)(111,114,141,144)(112,143,142,113)(115,120,145,150)(116,149,146,119)(117,118,147,148) );
G=PermutationGroup([(1,95,30,153,123,86,60,13),(2,96,21,154,124,87,51,14),(3,97,22,155,125,88,52,15),(4,98,23,156,126,89,53,16),(5,99,24,157,127,90,54,17),(6,100,25,158,128,81,55,18),(7,91,26,159,129,82,56,19),(8,92,27,160,130,83,57,20),(9,93,28,151,121,84,58,11),(10,94,29,152,122,85,59,12),(31,144,104,41,61,114,134,71),(32,145,105,42,62,115,135,72),(33,146,106,43,63,116,136,73),(34,147,107,44,64,117,137,74),(35,148,108,45,65,118,138,75),(36,149,109,46,66,119,139,76),(37,150,110,47,67,120,140,77),(38,141,101,48,68,111,131,78),(39,142,102,49,69,112,132,79),(40,143,103,50,70,113,133,80)], [(1,148),(2,119),(3,150),(4,111),(5,142),(6,113),(7,144),(8,115),(9,146),(10,117),(11,106),(12,137),(13,108),(14,139),(15,110),(16,131),(17,102),(18,133),(19,104),(20,135),(21,46),(22,77),(23,48),(24,79),(25,50),(26,71),(27,42),(28,73),(29,44),(30,75),(31,91),(32,83),(33,93),(34,85),(35,95),(36,87),(37,97),(38,89),(39,99),(40,81),(41,56),(43,58),(45,60),(47,52),(49,54),(51,76),(53,78),(55,80),(57,72),(59,74),(61,82),(62,92),(63,84),(64,94),(65,86),(66,96),(67,88),(68,98),(69,90),(70,100),(101,156),(103,158),(105,160),(107,152),(109,154),(112,127),(114,129),(116,121),(118,123),(120,125),(122,147),(124,149),(126,141),(128,143),(130,145),(132,157),(134,159),(136,151),(138,153),(140,155)], [(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)], [(1,10,123,122),(2,121,124,9),(3,8,125,130),(4,129,126,7),(5,6,127,128),(11,14,151,154),(12,153,152,13),(15,20,155,160),(16,159,156,19),(17,18,157,158),(21,58,51,28),(22,27,52,57),(23,56,53,26),(24,25,54,55),(29,60,59,30),(31,68,61,38),(32,37,62,67),(33,66,63,36),(34,35,64,65),(39,70,69,40),(41,78,71,48),(42,47,72,77),(43,76,73,46),(44,45,74,75),(49,80,79,50),(81,99,100,90),(82,89,91,98),(83,97,92,88),(84,87,93,96),(85,95,94,86),(101,104,131,134),(102,133,132,103),(105,110,135,140),(106,139,136,109),(107,108,137,138),(111,114,141,144),(112,143,142,113),(115,120,145,150),(116,149,146,119),(117,118,147,148)])
Matrix representation ►G ⊆ GL4(𝔽41) generated by
| 0 | 0 | 33 | 8 |
| 0 | 0 | 33 | 0 |
| 0 | 36 | 24 | 0 |
| 5 | 36 | 0 | 24 |
| 5 | 29 | 31 | 28 |
| 12 | 36 | 10 | 18 |
| 6 | 18 | 34 | 12 |
| 24 | 24 | 27 | 7 |
| 39 | 16 | 23 | 15 |
| 25 | 25 | 6 | 38 |
| 7 | 7 | 32 | 25 |
| 1 | 14 | 5 | 27 |
| 39 | 13 | 21 | 15 |
| 25 | 2 | 18 | 38 |
| 7 | 35 | 14 | 25 |
| 1 | 33 | 25 | 27 |
G:=sub<GL(4,GF(41))| [0,0,0,5,0,0,36,36,33,33,24,0,8,0,0,24],[5,12,6,24,29,36,18,24,31,10,34,27,28,18,12,7],[39,25,7,1,16,25,7,14,23,6,32,5,15,38,25,27],[39,25,7,1,13,2,35,33,21,18,14,25,15,38,25,27] >;
50 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 5A | 5B | 8A | 8B | 8C | 8D | 8E | 10A | 10B | 10C | 10D | 10E | 10F | 10G | 10H | 20A | 20B | 20C | 20D | 20E | 20F | 20G | 20H | 20I | 20J | 40A | ··· | 40H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 8 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 2 | 4 | 4 | 10 | 10 | 2 | 2 | 4 | 4 | 10 | 10 | 20 | 20 | 20 | 20 | 2 | 2 | 2 | 2 | 4 | 20 | 20 | 2 | 2 | 4 | 4 | 8 | 8 | 8 | 8 | 2 | 2 | 2 | 2 | 4 | 4 | 8 | 8 | 8 | 8 | 4 | ··· | 4 |
50 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 | C2 | D4 | D4 | D4 | D5 | D10 | D10 | D10 | D10 | D10 | Q8○D8 | D4×D5 | D4×D5 | D8.10D10 |
| kernel | D8.10D10 | D20.3C4 | C2×Dic20 | D8⋊3D5 | SD16⋊D5 | D5×Q16 | D4.9D10 | C5×C4○D8 | D4.10D10 | Dic10 | D20 | C5⋊D4 | C4○D8 | C2×C8 | D8 | SD16 | Q16 | C4○D4 | C5 | C4 | C22 | C1 |
| # reps | 1 | 1 | 1 | 2 | 4 | 2 | 2 | 1 | 2 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 2 | 4 | 2 | 2 | 2 | 8 |
In GAP, Magma, Sage, TeX
D_8._{10}D_{10} % in TeX
G:=Group("D8.10D10"); // GroupNames label
G:=SmallGroup(320,1443);
// by ID
G=gap.SmallGroup(320,1443);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,477,387,184,570,185,136,438,235,102,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^8=b^2=c^10=1,d^2=a^4,b*a*b=a^-1,a*c=c*a,a*d=d*a,c*b*c^-1=a^4*b,b*d=d*b,d*c*d^-1=a^4*c^-1>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀