metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D10.13D8, C2.D8⋊2D5, C2.13(D5×D8), C4⋊C4.47D10, C10.29(C2×D8), (C2×C8).27D10, C4⋊D20.8C2, D10⋊1C8⋊25C2, D20⋊5C4⋊26C2, D20⋊6C4⋊20C2, C4.80(C4○D20), C20.37(C4○D4), C10.D8⋊21C2, (C2×Dic5).56D4, C22.228(D4×D5), C5⋊3(C22.D8), (C2×C20).298C23, (C2×C40).241C22, C4.28(Q8⋊2D5), (C2×D20).85C22, (C22×D5).122D4, C2.22(Q16⋊D5), C10.70(C8.C22), C4⋊Dic5.124C22, C2.15(D10.13D4), C10.45(C22.D4), (D5×C4⋊C4)⋊7C2, (C5×C2.D8)⋊11C2, (C2×C4×D5).42C22, (C2×C10).303(C2×D4), (C5×C4⋊C4).91C22, (C2×C5⋊2C8).69C22, (C2×C4).401(C22×D5), SmallGroup(320,509)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D10.13D8
G = < a,b,c,d | a10=b2=c8=1, d2=a5, bab=a-1, ac=ca, ad=da, cbc-1=dbd-1=a5b, dcd-1=c-1 >
Subgroups: 574 in 114 conjugacy classes, 39 normal (37 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, C23, D5, C10, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, C22×C4, C2×D4, Dic5, C20, C20, D10, D10, C2×C10, C22⋊C8, D4⋊C4, C2.D8, C2.D8, C2×C4⋊C4, C4⋊D4, C5⋊2C8, C40, C4×D5, D20, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C22×D5, C22×D5, C22.D8, C2×C5⋊2C8, C10.D4, C4⋊Dic5, D10⋊C4, C5×C4⋊C4, C2×C40, C2×C4×D5, C2×C4×D5, C2×D20, C2×D20, C10.D8, D20⋊6C4, D10⋊1C8, D20⋊5C4, C5×C2.D8, D5×C4⋊C4, C4⋊D20, D10.13D8
Quotients: C1, C2, C22, D4, C23, D5, D8, C2×D4, C4○D4, D10, C22.D4, C2×D8, C8.C22, C22×D5, C22.D8, C4○D20, D4×D5, Q8⋊2D5, D10.13D4, D5×D8, Q16⋊D5, D10.13D8
►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)
(1 152)(2 151)(3 160)(4 159)(5 158)(6 157)(7 156)(8 155)(9 154)(10 153)(11 134)(12 133)(13 132)(14 131)(15 140)(16 139)(17 138)(18 137)(19 136)(20 135)(21 44)(22 43)(23 42)(24 41)(25 50)(26 49)(27 48)(28 47)(29 46)(30 45)(31 145)(32 144)(33 143)(34 142)(35 141)(36 150)(37 149)(38 148)(39 147)(40 146)(51 99)(52 98)(53 97)(54 96)(55 95)(56 94)(57 93)(58 92)(59 91)(60 100)(61 120)(62 119)(63 118)(64 117)(65 116)(66 115)(67 114)(68 113)(69 112)(70 111)(71 110)(72 109)(73 108)(74 107)(75 106)(76 105)(77 104)(78 103)(79 102)(80 101)(81 123)(82 122)(83 121)(84 130)(85 129)(86 128)(87 127)(88 126)(89 125)(90 124)
(1 53 13 73 33 90 50 64)(2 54 14 74 34 81 41 65)(3 55 15 75 35 82 42 66)(4 56 16 76 36 83 43 67)(5 57 17 77 37 84 44 68)(6 58 18 78 38 85 45 69)(7 59 19 79 39 86 46 70)(8 60 20 80 40 87 47 61)(9 51 11 71 31 88 48 62)(10 52 12 72 32 89 49 63)(21 118 158 98 138 109 149 125)(22 119 159 99 139 110 150 126)(23 120 160 100 140 101 141 127)(24 111 151 91 131 102 142 128)(25 112 152 92 132 103 143 129)(26 113 153 93 133 104 144 130)(27 114 154 94 134 105 145 121)(28 115 155 95 135 106 146 122)(29 116 156 96 136 107 147 123)(30 117 157 97 137 108 148 124)
(1 138 6 133)(2 139 7 134)(3 140 8 135)(4 131 9 136)(5 132 10 137)(11 156 16 151)(12 157 17 152)(13 158 18 153)(14 159 19 154)(15 160 20 155)(21 38 26 33)(22 39 27 34)(23 40 28 35)(24 31 29 36)(25 32 30 37)(41 150 46 145)(42 141 47 146)(43 142 48 147)(44 143 49 148)(45 144 50 149)(51 96 56 91)(52 97 57 92)(53 98 58 93)(54 99 59 94)(55 100 60 95)(61 106 66 101)(62 107 67 102)(63 108 68 103)(64 109 69 104)(65 110 70 105)(71 116 76 111)(72 117 77 112)(73 118 78 113)(74 119 79 114)(75 120 80 115)(81 126 86 121)(82 127 87 122)(83 128 88 123)(84 129 89 124)(85 130 90 125)
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), (1,152)(2,151)(3,160)(4,159)(5,158)(6,157)(7,156)(8,155)(9,154)(10,153)(11,134)(12,133)(13,132)(14,131)(15,140)(16,139)(17,138)(18,137)(19,136)(20,135)(21,44)(22,43)(23,42)(24,41)(25,50)(26,49)(27,48)(28,47)(29,46)(30,45)(31,145)(32,144)(33,143)(34,142)(35,141)(36,150)(37,149)(38,148)(39,147)(40,146)(51,99)(52,98)(53,97)(54,96)(55,95)(56,94)(57,93)(58,92)(59,91)(60,100)(61,120)(62,119)(63,118)(64,117)(65,116)(66,115)(67,114)(68,113)(69,112)(70,111)(71,110)(72,109)(73,108)(74,107)(75,106)(76,105)(77,104)(78,103)(79,102)(80,101)(81,123)(82,122)(83,121)(84,130)(85,129)(86,128)(87,127)(88,126)(89,125)(90,124), (1,53,13,73,33,90,50,64)(2,54,14,74,34,81,41,65)(3,55,15,75,35,82,42,66)(4,56,16,76,36,83,43,67)(5,57,17,77,37,84,44,68)(6,58,18,78,38,85,45,69)(7,59,19,79,39,86,46,70)(8,60,20,80,40,87,47,61)(9,51,11,71,31,88,48,62)(10,52,12,72,32,89,49,63)(21,118,158,98,138,109,149,125)(22,119,159,99,139,110,150,126)(23,120,160,100,140,101,141,127)(24,111,151,91,131,102,142,128)(25,112,152,92,132,103,143,129)(26,113,153,93,133,104,144,130)(27,114,154,94,134,105,145,121)(28,115,155,95,135,106,146,122)(29,116,156,96,136,107,147,123)(30,117,157,97,137,108,148,124), (1,138,6,133)(2,139,7,134)(3,140,8,135)(4,131,9,136)(5,132,10,137)(11,156,16,151)(12,157,17,152)(13,158,18,153)(14,159,19,154)(15,160,20,155)(21,38,26,33)(22,39,27,34)(23,40,28,35)(24,31,29,36)(25,32,30,37)(41,150,46,145)(42,141,47,146)(43,142,48,147)(44,143,49,148)(45,144,50,149)(51,96,56,91)(52,97,57,92)(53,98,58,93)(54,99,59,94)(55,100,60,95)(61,106,66,101)(62,107,67,102)(63,108,68,103)(64,109,69,104)(65,110,70,105)(71,116,76,111)(72,117,77,112)(73,118,78,113)(74,119,79,114)(75,120,80,115)(81,126,86,121)(82,127,87,122)(83,128,88,123)(84,129,89,124)(85,130,90,125)>;
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), (1,152)(2,151)(3,160)(4,159)(5,158)(6,157)(7,156)(8,155)(9,154)(10,153)(11,134)(12,133)(13,132)(14,131)(15,140)(16,139)(17,138)(18,137)(19,136)(20,135)(21,44)(22,43)(23,42)(24,41)(25,50)(26,49)(27,48)(28,47)(29,46)(30,45)(31,145)(32,144)(33,143)(34,142)(35,141)(36,150)(37,149)(38,148)(39,147)(40,146)(51,99)(52,98)(53,97)(54,96)(55,95)(56,94)(57,93)(58,92)(59,91)(60,100)(61,120)(62,119)(63,118)(64,117)(65,116)(66,115)(67,114)(68,113)(69,112)(70,111)(71,110)(72,109)(73,108)(74,107)(75,106)(76,105)(77,104)(78,103)(79,102)(80,101)(81,123)(82,122)(83,121)(84,130)(85,129)(86,128)(87,127)(88,126)(89,125)(90,124), (1,53,13,73,33,90,50,64)(2,54,14,74,34,81,41,65)(3,55,15,75,35,82,42,66)(4,56,16,76,36,83,43,67)(5,57,17,77,37,84,44,68)(6,58,18,78,38,85,45,69)(7,59,19,79,39,86,46,70)(8,60,20,80,40,87,47,61)(9,51,11,71,31,88,48,62)(10,52,12,72,32,89,49,63)(21,118,158,98,138,109,149,125)(22,119,159,99,139,110,150,126)(23,120,160,100,140,101,141,127)(24,111,151,91,131,102,142,128)(25,112,152,92,132,103,143,129)(26,113,153,93,133,104,144,130)(27,114,154,94,134,105,145,121)(28,115,155,95,135,106,146,122)(29,116,156,96,136,107,147,123)(30,117,157,97,137,108,148,124), (1,138,6,133)(2,139,7,134)(3,140,8,135)(4,131,9,136)(5,132,10,137)(11,156,16,151)(12,157,17,152)(13,158,18,153)(14,159,19,154)(15,160,20,155)(21,38,26,33)(22,39,27,34)(23,40,28,35)(24,31,29,36)(25,32,30,37)(41,150,46,145)(42,141,47,146)(43,142,48,147)(44,143,49,148)(45,144,50,149)(51,96,56,91)(52,97,57,92)(53,98,58,93)(54,99,59,94)(55,100,60,95)(61,106,66,101)(62,107,67,102)(63,108,68,103)(64,109,69,104)(65,110,70,105)(71,116,76,111)(72,117,77,112)(73,118,78,113)(74,119,79,114)(75,120,80,115)(81,126,86,121)(82,127,87,122)(83,128,88,123)(84,129,89,124)(85,130,90,125) );
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)], [(1,152),(2,151),(3,160),(4,159),(5,158),(6,157),(7,156),(8,155),(9,154),(10,153),(11,134),(12,133),(13,132),(14,131),(15,140),(16,139),(17,138),(18,137),(19,136),(20,135),(21,44),(22,43),(23,42),(24,41),(25,50),(26,49),(27,48),(28,47),(29,46),(30,45),(31,145),(32,144),(33,143),(34,142),(35,141),(36,150),(37,149),(38,148),(39,147),(40,146),(51,99),(52,98),(53,97),(54,96),(55,95),(56,94),(57,93),(58,92),(59,91),(60,100),(61,120),(62,119),(63,118),(64,117),(65,116),(66,115),(67,114),(68,113),(69,112),(70,111),(71,110),(72,109),(73,108),(74,107),(75,106),(76,105),(77,104),(78,103),(79,102),(80,101),(81,123),(82,122),(83,121),(84,130),(85,129),(86,128),(87,127),(88,126),(89,125),(90,124)], [(1,53,13,73,33,90,50,64),(2,54,14,74,34,81,41,65),(3,55,15,75,35,82,42,66),(4,56,16,76,36,83,43,67),(5,57,17,77,37,84,44,68),(6,58,18,78,38,85,45,69),(7,59,19,79,39,86,46,70),(8,60,20,80,40,87,47,61),(9,51,11,71,31,88,48,62),(10,52,12,72,32,89,49,63),(21,118,158,98,138,109,149,125),(22,119,159,99,139,110,150,126),(23,120,160,100,140,101,141,127),(24,111,151,91,131,102,142,128),(25,112,152,92,132,103,143,129),(26,113,153,93,133,104,144,130),(27,114,154,94,134,105,145,121),(28,115,155,95,135,106,146,122),(29,116,156,96,136,107,147,123),(30,117,157,97,137,108,148,124)], [(1,138,6,133),(2,139,7,134),(3,140,8,135),(4,131,9,136),(5,132,10,137),(11,156,16,151),(12,157,17,152),(13,158,18,153),(14,159,19,154),(15,160,20,155),(21,38,26,33),(22,39,27,34),(23,40,28,35),(24,31,29,36),(25,32,30,37),(41,150,46,145),(42,141,47,146),(43,142,48,147),(44,143,49,148),(45,144,50,149),(51,96,56,91),(52,97,57,92),(53,98,58,93),(54,99,59,94),(55,100,60,95),(61,106,66,101),(62,107,67,102),(63,108,68,103),(64,109,69,104),(65,110,70,105),(71,116,76,111),(72,117,77,112),(73,118,78,113),(74,119,79,114),(75,120,80,115),(81,126,86,121),(82,127,87,122),(83,128,88,123),(84,129,89,124),(85,130,90,125)]])
47 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 5A | 5B | 8A | 8B | 8C | 8D | 10A | ··· | 10F | 20A | 20B | 20C | 20D | 20E | ··· | 20L | 40A | ··· | 40H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 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 | 10 | 10 | 40 | 2 | 2 | 4 | 4 | 8 | 20 | 20 | 20 | 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 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D5 | C4○D4 | D8 | D10 | D10 | C4○D20 | C8.C22 | Q8⋊2D5 | D4×D5 | D5×D8 | Q16⋊D5 |
| kernel | D10.13D8 | C10.D8 | D20⋊6C4 | D10⋊1C8 | D20⋊5C4 | C5×C2.D8 | D5×C4⋊C4 | C4⋊D20 | C2×Dic5 | C22×D5 | C2.D8 | C20 | D10 | C4⋊C4 | C2×C8 | C4 | C10 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 4 | 4 | 4 | 2 | 8 | 1 | 2 | 2 | 4 | 4 |
Matrix representation of D10.13D8 ►in GL6(𝔽41)
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 35 | 35 | 0 | 0 |
| 0 | 0 | 6 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 9 | 0 | 0 | 0 | 0 |
| 32 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 35 | 35 | 0 | 0 |
| 0 | 0 | 40 | 6 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 7 |
| 0 | 0 | 0 | 0 | 35 | 24 |
| 0 | 32 | 0 | 0 | 0 | 0 |
| 32 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 4 |
| 0 | 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,35,6,0,0,0,0,35,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,32,0,0,0,0,9,0,0,0,0,0,0,0,35,40,0,0,0,0,35,6,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,35,0,0,0,0,7,24],[0,32,0,0,0,0,32,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,4,1] >;
D10.13D8 in GAP, Magma, Sage, TeX
D_{10}._{13}D_8 % in TeX
G:=Group("D10.13D8"); // GroupNames label
G:=SmallGroup(320,509);
// by ID
G=gap.SmallGroup(320,509);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,64,254,219,268,851,102,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^10=b^2=c^8=1,d^2=a^5,b*a*b=a^-1,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d^-1=a^5*b,d*c*d^-1=c^-1>;
// generators/relations