metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Dic10.37D4, (C2×C10)⋊2Q16, C4⋊C4.69D10, (C2×C20).79D4, C4.103(D4×D5), C22⋊Q8.5D5, C20.156(C2×D4), (C2×Q8).31D10, C10.38(C2×Q16), C5⋊3(C22⋊Q16), C10.49C22≀C2, C10.Q16⋊38C2, C22⋊2(C5⋊Q16), (C22×C10).96D4, (C2×C20).369C23, C20.55D4.9C2, (C22×C4).130D10, C23.63(C5⋊D4), (Q8×C10).49C22, C2.17(C23⋊D10), C2.15(D4.9D10), C10.117(C8.C22), (C22×C20).173C22, (C22×Dic10).14C2, (C2×Dic10).279C22, (C2×C5⋊Q16)⋊9C2, C2.9(C2×C5⋊Q16), (C5×C22⋊Q8).4C2, (C2×C10).500(C2×D4), (C2×C4).57(C5⋊D4), (C5×C4⋊C4).116C22, (C2×C4).469(C22×D5), C22.175(C2×C5⋊D4), (C2×C5⋊2C8).117C22, SmallGroup(320,677)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic10.37D4
G = < a,b,c,d | a20=c4=d2=1, b2=a10, bab-1=a-1, cac-1=a11, ad=da, cbc-1=a5b, bd=db, dcd=a10c-1 >
Subgroups: 526 in 148 conjugacy classes, 47 normal (27 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C5, C8, C2×C4, C2×C4, Q8, C23, C10, C10, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, Q16, C22×C4, C22×C4, C2×Q8, C2×Q8, Dic5, C20, C20, C2×C10, C2×C10, C2×C10, C22⋊C8, Q8⋊C4, C22⋊Q8, C2×Q16, C22×Q8, C5⋊2C8, Dic10, Dic10, C2×Dic5, C2×C20, C2×C20, C5×Q8, C22×C10, C22⋊Q16, C2×C5⋊2C8, C5⋊Q16, C5×C22⋊C4, C5×C4⋊C4, C5×C4⋊C4, C2×Dic10, C2×Dic10, C22×Dic5, C22×C20, Q8×C10, C10.Q16, C20.55D4, C2×C5⋊Q16, C5×C22⋊Q8, C22×Dic10, Dic10.37D4
Quotients: C1, C2, C22, D4, C23, D5, Q16, C2×D4, D10, C22≀C2, C2×Q16, C8.C22, C5⋊D4, C22×D5, C22⋊Q16, C5⋊Q16, D4×D5, C2×C5⋊D4, C23⋊D10, C2×C5⋊Q16, D4.9D10, Dic10.37D4
►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 53 11 43)(2 52 12 42)(3 51 13 41)(4 50 14 60)(5 49 15 59)(6 48 16 58)(7 47 17 57)(8 46 18 56)(9 45 19 55)(10 44 20 54)(21 147 31 157)(22 146 32 156)(23 145 33 155)(24 144 34 154)(25 143 35 153)(26 142 36 152)(27 141 37 151)(28 160 38 150)(29 159 39 149)(30 158 40 148)(61 131 71 121)(62 130 72 140)(63 129 73 139)(64 128 74 138)(65 127 75 137)(66 126 76 136)(67 125 77 135)(68 124 78 134)(69 123 79 133)(70 122 80 132)(81 115 91 105)(82 114 92 104)(83 113 93 103)(84 112 94 102)(85 111 95 101)(86 110 96 120)(87 109 97 119)(88 108 98 118)(89 107 99 117)(90 106 100 116)
(1 112 154 137)(2 103 155 128)(3 114 156 139)(4 105 157 130)(5 116 158 121)(6 107 159 132)(7 118 160 123)(8 109 141 134)(9 120 142 125)(10 111 143 136)(11 102 144 127)(12 113 145 138)(13 104 146 129)(14 115 147 140)(15 106 148 131)(16 117 149 122)(17 108 150 133)(18 119 151 124)(19 110 152 135)(20 101 153 126)(21 67 50 96)(22 78 51 87)(23 69 52 98)(24 80 53 89)(25 71 54 100)(26 62 55 91)(27 73 56 82)(28 64 57 93)(29 75 58 84)(30 66 59 95)(31 77 60 86)(32 68 41 97)(33 79 42 88)(34 70 43 99)(35 61 44 90)(36 72 45 81)(37 63 46 92)(38 74 47 83)(39 65 48 94)(40 76 49 85)
(61 100)(62 81)(63 82)(64 83)(65 84)(66 85)(67 86)(68 87)(69 88)(70 89)(71 90)(72 91)(73 92)(74 93)(75 94)(76 95)(77 96)(78 97)(79 98)(80 99)(101 136)(102 137)(103 138)(104 139)(105 140)(106 121)(107 122)(108 123)(109 124)(110 125)(111 126)(112 127)(113 128)(114 129)(115 130)(116 131)(117 132)(118 133)(119 134)(120 135)
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), (1,53,11,43)(2,52,12,42)(3,51,13,41)(4,50,14,60)(5,49,15,59)(6,48,16,58)(7,47,17,57)(8,46,18,56)(9,45,19,55)(10,44,20,54)(21,147,31,157)(22,146,32,156)(23,145,33,155)(24,144,34,154)(25,143,35,153)(26,142,36,152)(27,141,37,151)(28,160,38,150)(29,159,39,149)(30,158,40,148)(61,131,71,121)(62,130,72,140)(63,129,73,139)(64,128,74,138)(65,127,75,137)(66,126,76,136)(67,125,77,135)(68,124,78,134)(69,123,79,133)(70,122,80,132)(81,115,91,105)(82,114,92,104)(83,113,93,103)(84,112,94,102)(85,111,95,101)(86,110,96,120)(87,109,97,119)(88,108,98,118)(89,107,99,117)(90,106,100,116), (1,112,154,137)(2,103,155,128)(3,114,156,139)(4,105,157,130)(5,116,158,121)(6,107,159,132)(7,118,160,123)(8,109,141,134)(9,120,142,125)(10,111,143,136)(11,102,144,127)(12,113,145,138)(13,104,146,129)(14,115,147,140)(15,106,148,131)(16,117,149,122)(17,108,150,133)(18,119,151,124)(19,110,152,135)(20,101,153,126)(21,67,50,96)(22,78,51,87)(23,69,52,98)(24,80,53,89)(25,71,54,100)(26,62,55,91)(27,73,56,82)(28,64,57,93)(29,75,58,84)(30,66,59,95)(31,77,60,86)(32,68,41,97)(33,79,42,88)(34,70,43,99)(35,61,44,90)(36,72,45,81)(37,63,46,92)(38,74,47,83)(39,65,48,94)(40,76,49,85), (61,100)(62,81)(63,82)(64,83)(65,84)(66,85)(67,86)(68,87)(69,88)(70,89)(71,90)(72,91)(73,92)(74,93)(75,94)(76,95)(77,96)(78,97)(79,98)(80,99)(101,136)(102,137)(103,138)(104,139)(105,140)(106,121)(107,122)(108,123)(109,124)(110,125)(111,126)(112,127)(113,128)(114,129)(115,130)(116,131)(117,132)(118,133)(119,134)(120,135)>;
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), (1,53,11,43)(2,52,12,42)(3,51,13,41)(4,50,14,60)(5,49,15,59)(6,48,16,58)(7,47,17,57)(8,46,18,56)(9,45,19,55)(10,44,20,54)(21,147,31,157)(22,146,32,156)(23,145,33,155)(24,144,34,154)(25,143,35,153)(26,142,36,152)(27,141,37,151)(28,160,38,150)(29,159,39,149)(30,158,40,148)(61,131,71,121)(62,130,72,140)(63,129,73,139)(64,128,74,138)(65,127,75,137)(66,126,76,136)(67,125,77,135)(68,124,78,134)(69,123,79,133)(70,122,80,132)(81,115,91,105)(82,114,92,104)(83,113,93,103)(84,112,94,102)(85,111,95,101)(86,110,96,120)(87,109,97,119)(88,108,98,118)(89,107,99,117)(90,106,100,116), (1,112,154,137)(2,103,155,128)(3,114,156,139)(4,105,157,130)(5,116,158,121)(6,107,159,132)(7,118,160,123)(8,109,141,134)(9,120,142,125)(10,111,143,136)(11,102,144,127)(12,113,145,138)(13,104,146,129)(14,115,147,140)(15,106,148,131)(16,117,149,122)(17,108,150,133)(18,119,151,124)(19,110,152,135)(20,101,153,126)(21,67,50,96)(22,78,51,87)(23,69,52,98)(24,80,53,89)(25,71,54,100)(26,62,55,91)(27,73,56,82)(28,64,57,93)(29,75,58,84)(30,66,59,95)(31,77,60,86)(32,68,41,97)(33,79,42,88)(34,70,43,99)(35,61,44,90)(36,72,45,81)(37,63,46,92)(38,74,47,83)(39,65,48,94)(40,76,49,85), (61,100)(62,81)(63,82)(64,83)(65,84)(66,85)(67,86)(68,87)(69,88)(70,89)(71,90)(72,91)(73,92)(74,93)(75,94)(76,95)(77,96)(78,97)(79,98)(80,99)(101,136)(102,137)(103,138)(104,139)(105,140)(106,121)(107,122)(108,123)(109,124)(110,125)(111,126)(112,127)(113,128)(114,129)(115,130)(116,131)(117,132)(118,133)(119,134)(120,135) );
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)], [(1,53,11,43),(2,52,12,42),(3,51,13,41),(4,50,14,60),(5,49,15,59),(6,48,16,58),(7,47,17,57),(8,46,18,56),(9,45,19,55),(10,44,20,54),(21,147,31,157),(22,146,32,156),(23,145,33,155),(24,144,34,154),(25,143,35,153),(26,142,36,152),(27,141,37,151),(28,160,38,150),(29,159,39,149),(30,158,40,148),(61,131,71,121),(62,130,72,140),(63,129,73,139),(64,128,74,138),(65,127,75,137),(66,126,76,136),(67,125,77,135),(68,124,78,134),(69,123,79,133),(70,122,80,132),(81,115,91,105),(82,114,92,104),(83,113,93,103),(84,112,94,102),(85,111,95,101),(86,110,96,120),(87,109,97,119),(88,108,98,118),(89,107,99,117),(90,106,100,116)], [(1,112,154,137),(2,103,155,128),(3,114,156,139),(4,105,157,130),(5,116,158,121),(6,107,159,132),(7,118,160,123),(8,109,141,134),(9,120,142,125),(10,111,143,136),(11,102,144,127),(12,113,145,138),(13,104,146,129),(14,115,147,140),(15,106,148,131),(16,117,149,122),(17,108,150,133),(18,119,151,124),(19,110,152,135),(20,101,153,126),(21,67,50,96),(22,78,51,87),(23,69,52,98),(24,80,53,89),(25,71,54,100),(26,62,55,91),(27,73,56,82),(28,64,57,93),(29,75,58,84),(30,66,59,95),(31,77,60,86),(32,68,41,97),(33,79,42,88),(34,70,43,99),(35,61,44,90),(36,72,45,81),(37,63,46,92),(38,74,47,83),(39,65,48,94),(40,76,49,85)], [(61,100),(62,81),(63,82),(64,83),(65,84),(66,85),(67,86),(68,87),(69,88),(70,89),(71,90),(72,91),(73,92),(74,93),(75,94),(76,95),(77,96),(78,97),(79,98),(80,99),(101,136),(102,137),(103,138),(104,139),(105,140),(106,121),(107,122),(108,123),(109,124),(110,125),(111,126),(112,127),(113,128),(114,129),(115,130),(116,131),(117,132),(118,133),(119,134),(120,135)]])
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 | ··· | 20H | 20I | ··· | 20P |
| 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 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 8 | 8 | 20 | 20 | 20 | 20 | 2 | 2 | 20 | 20 | 20 | 20 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 8 | ··· | 8 |
47 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | - | + | + | + | - | + | - | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | D5 | Q16 | D10 | D10 | D10 | C5⋊D4 | C5⋊D4 | C8.C22 | D4×D5 | C5⋊Q16 | D4.9D10 |
| kernel | Dic10.37D4 | C10.Q16 | C20.55D4 | C2×C5⋊Q16 | C5×C22⋊Q8 | C22×Dic10 | Dic10 | C2×C20 | C22×C10 | C22⋊Q8 | C2×C10 | C4⋊C4 | C22×C4 | C2×Q8 | C2×C4 | C23 | C10 | C4 | C22 | C2 |
| # reps | 1 | 2 | 1 | 2 | 1 | 1 | 4 | 1 | 1 | 2 | 4 | 2 | 2 | 2 | 4 | 4 | 1 | 4 | 4 | 4 |
Matrix representation of Dic10.37D4 ►in GL6(𝔽41)
| 16 | 0 | 0 | 0 | 0 | 0 |
| 37 | 18 | 0 | 0 | 0 | 0 |
| 0 | 0 | 32 | 0 | 0 | 0 |
| 0 | 0 | 28 | 9 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 11 | 15 | 0 | 0 | 0 | 0 |
| 33 | 30 | 0 | 0 | 0 | 0 |
| 0 | 0 | 37 | 15 | 0 | 0 |
| 0 | 0 | 18 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 37 | 0 | 0 |
| 0 | 0 | 26 | 29 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
G:=sub<GL(6,GF(41))| [16,37,0,0,0,0,0,18,0,0,0,0,0,0,32,28,0,0,0,0,0,9,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[11,33,0,0,0,0,15,30,0,0,0,0,0,0,37,18,0,0,0,0,15,4,0,0,0,0,0,0,1,0,0,0,0,0,0,40],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,26,0,0,0,0,37,29,0,0,0,0,0,0,0,1,0,0,0,0,40,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40] >;
Dic10.37D4 in GAP, Magma, Sage, TeX
{\rm Dic}_{10}._{37}D_4 % in TeX
G:=Group("Dic10.37D4"); // GroupNames label
G:=SmallGroup(320,677);
// by ID
G=gap.SmallGroup(320,677);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,254,219,184,1123,297,136,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^20=c^4=d^2=1,b^2=a^10,b*a*b^-1=a^-1,c*a*c^-1=a^11,a*d=d*a,c*b*c^-1=a^5*b,b*d=d*b,d*c*d=a^10*c^-1>;
// generators/relations