metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D4⋊2Dic10, Dic5.13SD16, C20⋊Q8⋊3C2, (C5×D4)⋊2Q8, C4⋊C4.3D10, C20.2(C2×Q8), C40⋊6C4⋊10C2, C5⋊1(D4⋊2Q8), C2.9(D5×SD16), D4⋊C4.5D5, (C2×C8).112D10, C20.Q8⋊2C2, C20.8Q8⋊9C2, (D4×Dic5).4C2, C4.2(C2×Dic10), (C2×D4).127D10, C2.8(D8⋊D5), C10.20(C2×SD16), D4⋊Dic5.2C2, C22.163(D4×D5), C10.8(C22⋊Q8), C20.146(C4○D4), C4.75(D4⋊2D5), C10.25(C8⋊C22), (C2×C40).123C22, (C2×C20).201C23, (C2×Dic5).191D4, (D4×C10).22C22, C4⋊Dic5.61C22, (C4×Dic5).11C22, C2.13(Dic5.14D4), (C5×C4⋊C4).6C22, (C5×D4⋊C4).5C2, (C2×C10).214(C2×D4), (C2×C5⋊2C8).7C22, (C2×C4).308(C22×D5), SmallGroup(320,388)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D4⋊Dic10
G = < a,b,c,d | a4=b2=c20=1, d2=c10, bab=cac-1=a-1, ad=da, cbc-1=a-1b, dbd-1=a2b, dcd-1=c-1 >
Subgroups: 422 in 108 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, C4⋊C4, C2×C8, C2×C8, C22×C4, C2×D4, C2×Q8, Dic5, Dic5, C20, C20, C2×C10, C2×C10, D4⋊C4, D4⋊C4, C4⋊C8, C4.Q8, C4×D4, C4⋊Q8, C5⋊2C8, C40, Dic10, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C5×D4, C5×D4, C22×C10, D4⋊2Q8, C2×C5⋊2C8, C4×Dic5, C10.D4, C4⋊Dic5, C23.D5, C5×C4⋊C4, C2×C40, C2×Dic10, C22×Dic5, D4×C10, C20.Q8, C20.8Q8, C40⋊6C4, D4⋊Dic5, C5×D4⋊C4, C20⋊Q8, D4×Dic5, D4⋊Dic10
Quotients: C1, C2, C22, D4, Q8, C23, D5, SD16, C2×D4, C2×Q8, C4○D4, D10, C22⋊Q8, C2×SD16, C8⋊C22, Dic10, C22×D5, D4⋊2Q8, C2×Dic10, D4×D5, D4⋊2D5, Dic5.14D4, D8⋊D5, D5×SD16, D4⋊Dic10
►On 160 points
Generators in S160
(1 52 68 99)(2 100 69 53)(3 54 70 81)(4 82 71 55)(5 56 72 83)(6 84 73 57)(7 58 74 85)(8 86 75 59)(9 60 76 87)(10 88 77 41)(11 42 78 89)(12 90 79 43)(13 44 80 91)(14 92 61 45)(15 46 62 93)(16 94 63 47)(17 48 64 95)(18 96 65 49)(19 50 66 97)(20 98 67 51)(21 134 159 117)(22 118 160 135)(23 136 141 119)(24 120 142 137)(25 138 143 101)(26 102 144 139)(27 140 145 103)(28 104 146 121)(29 122 147 105)(30 106 148 123)(31 124 149 107)(32 108 150 125)(33 126 151 109)(34 110 152 127)(35 128 153 111)(36 112 154 129)(37 130 155 113)(38 114 156 131)(39 132 157 115)(40 116 158 133)
(1 42)(2 79)(3 44)(4 61)(5 46)(6 63)(7 48)(8 65)(9 50)(10 67)(11 52)(12 69)(13 54)(14 71)(15 56)(16 73)(17 58)(18 75)(19 60)(20 77)(21 107)(22 32)(23 109)(24 34)(25 111)(26 36)(27 113)(28 38)(29 115)(30 40)(31 117)(33 119)(35 101)(37 103)(39 105)(41 51)(43 53)(45 55)(47 57)(49 59)(62 83)(64 85)(66 87)(68 89)(70 91)(72 93)(74 95)(76 97)(78 99)(80 81)(82 92)(84 94)(86 96)(88 98)(90 100)(102 129)(104 131)(106 133)(108 135)(110 137)(112 139)(114 121)(116 123)(118 125)(120 127)(122 157)(124 159)(126 141)(128 143)(130 145)(132 147)(134 149)(136 151)(138 153)(140 155)(142 152)(144 154)(146 156)(148 158)(150 160)
(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 39 11 29)(2 38 12 28)(3 37 13 27)(4 36 14 26)(5 35 15 25)(6 34 16 24)(7 33 17 23)(8 32 18 22)(9 31 19 21)(10 30 20 40)(41 123 51 133)(42 122 52 132)(43 121 53 131)(44 140 54 130)(45 139 55 129)(46 138 56 128)(47 137 57 127)(48 136 58 126)(49 135 59 125)(50 134 60 124)(61 144 71 154)(62 143 72 153)(63 142 73 152)(64 141 74 151)(65 160 75 150)(66 159 76 149)(67 158 77 148)(68 157 78 147)(69 156 79 146)(70 155 80 145)(81 113 91 103)(82 112 92 102)(83 111 93 101)(84 110 94 120)(85 109 95 119)(86 108 96 118)(87 107 97 117)(88 106 98 116)(89 105 99 115)(90 104 100 114)
G:=sub<Sym(160)| (1,52,68,99)(2,100,69,53)(3,54,70,81)(4,82,71,55)(5,56,72,83)(6,84,73,57)(7,58,74,85)(8,86,75,59)(9,60,76,87)(10,88,77,41)(11,42,78,89)(12,90,79,43)(13,44,80,91)(14,92,61,45)(15,46,62,93)(16,94,63,47)(17,48,64,95)(18,96,65,49)(19,50,66,97)(20,98,67,51)(21,134,159,117)(22,118,160,135)(23,136,141,119)(24,120,142,137)(25,138,143,101)(26,102,144,139)(27,140,145,103)(28,104,146,121)(29,122,147,105)(30,106,148,123)(31,124,149,107)(32,108,150,125)(33,126,151,109)(34,110,152,127)(35,128,153,111)(36,112,154,129)(37,130,155,113)(38,114,156,131)(39,132,157,115)(40,116,158,133), (1,42)(2,79)(3,44)(4,61)(5,46)(6,63)(7,48)(8,65)(9,50)(10,67)(11,52)(12,69)(13,54)(14,71)(15,56)(16,73)(17,58)(18,75)(19,60)(20,77)(21,107)(22,32)(23,109)(24,34)(25,111)(26,36)(27,113)(28,38)(29,115)(30,40)(31,117)(33,119)(35,101)(37,103)(39,105)(41,51)(43,53)(45,55)(47,57)(49,59)(62,83)(64,85)(66,87)(68,89)(70,91)(72,93)(74,95)(76,97)(78,99)(80,81)(82,92)(84,94)(86,96)(88,98)(90,100)(102,129)(104,131)(106,133)(108,135)(110,137)(112,139)(114,121)(116,123)(118,125)(120,127)(122,157)(124,159)(126,141)(128,143)(130,145)(132,147)(134,149)(136,151)(138,153)(140,155)(142,152)(144,154)(146,156)(148,158)(150,160), (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), (1,39,11,29)(2,38,12,28)(3,37,13,27)(4,36,14,26)(5,35,15,25)(6,34,16,24)(7,33,17,23)(8,32,18,22)(9,31,19,21)(10,30,20,40)(41,123,51,133)(42,122,52,132)(43,121,53,131)(44,140,54,130)(45,139,55,129)(46,138,56,128)(47,137,57,127)(48,136,58,126)(49,135,59,125)(50,134,60,124)(61,144,71,154)(62,143,72,153)(63,142,73,152)(64,141,74,151)(65,160,75,150)(66,159,76,149)(67,158,77,148)(68,157,78,147)(69,156,79,146)(70,155,80,145)(81,113,91,103)(82,112,92,102)(83,111,93,101)(84,110,94,120)(85,109,95,119)(86,108,96,118)(87,107,97,117)(88,106,98,116)(89,105,99,115)(90,104,100,114)>;
G:=Group( (1,52,68,99)(2,100,69,53)(3,54,70,81)(4,82,71,55)(5,56,72,83)(6,84,73,57)(7,58,74,85)(8,86,75,59)(9,60,76,87)(10,88,77,41)(11,42,78,89)(12,90,79,43)(13,44,80,91)(14,92,61,45)(15,46,62,93)(16,94,63,47)(17,48,64,95)(18,96,65,49)(19,50,66,97)(20,98,67,51)(21,134,159,117)(22,118,160,135)(23,136,141,119)(24,120,142,137)(25,138,143,101)(26,102,144,139)(27,140,145,103)(28,104,146,121)(29,122,147,105)(30,106,148,123)(31,124,149,107)(32,108,150,125)(33,126,151,109)(34,110,152,127)(35,128,153,111)(36,112,154,129)(37,130,155,113)(38,114,156,131)(39,132,157,115)(40,116,158,133), (1,42)(2,79)(3,44)(4,61)(5,46)(6,63)(7,48)(8,65)(9,50)(10,67)(11,52)(12,69)(13,54)(14,71)(15,56)(16,73)(17,58)(18,75)(19,60)(20,77)(21,107)(22,32)(23,109)(24,34)(25,111)(26,36)(27,113)(28,38)(29,115)(30,40)(31,117)(33,119)(35,101)(37,103)(39,105)(41,51)(43,53)(45,55)(47,57)(49,59)(62,83)(64,85)(66,87)(68,89)(70,91)(72,93)(74,95)(76,97)(78,99)(80,81)(82,92)(84,94)(86,96)(88,98)(90,100)(102,129)(104,131)(106,133)(108,135)(110,137)(112,139)(114,121)(116,123)(118,125)(120,127)(122,157)(124,159)(126,141)(128,143)(130,145)(132,147)(134,149)(136,151)(138,153)(140,155)(142,152)(144,154)(146,156)(148,158)(150,160), (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), (1,39,11,29)(2,38,12,28)(3,37,13,27)(4,36,14,26)(5,35,15,25)(6,34,16,24)(7,33,17,23)(8,32,18,22)(9,31,19,21)(10,30,20,40)(41,123,51,133)(42,122,52,132)(43,121,53,131)(44,140,54,130)(45,139,55,129)(46,138,56,128)(47,137,57,127)(48,136,58,126)(49,135,59,125)(50,134,60,124)(61,144,71,154)(62,143,72,153)(63,142,73,152)(64,141,74,151)(65,160,75,150)(66,159,76,149)(67,158,77,148)(68,157,78,147)(69,156,79,146)(70,155,80,145)(81,113,91,103)(82,112,92,102)(83,111,93,101)(84,110,94,120)(85,109,95,119)(86,108,96,118)(87,107,97,117)(88,106,98,116)(89,105,99,115)(90,104,100,114) );
G=PermutationGroup([[(1,52,68,99),(2,100,69,53),(3,54,70,81),(4,82,71,55),(5,56,72,83),(6,84,73,57),(7,58,74,85),(8,86,75,59),(9,60,76,87),(10,88,77,41),(11,42,78,89),(12,90,79,43),(13,44,80,91),(14,92,61,45),(15,46,62,93),(16,94,63,47),(17,48,64,95),(18,96,65,49),(19,50,66,97),(20,98,67,51),(21,134,159,117),(22,118,160,135),(23,136,141,119),(24,120,142,137),(25,138,143,101),(26,102,144,139),(27,140,145,103),(28,104,146,121),(29,122,147,105),(30,106,148,123),(31,124,149,107),(32,108,150,125),(33,126,151,109),(34,110,152,127),(35,128,153,111),(36,112,154,129),(37,130,155,113),(38,114,156,131),(39,132,157,115),(40,116,158,133)], [(1,42),(2,79),(3,44),(4,61),(5,46),(6,63),(7,48),(8,65),(9,50),(10,67),(11,52),(12,69),(13,54),(14,71),(15,56),(16,73),(17,58),(18,75),(19,60),(20,77),(21,107),(22,32),(23,109),(24,34),(25,111),(26,36),(27,113),(28,38),(29,115),(30,40),(31,117),(33,119),(35,101),(37,103),(39,105),(41,51),(43,53),(45,55),(47,57),(49,59),(62,83),(64,85),(66,87),(68,89),(70,91),(72,93),(74,95),(76,97),(78,99),(80,81),(82,92),(84,94),(86,96),(88,98),(90,100),(102,129),(104,131),(106,133),(108,135),(110,137),(112,139),(114,121),(116,123),(118,125),(120,127),(122,157),(124,159),(126,141),(128,143),(130,145),(132,147),(134,149),(136,151),(138,153),(140,155),(142,152),(144,154),(146,156),(148,158),(150,160)], [(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)], [(1,39,11,29),(2,38,12,28),(3,37,13,27),(4,36,14,26),(5,35,15,25),(6,34,16,24),(7,33,17,23),(8,32,18,22),(9,31,19,21),(10,30,20,40),(41,123,51,133),(42,122,52,132),(43,121,53,131),(44,140,54,130),(45,139,55,129),(46,138,56,128),(47,137,57,127),(48,136,58,126),(49,135,59,125),(50,134,60,124),(61,144,71,154),(62,143,72,153),(63,142,73,152),(64,141,74,151),(65,160,75,150),(66,159,76,149),(67,158,77,148),(68,157,78,147),(69,156,79,146),(70,155,80,145),(81,113,91,103),(82,112,92,102),(83,111,93,101),(84,110,94,120),(85,109,95,119),(86,108,96,118),(87,107,97,117),(88,106,98,116),(89,105,99,115),(90,104,100,114)]])
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 | Q8 | D5 | SD16 | C4○D4 | D10 | D10 | D10 | Dic10 | C8⋊C22 | D4⋊2D5 | D4×D5 | D8⋊D5 | D5×SD16 |
| kernel | D4⋊Dic10 | C20.Q8 | C20.8Q8 | C40⋊6C4 | D4⋊Dic5 | C5×D4⋊C4 | C20⋊Q8 | D4×Dic5 | C2×Dic5 | C5×D4 | D4⋊C4 | Dic5 | C20 | C4⋊C4 | C2×C8 | C2×D4 | 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 D4⋊Dic10 ►in GL4(𝔽41) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 40 | 0 |
| 40 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
| 9 | 11 | 0 | 0 |
| 30 | 14 | 0 | 0 |
| 0 | 0 | 26 | 26 |
| 0 | 0 | 26 | 15 |
| 22 | 22 | 0 | 0 |
| 32 | 19 | 0 | 0 |
| 0 | 0 | 0 | 40 |
| 0 | 0 | 1 | 0 |
G:=sub<GL(4,GF(41))| [1,0,0,0,0,1,0,0,0,0,0,40,0,0,1,0],[40,0,0,0,0,40,0,0,0,0,0,1,0,0,1,0],[9,30,0,0,11,14,0,0,0,0,26,26,0,0,26,15],[22,32,0,0,22,19,0,0,0,0,0,1,0,0,40,0] >;
D4⋊Dic10 in GAP, Magma, Sage, TeX
D_4\rtimes {\rm Dic}_{10} % in TeX
G:=Group("D4:Dic10"); // GroupNames label
G:=SmallGroup(320,388);
// by ID
G=gap.SmallGroup(320,388);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,254,219,226,851,438,102,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^2=c^20=1,d^2=c^10,b*a*b=c*a*c^-1=a^-1,a*d=d*a,c*b*c^-1=a^-1*b,d*b*d^-1=a^2*b,d*c*d^-1=c^-1>;
// generators/relations