direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D4×Dic10, C42.101D10, C10.552- 1+4, C5⋊1(D4×Q8), (C5×D4)⋊5Q8, C20⋊1(C2×Q8), C20⋊Q8⋊14C2, (C4×D4).10D5, C4⋊1(C2×Dic10), C4.138(D4×D5), C4⋊C4.276D10, C20⋊2Q8⋊21C2, (D4×C20).11C2, C20.344(C2×D4), (C4×Dic10)⋊24C2, (C2×D4).241D10, C20.48D4⋊6C2, (C2×C10).81C24, (D4×Dic5).11C2, Dic5.40(C2×D4), C22⋊1(C2×Dic10), C10.45(C22×D4), C10.12(C22×Q8), (C4×C20).144C22, (C2×C20).153C23, C22⋊C4.104D10, (C22×Dic10)⋊8C2, (C22×C4).201D10, Dic5.14D4⋊6C2, C23.D5.7C22, (D4×C10).248C22, C4⋊Dic5.197C22, (C22×C20).76C22, (C4×Dic5).79C22, C2.14(C22×Dic10), C10.D4.5C22, C23.162(C22×D5), C22.109(C23×D5), (C22×C10).151C23, (C2×Dic5).209C23, C2.13(D4.10D10), (C2×Dic10).242C22, (C22×Dic5).90C22, C2.18(C2×D4×D5), (C2×C10)⋊1(C2×Q8), (C5×C4⋊C4).317C22, (C2×C4).152(C22×D5), (C5×C22⋊C4).103C22, SmallGroup(320,1209)
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=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >
Subgroups: 886 in 280 conjugacy classes, 123 normal (29 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C5, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C10, C10, C42, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×Q8, Dic5, Dic5, C20, C20, C2×C10, C2×C10, C2×C10, C4×D4, C4×D4, C4×Q8, C22⋊Q8, C4⋊Q8, C22×Q8, Dic10, Dic10, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C2×C20, C5×D4, C22×C10, D4×Q8, C4×Dic5, C10.D4, C4⋊Dic5, C4⋊Dic5, C23.D5, C4×C20, C5×C22⋊C4, C5×C4⋊C4, C2×Dic10, C2×Dic10, C2×Dic10, C22×Dic5, C22×C20, D4×C10, C4×Dic10, C20⋊2Q8, Dic5.14D4, C20⋊Q8, C20.48D4, D4×Dic5, D4×C20, C22×Dic10, D4×Dic10
Quotients: C1, C2, C22, D4, Q8, C23, D5, C2×D4, C2×Q8, C24, D10, C22×D4, C22×Q8, 2- 1+4, Dic10, C22×D5, D4×Q8, C2×Dic10, D4×D5, C23×D5, C22×Dic10, C2×D4×D5, D4.10D10, D4×Dic10
►On 160 points
Generators in S160
(1 70 92 55)(2 71 93 56)(3 72 94 57)(4 73 95 58)(5 74 96 59)(6 75 97 60)(7 76 98 41)(8 77 99 42)(9 78 100 43)(10 79 81 44)(11 80 82 45)(12 61 83 46)(13 62 84 47)(14 63 85 48)(15 64 86 49)(16 65 87 50)(17 66 88 51)(18 67 89 52)(19 68 90 53)(20 69 91 54)(21 125 143 114)(22 126 144 115)(23 127 145 116)(24 128 146 117)(25 129 147 118)(26 130 148 119)(27 131 149 120)(28 132 150 101)(29 133 151 102)(30 134 152 103)(31 135 153 104)(32 136 154 105)(33 137 155 106)(34 138 156 107)(35 139 157 108)(36 140 158 109)(37 121 159 110)(38 122 160 111)(39 123 141 112)(40 124 142 113)
(1 45)(2 46)(3 47)(4 48)(5 49)(6 50)(7 51)(8 52)(9 53)(10 54)(11 55)(12 56)(13 57)(14 58)(15 59)(16 60)(17 41)(18 42)(19 43)(20 44)(21 104)(22 105)(23 106)(24 107)(25 108)(26 109)(27 110)(28 111)(29 112)(30 113)(31 114)(32 115)(33 116)(34 117)(35 118)(36 119)(37 120)(38 101)(39 102)(40 103)(61 93)(62 94)(63 95)(64 96)(65 97)(66 98)(67 99)(68 100)(69 81)(70 82)(71 83)(72 84)(73 85)(74 86)(75 87)(76 88)(77 89)(78 90)(79 91)(80 92)(121 149)(122 150)(123 151)(124 152)(125 153)(126 154)(127 155)(128 156)(129 157)(130 158)(131 159)(132 160)(133 141)(134 142)(135 143)(136 144)(137 145)(138 146)(139 147)(140 148)
(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 30 11 40)(2 29 12 39)(3 28 13 38)(4 27 14 37)(5 26 15 36)(6 25 16 35)(7 24 17 34)(8 23 18 33)(9 22 19 32)(10 21 20 31)(41 117 51 107)(42 116 52 106)(43 115 53 105)(44 114 54 104)(45 113 55 103)(46 112 56 102)(47 111 57 101)(48 110 58 120)(49 109 59 119)(50 108 60 118)(61 123 71 133)(62 122 72 132)(63 121 73 131)(64 140 74 130)(65 139 75 129)(66 138 76 128)(67 137 77 127)(68 136 78 126)(69 135 79 125)(70 134 80 124)(81 143 91 153)(82 142 92 152)(83 141 93 151)(84 160 94 150)(85 159 95 149)(86 158 96 148)(87 157 97 147)(88 156 98 146)(89 155 99 145)(90 154 100 144)
G:=sub<Sym(160)| (1,70,92,55)(2,71,93,56)(3,72,94,57)(4,73,95,58)(5,74,96,59)(6,75,97,60)(7,76,98,41)(8,77,99,42)(9,78,100,43)(10,79,81,44)(11,80,82,45)(12,61,83,46)(13,62,84,47)(14,63,85,48)(15,64,86,49)(16,65,87,50)(17,66,88,51)(18,67,89,52)(19,68,90,53)(20,69,91,54)(21,125,143,114)(22,126,144,115)(23,127,145,116)(24,128,146,117)(25,129,147,118)(26,130,148,119)(27,131,149,120)(28,132,150,101)(29,133,151,102)(30,134,152,103)(31,135,153,104)(32,136,154,105)(33,137,155,106)(34,138,156,107)(35,139,157,108)(36,140,158,109)(37,121,159,110)(38,122,160,111)(39,123,141,112)(40,124,142,113), (1,45)(2,46)(3,47)(4,48)(5,49)(6,50)(7,51)(8,52)(9,53)(10,54)(11,55)(12,56)(13,57)(14,58)(15,59)(16,60)(17,41)(18,42)(19,43)(20,44)(21,104)(22,105)(23,106)(24,107)(25,108)(26,109)(27,110)(28,111)(29,112)(30,113)(31,114)(32,115)(33,116)(34,117)(35,118)(36,119)(37,120)(38,101)(39,102)(40,103)(61,93)(62,94)(63,95)(64,96)(65,97)(66,98)(67,99)(68,100)(69,81)(70,82)(71,83)(72,84)(73,85)(74,86)(75,87)(76,88)(77,89)(78,90)(79,91)(80,92)(121,149)(122,150)(123,151)(124,152)(125,153)(126,154)(127,155)(128,156)(129,157)(130,158)(131,159)(132,160)(133,141)(134,142)(135,143)(136,144)(137,145)(138,146)(139,147)(140,148), (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), (1,30,11,40)(2,29,12,39)(3,28,13,38)(4,27,14,37)(5,26,15,36)(6,25,16,35)(7,24,17,34)(8,23,18,33)(9,22,19,32)(10,21,20,31)(41,117,51,107)(42,116,52,106)(43,115,53,105)(44,114,54,104)(45,113,55,103)(46,112,56,102)(47,111,57,101)(48,110,58,120)(49,109,59,119)(50,108,60,118)(61,123,71,133)(62,122,72,132)(63,121,73,131)(64,140,74,130)(65,139,75,129)(66,138,76,128)(67,137,77,127)(68,136,78,126)(69,135,79,125)(70,134,80,124)(81,143,91,153)(82,142,92,152)(83,141,93,151)(84,160,94,150)(85,159,95,149)(86,158,96,148)(87,157,97,147)(88,156,98,146)(89,155,99,145)(90,154,100,144)>;
G:=Group( (1,70,92,55)(2,71,93,56)(3,72,94,57)(4,73,95,58)(5,74,96,59)(6,75,97,60)(7,76,98,41)(8,77,99,42)(9,78,100,43)(10,79,81,44)(11,80,82,45)(12,61,83,46)(13,62,84,47)(14,63,85,48)(15,64,86,49)(16,65,87,50)(17,66,88,51)(18,67,89,52)(19,68,90,53)(20,69,91,54)(21,125,143,114)(22,126,144,115)(23,127,145,116)(24,128,146,117)(25,129,147,118)(26,130,148,119)(27,131,149,120)(28,132,150,101)(29,133,151,102)(30,134,152,103)(31,135,153,104)(32,136,154,105)(33,137,155,106)(34,138,156,107)(35,139,157,108)(36,140,158,109)(37,121,159,110)(38,122,160,111)(39,123,141,112)(40,124,142,113), (1,45)(2,46)(3,47)(4,48)(5,49)(6,50)(7,51)(8,52)(9,53)(10,54)(11,55)(12,56)(13,57)(14,58)(15,59)(16,60)(17,41)(18,42)(19,43)(20,44)(21,104)(22,105)(23,106)(24,107)(25,108)(26,109)(27,110)(28,111)(29,112)(30,113)(31,114)(32,115)(33,116)(34,117)(35,118)(36,119)(37,120)(38,101)(39,102)(40,103)(61,93)(62,94)(63,95)(64,96)(65,97)(66,98)(67,99)(68,100)(69,81)(70,82)(71,83)(72,84)(73,85)(74,86)(75,87)(76,88)(77,89)(78,90)(79,91)(80,92)(121,149)(122,150)(123,151)(124,152)(125,153)(126,154)(127,155)(128,156)(129,157)(130,158)(131,159)(132,160)(133,141)(134,142)(135,143)(136,144)(137,145)(138,146)(139,147)(140,148), (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), (1,30,11,40)(2,29,12,39)(3,28,13,38)(4,27,14,37)(5,26,15,36)(6,25,16,35)(7,24,17,34)(8,23,18,33)(9,22,19,32)(10,21,20,31)(41,117,51,107)(42,116,52,106)(43,115,53,105)(44,114,54,104)(45,113,55,103)(46,112,56,102)(47,111,57,101)(48,110,58,120)(49,109,59,119)(50,108,60,118)(61,123,71,133)(62,122,72,132)(63,121,73,131)(64,140,74,130)(65,139,75,129)(66,138,76,128)(67,137,77,127)(68,136,78,126)(69,135,79,125)(70,134,80,124)(81,143,91,153)(82,142,92,152)(83,141,93,151)(84,160,94,150)(85,159,95,149)(86,158,96,148)(87,157,97,147)(88,156,98,146)(89,155,99,145)(90,154,100,144) );
G=PermutationGroup([[(1,70,92,55),(2,71,93,56),(3,72,94,57),(4,73,95,58),(5,74,96,59),(6,75,97,60),(7,76,98,41),(8,77,99,42),(9,78,100,43),(10,79,81,44),(11,80,82,45),(12,61,83,46),(13,62,84,47),(14,63,85,48),(15,64,86,49),(16,65,87,50),(17,66,88,51),(18,67,89,52),(19,68,90,53),(20,69,91,54),(21,125,143,114),(22,126,144,115),(23,127,145,116),(24,128,146,117),(25,129,147,118),(26,130,148,119),(27,131,149,120),(28,132,150,101),(29,133,151,102),(30,134,152,103),(31,135,153,104),(32,136,154,105),(33,137,155,106),(34,138,156,107),(35,139,157,108),(36,140,158,109),(37,121,159,110),(38,122,160,111),(39,123,141,112),(40,124,142,113)], [(1,45),(2,46),(3,47),(4,48),(5,49),(6,50),(7,51),(8,52),(9,53),(10,54),(11,55),(12,56),(13,57),(14,58),(15,59),(16,60),(17,41),(18,42),(19,43),(20,44),(21,104),(22,105),(23,106),(24,107),(25,108),(26,109),(27,110),(28,111),(29,112),(30,113),(31,114),(32,115),(33,116),(34,117),(35,118),(36,119),(37,120),(38,101),(39,102),(40,103),(61,93),(62,94),(63,95),(64,96),(65,97),(66,98),(67,99),(68,100),(69,81),(70,82),(71,83),(72,84),(73,85),(74,86),(75,87),(76,88),(77,89),(78,90),(79,91),(80,92),(121,149),(122,150),(123,151),(124,152),(125,153),(126,154),(127,155),(128,156),(129,157),(130,158),(131,159),(132,160),(133,141),(134,142),(135,143),(136,144),(137,145),(138,146),(139,147),(140,148)], [(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)], [(1,30,11,40),(2,29,12,39),(3,28,13,38),(4,27,14,37),(5,26,15,36),(6,25,16,35),(7,24,17,34),(8,23,18,33),(9,22,19,32),(10,21,20,31),(41,117,51,107),(42,116,52,106),(43,115,53,105),(44,114,54,104),(45,113,55,103),(46,112,56,102),(47,111,57,101),(48,110,58,120),(49,109,59,119),(50,108,60,118),(61,123,71,133),(62,122,72,132),(63,121,73,131),(64,140,74,130),(65,139,75,129),(66,138,76,128),(67,137,77,127),(68,136,78,126),(69,135,79,125),(70,134,80,124),(81,143,91,153),(82,142,92,152),(83,141,93,151),(84,160,94,150),(85,159,95,149),(86,158,96,148),(87,157,97,147),(88,156,98,146),(89,155,99,145),(90,154,100,144)]])
65 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | ··· | 4Q | 5A | 5B | 10A | ··· | 10F | 10G | ··· | 10N | 20A | ··· | 20H | 20I | ··· | 20X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 5 | 5 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 10 | 10 | 10 | 10 | 20 | ··· | 20 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
65 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | - | + | + | + | + | + | + | - | - | + | - |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | Q8 | D5 | D10 | D10 | D10 | D10 | D10 | Dic10 | 2- 1+4 | D4×D5 | D4.10D10 |
| kernel | D4×Dic10 | C4×Dic10 | C20⋊2Q8 | Dic5.14D4 | C20⋊Q8 | C20.48D4 | D4×Dic5 | D4×C20 | C22×Dic10 | Dic10 | C5×D4 | C4×D4 | C42 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×D4 | D4 | C10 | C4 | C2 |
| # reps | 1 | 1 | 1 | 4 | 2 | 2 | 2 | 1 | 2 | 4 | 4 | 2 | 2 | 4 | 2 | 4 | 2 | 16 | 1 | 4 | 4 |
Matrix representation of D4×Dic10 ►in GL4(𝔽41) generated by
| 40 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 40 | 9 |
| 0 | 0 | 18 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 40 | 9 |
| 0 | 0 | 0 | 1 |
| 9 | 11 | 0 | 0 |
| 30 | 14 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 24 | 0 | 0 |
| 17 | 40 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 40 |
G:=sub<GL(4,GF(41))| [40,0,0,0,0,40,0,0,0,0,40,18,0,0,9,1],[1,0,0,0,0,1,0,0,0,0,40,0,0,0,9,1],[9,30,0,0,11,14,0,0,0,0,1,0,0,0,0,1],[1,17,0,0,24,40,0,0,0,0,40,0,0,0,0,40] >;
D4×Dic10 in GAP, Magma, Sage, TeX
D_4\times {\rm Dic}_{10} % in TeX
G:=Group("D4xDic10"); // GroupNames label
G:=SmallGroup(320,1209);
// by ID
G=gap.SmallGroup(320,1209);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,387,675,80,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^2=c^20=1,d^2=c^10,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations