metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D4.2Dic10, C4⋊C4.7D10, C40⋊5C4⋊7C2, (C2×C8).7D10, C5⋊2(D4.Q8), (C5×D4).2Q8, C20.4(C2×Q8), D4⋊C4.3D5, (C2×C40).7C22, C20.Q8⋊3C2, C4.Dic10⋊3C2, C20.8Q8⋊5C2, (D4×Dic5).6C2, C4.4(C2×Dic10), (C2×D4).129D10, C10.21(C4○D8), C2.6(D8⋊3D5), C2.9(D40⋊C2), D4⋊Dic5.5C2, C22.168(D4×D5), C20.148(C4○D4), C4.77(D4⋊2D5), C10.54(C8⋊C22), (C2×C20).206C23, (C2×Dic5).194D4, (D4×C10).27C22, C10.10(C22⋊Q8), C4⋊Dic5.65C22, (C4×Dic5).16C22, C2.15(Dic5.14D4), (C5×D4⋊C4).3C2, (C2×C10).219(C2×D4), (C5×C4⋊C4).11C22, (C2×C5⋊2C8).12C22, (C2×C4).313(C22×D5), SmallGroup(320,393)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D4.2Dic10
G = < a,b,c,d | a4=b2=c20=1, d2=a2c10, bab=cac-1=a-1, ad=da, cbc-1=a-1b, bd=db, dcd-1=c-1 >
Subgroups: 374 in 102 conjugacy classes, 41 normal (37 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, D4, C23, C10, C10, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, C22×C4, C2×D4, Dic5, C20, C20, C2×C10, C2×C10, D4⋊C4, D4⋊C4, C4⋊C8, C4.Q8, C2.D8, C4×D4, C42.C2, C5⋊2C8, C40, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C5×D4, C5×D4, C22×C10, D4.Q8, C2×C5⋊2C8, C4×Dic5, C10.D4, C4⋊Dic5, C4⋊Dic5, C23.D5, C5×C4⋊C4, C2×C40, C22×Dic5, D4×C10, C20.Q8, C20.8Q8, C40⋊5C4, D4⋊Dic5, C5×D4⋊C4, C4.Dic10, D4×Dic5, D4.2Dic10
Quotients: C1, C2, C22, D4, Q8, C23, D5, C2×D4, C2×Q8, C4○D4, D10, C22⋊Q8, C4○D8, C8⋊C22, Dic10, C22×D5, D4.Q8, C2×Dic10, D4×D5, D4⋊2D5, Dic5.14D4, D8⋊3D5, D40⋊C2, D4.2Dic10
►On 160 points
Generators in S160
(1 112 121 149)(2 150 122 113)(3 114 123 151)(4 152 124 115)(5 116 125 153)(6 154 126 117)(7 118 127 155)(8 156 128 119)(9 120 129 157)(10 158 130 101)(11 102 131 159)(12 160 132 103)(13 104 133 141)(14 142 134 105)(15 106 135 143)(16 144 136 107)(17 108 137 145)(18 146 138 109)(19 110 139 147)(20 148 140 111)(21 69 94 53)(22 54 95 70)(23 71 96 55)(24 56 97 72)(25 73 98 57)(26 58 99 74)(27 75 100 59)(28 60 81 76)(29 77 82 41)(30 42 83 78)(31 79 84 43)(32 44 85 80)(33 61 86 45)(34 46 87 62)(35 63 88 47)(36 48 89 64)(37 65 90 49)(38 50 91 66)(39 67 92 51)(40 52 93 68)
(1 102)(2 132)(3 104)(4 134)(5 106)(6 136)(7 108)(8 138)(9 110)(10 140)(11 112)(12 122)(13 114)(14 124)(15 116)(16 126)(17 118)(18 128)(19 120)(20 130)(21 43)(22 32)(23 45)(24 34)(25 47)(26 36)(27 49)(28 38)(29 51)(30 40)(31 53)(33 55)(35 57)(37 59)(39 41)(42 68)(44 70)(46 72)(48 74)(50 76)(52 78)(54 80)(56 62)(58 64)(60 66)(61 96)(63 98)(65 100)(67 82)(69 84)(71 86)(73 88)(75 90)(77 92)(79 94)(81 91)(83 93)(85 95)(87 97)(89 99)(101 111)(103 113)(105 115)(107 117)(109 119)(121 159)(123 141)(125 143)(127 145)(129 147)(131 149)(133 151)(135 153)(137 155)(139 157)(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 45 131 71)(2 44 132 70)(3 43 133 69)(4 42 134 68)(5 41 135 67)(6 60 136 66)(7 59 137 65)(8 58 138 64)(9 57 139 63)(10 56 140 62)(11 55 121 61)(12 54 122 80)(13 53 123 79)(14 52 124 78)(15 51 125 77)(16 50 126 76)(17 49 127 75)(18 48 128 74)(19 47 129 73)(20 46 130 72)(21 151 84 104)(22 150 85 103)(23 149 86 102)(24 148 87 101)(25 147 88 120)(26 146 89 119)(27 145 90 118)(28 144 91 117)(29 143 92 116)(30 142 93 115)(31 141 94 114)(32 160 95 113)(33 159 96 112)(34 158 97 111)(35 157 98 110)(36 156 99 109)(37 155 100 108)(38 154 81 107)(39 153 82 106)(40 152 83 105)
G:=sub<Sym(160)| (1,112,121,149)(2,150,122,113)(3,114,123,151)(4,152,124,115)(5,116,125,153)(6,154,126,117)(7,118,127,155)(8,156,128,119)(9,120,129,157)(10,158,130,101)(11,102,131,159)(12,160,132,103)(13,104,133,141)(14,142,134,105)(15,106,135,143)(16,144,136,107)(17,108,137,145)(18,146,138,109)(19,110,139,147)(20,148,140,111)(21,69,94,53)(22,54,95,70)(23,71,96,55)(24,56,97,72)(25,73,98,57)(26,58,99,74)(27,75,100,59)(28,60,81,76)(29,77,82,41)(30,42,83,78)(31,79,84,43)(32,44,85,80)(33,61,86,45)(34,46,87,62)(35,63,88,47)(36,48,89,64)(37,65,90,49)(38,50,91,66)(39,67,92,51)(40,52,93,68), (1,102)(2,132)(3,104)(4,134)(5,106)(6,136)(7,108)(8,138)(9,110)(10,140)(11,112)(12,122)(13,114)(14,124)(15,116)(16,126)(17,118)(18,128)(19,120)(20,130)(21,43)(22,32)(23,45)(24,34)(25,47)(26,36)(27,49)(28,38)(29,51)(30,40)(31,53)(33,55)(35,57)(37,59)(39,41)(42,68)(44,70)(46,72)(48,74)(50,76)(52,78)(54,80)(56,62)(58,64)(60,66)(61,96)(63,98)(65,100)(67,82)(69,84)(71,86)(73,88)(75,90)(77,92)(79,94)(81,91)(83,93)(85,95)(87,97)(89,99)(101,111)(103,113)(105,115)(107,117)(109,119)(121,159)(123,141)(125,143)(127,145)(129,147)(131,149)(133,151)(135,153)(137,155)(139,157)(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), (1,45,131,71)(2,44,132,70)(3,43,133,69)(4,42,134,68)(5,41,135,67)(6,60,136,66)(7,59,137,65)(8,58,138,64)(9,57,139,63)(10,56,140,62)(11,55,121,61)(12,54,122,80)(13,53,123,79)(14,52,124,78)(15,51,125,77)(16,50,126,76)(17,49,127,75)(18,48,128,74)(19,47,129,73)(20,46,130,72)(21,151,84,104)(22,150,85,103)(23,149,86,102)(24,148,87,101)(25,147,88,120)(26,146,89,119)(27,145,90,118)(28,144,91,117)(29,143,92,116)(30,142,93,115)(31,141,94,114)(32,160,95,113)(33,159,96,112)(34,158,97,111)(35,157,98,110)(36,156,99,109)(37,155,100,108)(38,154,81,107)(39,153,82,106)(40,152,83,105)>;
G:=Group( (1,112,121,149)(2,150,122,113)(3,114,123,151)(4,152,124,115)(5,116,125,153)(6,154,126,117)(7,118,127,155)(8,156,128,119)(9,120,129,157)(10,158,130,101)(11,102,131,159)(12,160,132,103)(13,104,133,141)(14,142,134,105)(15,106,135,143)(16,144,136,107)(17,108,137,145)(18,146,138,109)(19,110,139,147)(20,148,140,111)(21,69,94,53)(22,54,95,70)(23,71,96,55)(24,56,97,72)(25,73,98,57)(26,58,99,74)(27,75,100,59)(28,60,81,76)(29,77,82,41)(30,42,83,78)(31,79,84,43)(32,44,85,80)(33,61,86,45)(34,46,87,62)(35,63,88,47)(36,48,89,64)(37,65,90,49)(38,50,91,66)(39,67,92,51)(40,52,93,68), (1,102)(2,132)(3,104)(4,134)(5,106)(6,136)(7,108)(8,138)(9,110)(10,140)(11,112)(12,122)(13,114)(14,124)(15,116)(16,126)(17,118)(18,128)(19,120)(20,130)(21,43)(22,32)(23,45)(24,34)(25,47)(26,36)(27,49)(28,38)(29,51)(30,40)(31,53)(33,55)(35,57)(37,59)(39,41)(42,68)(44,70)(46,72)(48,74)(50,76)(52,78)(54,80)(56,62)(58,64)(60,66)(61,96)(63,98)(65,100)(67,82)(69,84)(71,86)(73,88)(75,90)(77,92)(79,94)(81,91)(83,93)(85,95)(87,97)(89,99)(101,111)(103,113)(105,115)(107,117)(109,119)(121,159)(123,141)(125,143)(127,145)(129,147)(131,149)(133,151)(135,153)(137,155)(139,157)(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), (1,45,131,71)(2,44,132,70)(3,43,133,69)(4,42,134,68)(5,41,135,67)(6,60,136,66)(7,59,137,65)(8,58,138,64)(9,57,139,63)(10,56,140,62)(11,55,121,61)(12,54,122,80)(13,53,123,79)(14,52,124,78)(15,51,125,77)(16,50,126,76)(17,49,127,75)(18,48,128,74)(19,47,129,73)(20,46,130,72)(21,151,84,104)(22,150,85,103)(23,149,86,102)(24,148,87,101)(25,147,88,120)(26,146,89,119)(27,145,90,118)(28,144,91,117)(29,143,92,116)(30,142,93,115)(31,141,94,114)(32,160,95,113)(33,159,96,112)(34,158,97,111)(35,157,98,110)(36,156,99,109)(37,155,100,108)(38,154,81,107)(39,153,82,106)(40,152,83,105) );
G=PermutationGroup([[(1,112,121,149),(2,150,122,113),(3,114,123,151),(4,152,124,115),(5,116,125,153),(6,154,126,117),(7,118,127,155),(8,156,128,119),(9,120,129,157),(10,158,130,101),(11,102,131,159),(12,160,132,103),(13,104,133,141),(14,142,134,105),(15,106,135,143),(16,144,136,107),(17,108,137,145),(18,146,138,109),(19,110,139,147),(20,148,140,111),(21,69,94,53),(22,54,95,70),(23,71,96,55),(24,56,97,72),(25,73,98,57),(26,58,99,74),(27,75,100,59),(28,60,81,76),(29,77,82,41),(30,42,83,78),(31,79,84,43),(32,44,85,80),(33,61,86,45),(34,46,87,62),(35,63,88,47),(36,48,89,64),(37,65,90,49),(38,50,91,66),(39,67,92,51),(40,52,93,68)], [(1,102),(2,132),(3,104),(4,134),(5,106),(6,136),(7,108),(8,138),(9,110),(10,140),(11,112),(12,122),(13,114),(14,124),(15,116),(16,126),(17,118),(18,128),(19,120),(20,130),(21,43),(22,32),(23,45),(24,34),(25,47),(26,36),(27,49),(28,38),(29,51),(30,40),(31,53),(33,55),(35,57),(37,59),(39,41),(42,68),(44,70),(46,72),(48,74),(50,76),(52,78),(54,80),(56,62),(58,64),(60,66),(61,96),(63,98),(65,100),(67,82),(69,84),(71,86),(73,88),(75,90),(77,92),(79,94),(81,91),(83,93),(85,95),(87,97),(89,99),(101,111),(103,113),(105,115),(107,117),(109,119),(121,159),(123,141),(125,143),(127,145),(129,147),(131,149),(133,151),(135,153),(137,155),(139,157),(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)], [(1,45,131,71),(2,44,132,70),(3,43,133,69),(4,42,134,68),(5,41,135,67),(6,60,136,66),(7,59,137,65),(8,58,138,64),(9,57,139,63),(10,56,140,62),(11,55,121,61),(12,54,122,80),(13,53,123,79),(14,52,124,78),(15,51,125,77),(16,50,126,76),(17,49,127,75),(18,48,128,74),(19,47,129,73),(20,46,130,72),(21,151,84,104),(22,150,85,103),(23,149,86,102),(24,148,87,101),(25,147,88,120),(26,146,89,119),(27,145,90,118),(28,144,91,117),(29,143,92,116),(30,142,93,115),(31,141,94,114),(32,160,95,113),(33,159,96,112),(34,158,97,111),(35,157,98,110),(36,156,99,109),(37,155,100,108),(38,154,81,107),(39,153,82,106),(40,152,83,105)]])
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 | C4○D4 | D10 | D10 | D10 | C4○D8 | Dic10 | C8⋊C22 | D4⋊2D5 | D4×D5 | D8⋊3D5 | D40⋊C2 |
| kernel | D4.2Dic10 | C20.Q8 | C20.8Q8 | C40⋊5C4 | D4⋊Dic5 | C5×D4⋊C4 | C4.Dic10 | D4×Dic5 | C2×Dic5 | C5×D4 | D4⋊C4 | C20 | C4⋊C4 | C2×C8 | C2×D4 | C10 | D4 | C10 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 8 | 1 | 2 | 2 | 4 | 4 |
Matrix representation of D4.2Dic10 ►in GL6(𝔽41)
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 37 | 0 | 0 | 0 | 0 |
| 31 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 12 | 0 | 0 |
| 0 | 0 | 12 | 29 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 40 |
| 0 | 0 | 0 | 0 | 36 | 6 |
| 32 | 0 | 0 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 32 | 0 | 0 | 0 |
| 0 | 0 | 0 | 32 | 0 | 0 |
| 0 | 0 | 0 | 0 | 35 | 40 |
| 0 | 0 | 0 | 0 | 35 | 6 |
G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,40,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,40,0,0,0,0,0,0,0,40,0,0,0,0,40,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,31,0,0,0,0,37,0,0,0,0,0,0,0,12,12,0,0,0,0,12,29,0,0,0,0,0,0,1,36,0,0,0,0,40,6],[32,0,0,0,0,0,0,9,0,0,0,0,0,0,32,0,0,0,0,0,0,32,0,0,0,0,0,0,35,35,0,0,0,0,40,6] >;
D4.2Dic10 in GAP, Magma, Sage, TeX
D_4._2{\rm Dic}_{10} % in TeX
G:=Group("D4.2Dic10"); // GroupNames label
G:=SmallGroup(320,393);
// by ID
G=gap.SmallGroup(320,393);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,926,219,226,851,438,102,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^2=c^20=1,d^2=a^2*c^10,b*a*b=c*a*c^-1=a^-1,a*d=d*a,c*b*c^-1=a^-1*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations