metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D4⋊6Dic10, C42.107D10, C10.1012+ 1+4, (C5×D4)⋊7Q8, (C4×D4).14D5, C5⋊3(D4⋊3Q8), C20.43(C2×Q8), C4⋊C4.281D10, C20⋊2Q8⋊23C2, (D4×C20).15C2, (C4×Dic10)⋊29C2, (C2×D4).243D10, C20.48D4⋊9C2, (C2×C10).87C24, C4.Dic10⋊15C2, (D4×Dic5).13C2, C4.16(C2×Dic10), C20.292(C4○D4), C10.14(C22×Q8), (C2×C20).156C23, (C4×C20).149C22, C22⋊C4.108D10, (C22×C4).206D10, C4.117(D4⋊2D5), C2.13(D4⋊8D10), C22.2(C2×Dic10), Dic5.14D4⋊8C2, (D4×C10).251C22, C4⋊Dic5.198C22, (C22×C20).80C22, (C4×Dic5).82C22, (C2×Dic5).37C23, C2.16(C22×Dic10), C10.D4.6C22, C23.166(C22×D5), C22.115(C23×D5), C23.D5.10C22, (C22×C10).157C23, (C2×Dic10).28C22, (C22×Dic5).94C22, (C2×C10).4(C2×Q8), (C2×C4⋊Dic5)⋊24C2, C10.73(C2×C4○D4), C2.21(C2×D4⋊2D5), (C5×C4⋊C4).323C22, (C2×C4).731(C22×D5), (C5×C22⋊C4).105C22, SmallGroup(320,1215)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D4⋊6Dic10
G = < a,b,c,d | a4=b2=c20=1, d2=c10, bab=a-1, ac=ca, ad=da, bc=cb, dbd-1=a2b, dcd-1=c-1 >
Subgroups: 694 in 228 conjugacy classes, 115 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, C20, C20, C2×C10, C2×C10, C2×C10, C2×C4⋊C4, C4×D4, C4×D4, C4×Q8, C22⋊Q8, C42.C2, C4⋊Q8, Dic10, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C2×C20, C5×D4, C22×C10, D4⋊3Q8, C4×Dic5, C10.D4, C4⋊Dic5, C4⋊Dic5, C23.D5, C4×C20, C5×C22⋊C4, C5×C4⋊C4, C2×Dic10, C2×Dic10, C22×Dic5, C22×C20, D4×C10, C4×Dic10, C20⋊2Q8, Dic5.14D4, C4.Dic10, C20.48D4, C2×C4⋊Dic5, D4×Dic5, D4×C20, D4⋊6Dic10
Quotients: C1, C2, C22, Q8, C23, D5, C2×Q8, C4○D4, C24, D10, C22×Q8, C2×C4○D4, 2+ 1+4, Dic10, C22×D5, D4⋊3Q8, C2×Dic10, D4⋊2D5, C23×D5, C22×Dic10, C2×D4⋊2D5, D4⋊8D10, D4⋊6Dic10
►On 160 points
Generators in S160
(1 21 88 47)(2 22 89 48)(3 23 90 49)(4 24 91 50)(5 25 92 51)(6 26 93 52)(7 27 94 53)(8 28 95 54)(9 29 96 55)(10 30 97 56)(11 31 98 57)(12 32 99 58)(13 33 100 59)(14 34 81 60)(15 35 82 41)(16 36 83 42)(17 37 84 43)(18 38 85 44)(19 39 86 45)(20 40 87 46)(61 126 144 115)(62 127 145 116)(63 128 146 117)(64 129 147 118)(65 130 148 119)(66 131 149 120)(67 132 150 101)(68 133 151 102)(69 134 152 103)(70 135 153 104)(71 136 154 105)(72 137 155 106)(73 138 156 107)(74 139 157 108)(75 140 158 109)(76 121 159 110)(77 122 160 111)(78 123 141 112)(79 124 142 113)(80 125 143 114)
(1 31)(2 32)(3 33)(4 34)(5 35)(6 36)(7 37)(8 38)(9 39)(10 40)(11 21)(12 22)(13 23)(14 24)(15 25)(16 26)(17 27)(18 28)(19 29)(20 30)(41 92)(42 93)(43 94)(44 95)(45 96)(46 97)(47 98)(48 99)(49 100)(50 81)(51 82)(52 83)(53 84)(54 85)(55 86)(56 87)(57 88)(58 89)(59 90)(60 91)(61 105)(62 106)(63 107)(64 108)(65 109)(66 110)(67 111)(68 112)(69 113)(70 114)(71 115)(72 116)(73 117)(74 118)(75 119)(76 120)(77 101)(78 102)(79 103)(80 104)(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)
(1 150 11 160)(2 149 12 159)(3 148 13 158)(4 147 14 157)(5 146 15 156)(6 145 16 155)(7 144 17 154)(8 143 18 153)(9 142 19 152)(10 141 20 151)(21 101 31 111)(22 120 32 110)(23 119 33 109)(24 118 34 108)(25 117 35 107)(26 116 36 106)(27 115 37 105)(28 114 38 104)(29 113 39 103)(30 112 40 102)(41 138 51 128)(42 137 52 127)(43 136 53 126)(44 135 54 125)(45 134 55 124)(46 133 56 123)(47 132 57 122)(48 131 58 121)(49 130 59 140)(50 129 60 139)(61 84 71 94)(62 83 72 93)(63 82 73 92)(64 81 74 91)(65 100 75 90)(66 99 76 89)(67 98 77 88)(68 97 78 87)(69 96 79 86)(70 95 80 85)
G:=sub<Sym(160)| (1,21,88,47)(2,22,89,48)(3,23,90,49)(4,24,91,50)(5,25,92,51)(6,26,93,52)(7,27,94,53)(8,28,95,54)(9,29,96,55)(10,30,97,56)(11,31,98,57)(12,32,99,58)(13,33,100,59)(14,34,81,60)(15,35,82,41)(16,36,83,42)(17,37,84,43)(18,38,85,44)(19,39,86,45)(20,40,87,46)(61,126,144,115)(62,127,145,116)(63,128,146,117)(64,129,147,118)(65,130,148,119)(66,131,149,120)(67,132,150,101)(68,133,151,102)(69,134,152,103)(70,135,153,104)(71,136,154,105)(72,137,155,106)(73,138,156,107)(74,139,157,108)(75,140,158,109)(76,121,159,110)(77,122,160,111)(78,123,141,112)(79,124,142,113)(80,125,143,114), (1,31)(2,32)(3,33)(4,34)(5,35)(6,36)(7,37)(8,38)(9,39)(10,40)(11,21)(12,22)(13,23)(14,24)(15,25)(16,26)(17,27)(18,28)(19,29)(20,30)(41,92)(42,93)(43,94)(44,95)(45,96)(46,97)(47,98)(48,99)(49,100)(50,81)(51,82)(52,83)(53,84)(54,85)(55,86)(56,87)(57,88)(58,89)(59,90)(60,91)(61,105)(62,106)(63,107)(64,108)(65,109)(66,110)(67,111)(68,112)(69,113)(70,114)(71,115)(72,116)(73,117)(74,118)(75,119)(76,120)(77,101)(78,102)(79,103)(80,104)(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), (1,150,11,160)(2,149,12,159)(3,148,13,158)(4,147,14,157)(5,146,15,156)(6,145,16,155)(7,144,17,154)(8,143,18,153)(9,142,19,152)(10,141,20,151)(21,101,31,111)(22,120,32,110)(23,119,33,109)(24,118,34,108)(25,117,35,107)(26,116,36,106)(27,115,37,105)(28,114,38,104)(29,113,39,103)(30,112,40,102)(41,138,51,128)(42,137,52,127)(43,136,53,126)(44,135,54,125)(45,134,55,124)(46,133,56,123)(47,132,57,122)(48,131,58,121)(49,130,59,140)(50,129,60,139)(61,84,71,94)(62,83,72,93)(63,82,73,92)(64,81,74,91)(65,100,75,90)(66,99,76,89)(67,98,77,88)(68,97,78,87)(69,96,79,86)(70,95,80,85)>;
G:=Group( (1,21,88,47)(2,22,89,48)(3,23,90,49)(4,24,91,50)(5,25,92,51)(6,26,93,52)(7,27,94,53)(8,28,95,54)(9,29,96,55)(10,30,97,56)(11,31,98,57)(12,32,99,58)(13,33,100,59)(14,34,81,60)(15,35,82,41)(16,36,83,42)(17,37,84,43)(18,38,85,44)(19,39,86,45)(20,40,87,46)(61,126,144,115)(62,127,145,116)(63,128,146,117)(64,129,147,118)(65,130,148,119)(66,131,149,120)(67,132,150,101)(68,133,151,102)(69,134,152,103)(70,135,153,104)(71,136,154,105)(72,137,155,106)(73,138,156,107)(74,139,157,108)(75,140,158,109)(76,121,159,110)(77,122,160,111)(78,123,141,112)(79,124,142,113)(80,125,143,114), (1,31)(2,32)(3,33)(4,34)(5,35)(6,36)(7,37)(8,38)(9,39)(10,40)(11,21)(12,22)(13,23)(14,24)(15,25)(16,26)(17,27)(18,28)(19,29)(20,30)(41,92)(42,93)(43,94)(44,95)(45,96)(46,97)(47,98)(48,99)(49,100)(50,81)(51,82)(52,83)(53,84)(54,85)(55,86)(56,87)(57,88)(58,89)(59,90)(60,91)(61,105)(62,106)(63,107)(64,108)(65,109)(66,110)(67,111)(68,112)(69,113)(70,114)(71,115)(72,116)(73,117)(74,118)(75,119)(76,120)(77,101)(78,102)(79,103)(80,104)(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), (1,150,11,160)(2,149,12,159)(3,148,13,158)(4,147,14,157)(5,146,15,156)(6,145,16,155)(7,144,17,154)(8,143,18,153)(9,142,19,152)(10,141,20,151)(21,101,31,111)(22,120,32,110)(23,119,33,109)(24,118,34,108)(25,117,35,107)(26,116,36,106)(27,115,37,105)(28,114,38,104)(29,113,39,103)(30,112,40,102)(41,138,51,128)(42,137,52,127)(43,136,53,126)(44,135,54,125)(45,134,55,124)(46,133,56,123)(47,132,57,122)(48,131,58,121)(49,130,59,140)(50,129,60,139)(61,84,71,94)(62,83,72,93)(63,82,73,92)(64,81,74,91)(65,100,75,90)(66,99,76,89)(67,98,77,88)(68,97,78,87)(69,96,79,86)(70,95,80,85) );
G=PermutationGroup([[(1,21,88,47),(2,22,89,48),(3,23,90,49),(4,24,91,50),(5,25,92,51),(6,26,93,52),(7,27,94,53),(8,28,95,54),(9,29,96,55),(10,30,97,56),(11,31,98,57),(12,32,99,58),(13,33,100,59),(14,34,81,60),(15,35,82,41),(16,36,83,42),(17,37,84,43),(18,38,85,44),(19,39,86,45),(20,40,87,46),(61,126,144,115),(62,127,145,116),(63,128,146,117),(64,129,147,118),(65,130,148,119),(66,131,149,120),(67,132,150,101),(68,133,151,102),(69,134,152,103),(70,135,153,104),(71,136,154,105),(72,137,155,106),(73,138,156,107),(74,139,157,108),(75,140,158,109),(76,121,159,110),(77,122,160,111),(78,123,141,112),(79,124,142,113),(80,125,143,114)], [(1,31),(2,32),(3,33),(4,34),(5,35),(6,36),(7,37),(8,38),(9,39),(10,40),(11,21),(12,22),(13,23),(14,24),(15,25),(16,26),(17,27),(18,28),(19,29),(20,30),(41,92),(42,93),(43,94),(44,95),(45,96),(46,97),(47,98),(48,99),(49,100),(50,81),(51,82),(52,83),(53,84),(54,85),(55,86),(56,87),(57,88),(58,89),(59,90),(60,91),(61,105),(62,106),(63,107),(64,108),(65,109),(66,110),(67,111),(68,112),(69,113),(70,114),(71,115),(72,116),(73,117),(74,118),(75,119),(76,120),(77,101),(78,102),(79,103),(80,104),(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)], [(1,150,11,160),(2,149,12,159),(3,148,13,158),(4,147,14,157),(5,146,15,156),(6,145,16,155),(7,144,17,154),(8,143,18,153),(9,142,19,152),(10,141,20,151),(21,101,31,111),(22,120,32,110),(23,119,33,109),(24,118,34,108),(25,117,35,107),(26,116,36,106),(27,115,37,105),(28,114,38,104),(29,113,39,103),(30,112,40,102),(41,138,51,128),(42,137,52,127),(43,136,53,126),(44,135,54,125),(45,134,55,124),(46,133,56,123),(47,132,57,122),(48,131,58,121),(49,130,59,140),(50,129,60,139),(61,84,71,94),(62,83,72,93),(63,82,73,92),(64,81,74,91),(65,100,75,90),(66,99,76,89),(67,98,77,88),(68,97,78,87),(69,96,79,86),(70,95,80,85)]])
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 | Q8 | D5 | C4○D4 | D10 | D10 | D10 | D10 | D10 | Dic10 | 2+ 1+4 | D4⋊2D5 | D4⋊8D10 |
| kernel | D4⋊6Dic10 | C4×Dic10 | C20⋊2Q8 | Dic5.14D4 | C4.Dic10 | C20.48D4 | C2×C4⋊Dic5 | D4×Dic5 | D4×C20 | C5×D4 | C4×D4 | C20 | C42 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×D4 | D4 | C10 | C4 | C2 |
| # reps | 1 | 1 | 1 | 4 | 2 | 2 | 2 | 2 | 1 | 4 | 2 | 4 | 2 | 4 | 2 | 4 | 2 | 16 | 1 | 4 | 4 |
Matrix representation of D4⋊6Dic10 ►in GL6(𝔽41)
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 39 |
| 0 | 0 | 0 | 0 | 1 | 1 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 2 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 34 | 40 | 0 | 0 | 0 | 0 |
| 8 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 5 | 0 | 0 |
| 0 | 0 | 16 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 35 | 12 | 0 | 0 | 0 | 0 |
| 21 | 6 | 0 | 0 | 0 | 0 |
| 0 | 0 | 7 | 12 | 0 | 0 |
| 0 | 0 | 30 | 34 | 0 | 0 |
| 0 | 0 | 0 | 0 | 9 | 18 |
| 0 | 0 | 0 | 0 | 32 | 32 |
G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,1,0,0,0,0,39,1],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,2,40],[34,8,0,0,0,0,40,1,0,0,0,0,0,0,40,16,0,0,0,0,5,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[35,21,0,0,0,0,12,6,0,0,0,0,0,0,7,30,0,0,0,0,12,34,0,0,0,0,0,0,9,32,0,0,0,0,18,32] >;
D4⋊6Dic10 in GAP, Magma, Sage, TeX
D_4\rtimes_6{\rm Dic}_{10} % in TeX
G:=Group("D4:6Dic10"); // GroupNames label
G:=SmallGroup(320,1215);
// by ID
G=gap.SmallGroup(320,1215);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,387,1571,192,12550]);
// 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,d*b*d^-1=a^2*b,d*c*d^-1=c^-1>;
// generators/relations