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, C2.13(D4⋊8D10), C4.117(D4⋊2D5), 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
Subgroups: 694 in 228 conjugacy classes, 115 normal (29 characteristic)
C1, C2 [×3], C2 [×4], C4 [×4], C4 [×11], C22, C22 [×4], C22 [×4], C5, C2×C4 [×3], C2×C4 [×2], C2×C4 [×16], D4 [×4], Q8 [×4], C23 [×2], C10 [×3], C10 [×4], C42, C42 [×2], C22⋊C4 [×2], C22⋊C4 [×4], C4⋊C4, C4⋊C4 [×15], C22×C4 [×2], C22×C4 [×4], C2×D4, C2×Q8 [×3], Dic5 [×8], C20 [×4], C20 [×3], C2×C10, C2×C10 [×4], C2×C10 [×4], C2×C4⋊C4 [×2], C4×D4, C4×D4 [×2], C4×Q8, C22⋊Q8 [×6], C42.C2 [×2], C4⋊Q8, Dic10 [×4], C2×Dic5 [×8], C2×Dic5 [×4], C2×C20 [×3], C2×C20 [×2], C2×C20 [×4], C5×D4 [×4], C22×C10 [×2], D4⋊3Q8, C4×Dic5 [×2], C10.D4 [×6], C4⋊Dic5, C4⋊Dic5 [×8], C23.D5 [×4], C4×C20, C5×C22⋊C4 [×2], C5×C4⋊C4, C2×Dic10, C2×Dic10 [×2], C22×Dic5 [×4], C22×C20 [×2], D4×C10, C4×Dic10, C20⋊2Q8, Dic5.14D4 [×4], C4.Dic10 [×2], C20.48D4 [×2], C2×C4⋊Dic5 [×2], D4×Dic5 [×2], D4×C20, D4⋊6Dic10
Quotients:
C1, C2 [×15], C22 [×35], Q8 [×4], C23 [×15], D5, C2×Q8 [×6], C4○D4 [×2], C24, D10 [×7], C22×Q8, C2×C4○D4, 2+ (1+4), Dic10 [×4], C22×D5 [×7], D4⋊3Q8, C2×Dic10 [×6], D4⋊2D5 [×2], C23×D5, C22×Dic10, C2×D4⋊2D5, D4⋊8D10, D4⋊6Dic10
Generators and relations
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 >
►On 160 points
Generators in S160
(1 64 114 92)(2 65 115 93)(3 66 116 94)(4 67 117 95)(5 68 118 96)(6 69 119 97)(7 70 120 98)(8 71 101 99)(9 72 102 100)(10 73 103 81)(11 74 104 82)(12 75 105 83)(13 76 106 84)(14 77 107 85)(15 78 108 86)(16 79 109 87)(17 80 110 88)(18 61 111 89)(19 62 112 90)(20 63 113 91)(21 141 136 52)(22 142 137 53)(23 143 138 54)(24 144 139 55)(25 145 140 56)(26 146 121 57)(27 147 122 58)(28 148 123 59)(29 149 124 60)(30 150 125 41)(31 151 126 42)(32 152 127 43)(33 153 128 44)(34 154 129 45)(35 155 130 46)(36 156 131 47)(37 157 132 48)(38 158 133 49)(39 159 134 50)(40 160 135 51)
(1 74)(2 75)(3 76)(4 77)(5 78)(6 79)(7 80)(8 61)(9 62)(10 63)(11 64)(12 65)(13 66)(14 67)(15 68)(16 69)(17 70)(18 71)(19 72)(20 73)(21 42)(22 43)(23 44)(24 45)(25 46)(26 47)(27 48)(28 49)(29 50)(30 51)(31 52)(32 53)(33 54)(34 55)(35 56)(36 57)(37 58)(38 59)(39 60)(40 41)(81 113)(82 114)(83 115)(84 116)(85 117)(86 118)(87 119)(88 120)(89 101)(90 102)(91 103)(92 104)(93 105)(94 106)(95 107)(96 108)(97 109)(98 110)(99 111)(100 112)(121 156)(122 157)(123 158)(124 159)(125 160)(126 141)(127 142)(128 143)(129 144)(130 145)(131 146)(132 147)(133 148)(134 149)(135 150)(136 151)(137 152)(138 153)(139 154)(140 155)
(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)
(1 134 11 124)(2 133 12 123)(3 132 13 122)(4 131 14 121)(5 130 15 140)(6 129 16 139)(7 128 17 138)(8 127 18 137)(9 126 19 136)(10 125 20 135)(21 102 31 112)(22 101 32 111)(23 120 33 110)(24 119 34 109)(25 118 35 108)(26 117 36 107)(27 116 37 106)(28 115 38 105)(29 114 39 104)(30 113 40 103)(41 63 51 73)(42 62 52 72)(43 61 53 71)(44 80 54 70)(45 79 55 69)(46 78 56 68)(47 77 57 67)(48 76 58 66)(49 75 59 65)(50 74 60 64)(81 150 91 160)(82 149 92 159)(83 148 93 158)(84 147 94 157)(85 146 95 156)(86 145 96 155)(87 144 97 154)(88 143 98 153)(89 142 99 152)(90 141 100 151)
G:=sub<Sym(160)| (1,64,114,92)(2,65,115,93)(3,66,116,94)(4,67,117,95)(5,68,118,96)(6,69,119,97)(7,70,120,98)(8,71,101,99)(9,72,102,100)(10,73,103,81)(11,74,104,82)(12,75,105,83)(13,76,106,84)(14,77,107,85)(15,78,108,86)(16,79,109,87)(17,80,110,88)(18,61,111,89)(19,62,112,90)(20,63,113,91)(21,141,136,52)(22,142,137,53)(23,143,138,54)(24,144,139,55)(25,145,140,56)(26,146,121,57)(27,147,122,58)(28,148,123,59)(29,149,124,60)(30,150,125,41)(31,151,126,42)(32,152,127,43)(33,153,128,44)(34,154,129,45)(35,155,130,46)(36,156,131,47)(37,157,132,48)(38,158,133,49)(39,159,134,50)(40,160,135,51), (1,74)(2,75)(3,76)(4,77)(5,78)(6,79)(7,80)(8,61)(9,62)(10,63)(11,64)(12,65)(13,66)(14,67)(15,68)(16,69)(17,70)(18,71)(19,72)(20,73)(21,42)(22,43)(23,44)(24,45)(25,46)(26,47)(27,48)(28,49)(29,50)(30,51)(31,52)(32,53)(33,54)(34,55)(35,56)(36,57)(37,58)(38,59)(39,60)(40,41)(81,113)(82,114)(83,115)(84,116)(85,117)(86,118)(87,119)(88,120)(89,101)(90,102)(91,103)(92,104)(93,105)(94,106)(95,107)(96,108)(97,109)(98,110)(99,111)(100,112)(121,156)(122,157)(123,158)(124,159)(125,160)(126,141)(127,142)(128,143)(129,144)(130,145)(131,146)(132,147)(133,148)(134,149)(135,150)(136,151)(137,152)(138,153)(139,154)(140,155), (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), (1,134,11,124)(2,133,12,123)(3,132,13,122)(4,131,14,121)(5,130,15,140)(6,129,16,139)(7,128,17,138)(8,127,18,137)(9,126,19,136)(10,125,20,135)(21,102,31,112)(22,101,32,111)(23,120,33,110)(24,119,34,109)(25,118,35,108)(26,117,36,107)(27,116,37,106)(28,115,38,105)(29,114,39,104)(30,113,40,103)(41,63,51,73)(42,62,52,72)(43,61,53,71)(44,80,54,70)(45,79,55,69)(46,78,56,68)(47,77,57,67)(48,76,58,66)(49,75,59,65)(50,74,60,64)(81,150,91,160)(82,149,92,159)(83,148,93,158)(84,147,94,157)(85,146,95,156)(86,145,96,155)(87,144,97,154)(88,143,98,153)(89,142,99,152)(90,141,100,151)>;
G:=Group( (1,64,114,92)(2,65,115,93)(3,66,116,94)(4,67,117,95)(5,68,118,96)(6,69,119,97)(7,70,120,98)(8,71,101,99)(9,72,102,100)(10,73,103,81)(11,74,104,82)(12,75,105,83)(13,76,106,84)(14,77,107,85)(15,78,108,86)(16,79,109,87)(17,80,110,88)(18,61,111,89)(19,62,112,90)(20,63,113,91)(21,141,136,52)(22,142,137,53)(23,143,138,54)(24,144,139,55)(25,145,140,56)(26,146,121,57)(27,147,122,58)(28,148,123,59)(29,149,124,60)(30,150,125,41)(31,151,126,42)(32,152,127,43)(33,153,128,44)(34,154,129,45)(35,155,130,46)(36,156,131,47)(37,157,132,48)(38,158,133,49)(39,159,134,50)(40,160,135,51), (1,74)(2,75)(3,76)(4,77)(5,78)(6,79)(7,80)(8,61)(9,62)(10,63)(11,64)(12,65)(13,66)(14,67)(15,68)(16,69)(17,70)(18,71)(19,72)(20,73)(21,42)(22,43)(23,44)(24,45)(25,46)(26,47)(27,48)(28,49)(29,50)(30,51)(31,52)(32,53)(33,54)(34,55)(35,56)(36,57)(37,58)(38,59)(39,60)(40,41)(81,113)(82,114)(83,115)(84,116)(85,117)(86,118)(87,119)(88,120)(89,101)(90,102)(91,103)(92,104)(93,105)(94,106)(95,107)(96,108)(97,109)(98,110)(99,111)(100,112)(121,156)(122,157)(123,158)(124,159)(125,160)(126,141)(127,142)(128,143)(129,144)(130,145)(131,146)(132,147)(133,148)(134,149)(135,150)(136,151)(137,152)(138,153)(139,154)(140,155), (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), (1,134,11,124)(2,133,12,123)(3,132,13,122)(4,131,14,121)(5,130,15,140)(6,129,16,139)(7,128,17,138)(8,127,18,137)(9,126,19,136)(10,125,20,135)(21,102,31,112)(22,101,32,111)(23,120,33,110)(24,119,34,109)(25,118,35,108)(26,117,36,107)(27,116,37,106)(28,115,38,105)(29,114,39,104)(30,113,40,103)(41,63,51,73)(42,62,52,72)(43,61,53,71)(44,80,54,70)(45,79,55,69)(46,78,56,68)(47,77,57,67)(48,76,58,66)(49,75,59,65)(50,74,60,64)(81,150,91,160)(82,149,92,159)(83,148,93,158)(84,147,94,157)(85,146,95,156)(86,145,96,155)(87,144,97,154)(88,143,98,153)(89,142,99,152)(90,141,100,151) );
G=PermutationGroup([(1,64,114,92),(2,65,115,93),(3,66,116,94),(4,67,117,95),(5,68,118,96),(6,69,119,97),(7,70,120,98),(8,71,101,99),(9,72,102,100),(10,73,103,81),(11,74,104,82),(12,75,105,83),(13,76,106,84),(14,77,107,85),(15,78,108,86),(16,79,109,87),(17,80,110,88),(18,61,111,89),(19,62,112,90),(20,63,113,91),(21,141,136,52),(22,142,137,53),(23,143,138,54),(24,144,139,55),(25,145,140,56),(26,146,121,57),(27,147,122,58),(28,148,123,59),(29,149,124,60),(30,150,125,41),(31,151,126,42),(32,152,127,43),(33,153,128,44),(34,154,129,45),(35,155,130,46),(36,156,131,47),(37,157,132,48),(38,158,133,49),(39,159,134,50),(40,160,135,51)], [(1,74),(2,75),(3,76),(4,77),(5,78),(6,79),(7,80),(8,61),(9,62),(10,63),(11,64),(12,65),(13,66),(14,67),(15,68),(16,69),(17,70),(18,71),(19,72),(20,73),(21,42),(22,43),(23,44),(24,45),(25,46),(26,47),(27,48),(28,49),(29,50),(30,51),(31,52),(32,53),(33,54),(34,55),(35,56),(36,57),(37,58),(38,59),(39,60),(40,41),(81,113),(82,114),(83,115),(84,116),(85,117),(86,118),(87,119),(88,120),(89,101),(90,102),(91,103),(92,104),(93,105),(94,106),(95,107),(96,108),(97,109),(98,110),(99,111),(100,112),(121,156),(122,157),(123,158),(124,159),(125,160),(126,141),(127,142),(128,143),(129,144),(130,145),(131,146),(132,147),(133,148),(134,149),(135,150),(136,151),(137,152),(138,153),(139,154),(140,155)], [(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)], [(1,134,11,124),(2,133,12,123),(3,132,13,122),(4,131,14,121),(5,130,15,140),(6,129,16,139),(7,128,17,138),(8,127,18,137),(9,126,19,136),(10,125,20,135),(21,102,31,112),(22,101,32,111),(23,120,33,110),(24,119,34,109),(25,118,35,108),(26,117,36,107),(27,116,37,106),(28,115,38,105),(29,114,39,104),(30,113,40,103),(41,63,51,73),(42,62,52,72),(43,61,53,71),(44,80,54,70),(45,79,55,69),(46,78,56,68),(47,77,57,67),(48,76,58,66),(49,75,59,65),(50,74,60,64),(81,150,91,160),(82,149,92,159),(83,148,93,158),(84,147,94,157),(85,146,95,156),(86,145,96,155),(87,144,97,154),(88,143,98,153),(89,142,99,152),(90,141,100,151)])
Matrix representation ►G ⊆ 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] >;
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 |
In GAP, Magma, Sage, TeX
D_4\rtimes_6Dic_{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
×
⋊
ℤ
𝔽
○
≀