metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D4⋊5Dic10, C42.103D10, C10.132+ 1+4, (C5×D4)⋊6Q8, C20⋊Q8⋊15C2, C5⋊2(D4⋊3Q8), (C4×D4).11D5, C20.42(C2×Q8), C4⋊C4.278D10, (D4×C20).12C2, (C4×Dic10)⋊26C2, (C2×D4).242D10, C20.48D4⋊7C2, (C2×C10).83C24, C20.6Q8⋊14C2, C4.Dic10⋊14C2, (D4×Dic5).12C2, C4.15(C2×Dic10), C10.13(C22×Q8), (C4×C20).146C22, (C2×C20).154C23, C22⋊C4.106D10, (C22×C4).202D10, C4⋊Dic5.37C22, C2.16(D4⋊6D10), C22.1(C2×Dic10), Dic5.35(C4○D4), Dic5.14D4⋊7C2, C23.D5.8C22, (D4×C10).249C22, (C22×C20).77C22, (C2×Dic5).33C23, (C4×Dic5).80C22, C2.15(C22×Dic10), C23.163(C22×D5), C22.111(C23×D5), (C22×C10).153C23, (C2×Dic10).27C22, C10.D4.108C22, (C22×Dic5).91C22, C2.18(D5×C4○D4), (C2×C10).3(C2×Q8), C10.137(C2×C4○D4), (C2×C10.D4)⋊24C2, (C5×C4⋊C4).319C22, (C2×C4).154(C22×D5), (C5×C22⋊C4).104C22, SmallGroup(320,1211)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D4⋊5Dic10
G = < a,b,c,d | a4=b2=c20=1, d2=c10, bab=cac-1=a-1, ad=da, cbc-1=dbd-1=a2b, dcd-1=c-1 >
Subgroups: 694 in 228 conjugacy classes, 113 normal (43 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, C2×C4⋊C4, C4×D4, C4×D4, C4×Q8, C22⋊Q8, C42.C2, C4⋊Q8, Dic10, C2×Dic5, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C2×C20, C5×D4, C22×C10, D4⋊3Q8, C4×Dic5, C10.D4, 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, C20.6Q8, Dic5.14D4, C20⋊Q8, C4.Dic10, C2×C10.D4, C20.48D4, D4×Dic5, D4×C20, D4⋊5Dic10
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, C23×D5, C22×Dic10, D4⋊6D10, D5×C4○D4, D4⋊5Dic10
►On 160 points
Generators in S160
(1 111 93 136)(2 137 94 112)(3 113 95 138)(4 139 96 114)(5 115 97 140)(6 121 98 116)(7 117 99 122)(8 123 100 118)(9 119 81 124)(10 125 82 120)(11 101 83 126)(12 127 84 102)(13 103 85 128)(14 129 86 104)(15 105 87 130)(16 131 88 106)(17 107 89 132)(18 133 90 108)(19 109 91 134)(20 135 92 110)(21 78 60 159)(22 160 41 79)(23 80 42 141)(24 142 43 61)(25 62 44 143)(26 144 45 63)(27 64 46 145)(28 146 47 65)(29 66 48 147)(30 148 49 67)(31 68 50 149)(32 150 51 69)(33 70 52 151)(34 152 53 71)(35 72 54 153)(36 154 55 73)(37 74 56 155)(38 156 57 75)(39 76 58 157)(40 158 59 77)
(1 101)(2 127)(3 103)(4 129)(5 105)(6 131)(7 107)(8 133)(9 109)(10 135)(11 111)(12 137)(13 113)(14 139)(15 115)(16 121)(17 117)(18 123)(19 119)(20 125)(21 68)(22 150)(23 70)(24 152)(25 72)(26 154)(27 74)(28 156)(29 76)(30 158)(31 78)(32 160)(33 80)(34 142)(35 62)(36 144)(37 64)(38 146)(39 66)(40 148)(41 69)(42 151)(43 71)(44 153)(45 73)(46 155)(47 75)(48 157)(49 77)(50 159)(51 79)(52 141)(53 61)(54 143)(55 63)(56 145)(57 65)(58 147)(59 67)(60 149)(81 134)(82 110)(83 136)(84 112)(85 138)(86 114)(87 140)(88 116)(89 122)(90 118)(91 124)(92 120)(93 126)(94 102)(95 128)(96 104)(97 130)(98 106)(99 132)(100 108)
(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)
(1 156 11 146)(2 155 12 145)(3 154 13 144)(4 153 14 143)(5 152 15 142)(6 151 16 141)(7 150 17 160)(8 149 18 159)(9 148 19 158)(10 147 20 157)(21 123 31 133)(22 122 32 132)(23 121 33 131)(24 140 34 130)(25 139 35 129)(26 138 36 128)(27 137 37 127)(28 136 38 126)(29 135 39 125)(30 134 40 124)(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 97 71 87)(62 96 72 86)(63 95 73 85)(64 94 74 84)(65 93 75 83)(66 92 76 82)(67 91 77 81)(68 90 78 100)(69 89 79 99)(70 88 80 98)
G:=sub<Sym(160)| (1,111,93,136)(2,137,94,112)(3,113,95,138)(4,139,96,114)(5,115,97,140)(6,121,98,116)(7,117,99,122)(8,123,100,118)(9,119,81,124)(10,125,82,120)(11,101,83,126)(12,127,84,102)(13,103,85,128)(14,129,86,104)(15,105,87,130)(16,131,88,106)(17,107,89,132)(18,133,90,108)(19,109,91,134)(20,135,92,110)(21,78,60,159)(22,160,41,79)(23,80,42,141)(24,142,43,61)(25,62,44,143)(26,144,45,63)(27,64,46,145)(28,146,47,65)(29,66,48,147)(30,148,49,67)(31,68,50,149)(32,150,51,69)(33,70,52,151)(34,152,53,71)(35,72,54,153)(36,154,55,73)(37,74,56,155)(38,156,57,75)(39,76,58,157)(40,158,59,77), (1,101)(2,127)(3,103)(4,129)(5,105)(6,131)(7,107)(8,133)(9,109)(10,135)(11,111)(12,137)(13,113)(14,139)(15,115)(16,121)(17,117)(18,123)(19,119)(20,125)(21,68)(22,150)(23,70)(24,152)(25,72)(26,154)(27,74)(28,156)(29,76)(30,158)(31,78)(32,160)(33,80)(34,142)(35,62)(36,144)(37,64)(38,146)(39,66)(40,148)(41,69)(42,151)(43,71)(44,153)(45,73)(46,155)(47,75)(48,157)(49,77)(50,159)(51,79)(52,141)(53,61)(54,143)(55,63)(56,145)(57,65)(58,147)(59,67)(60,149)(81,134)(82,110)(83,136)(84,112)(85,138)(86,114)(87,140)(88,116)(89,122)(90,118)(91,124)(92,120)(93,126)(94,102)(95,128)(96,104)(97,130)(98,106)(99,132)(100,108), (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), (1,156,11,146)(2,155,12,145)(3,154,13,144)(4,153,14,143)(5,152,15,142)(6,151,16,141)(7,150,17,160)(8,149,18,159)(9,148,19,158)(10,147,20,157)(21,123,31,133)(22,122,32,132)(23,121,33,131)(24,140,34,130)(25,139,35,129)(26,138,36,128)(27,137,37,127)(28,136,38,126)(29,135,39,125)(30,134,40,124)(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)(61,97,71,87)(62,96,72,86)(63,95,73,85)(64,94,74,84)(65,93,75,83)(66,92,76,82)(67,91,77,81)(68,90,78,100)(69,89,79,99)(70,88,80,98)>;
G:=Group( (1,111,93,136)(2,137,94,112)(3,113,95,138)(4,139,96,114)(5,115,97,140)(6,121,98,116)(7,117,99,122)(8,123,100,118)(9,119,81,124)(10,125,82,120)(11,101,83,126)(12,127,84,102)(13,103,85,128)(14,129,86,104)(15,105,87,130)(16,131,88,106)(17,107,89,132)(18,133,90,108)(19,109,91,134)(20,135,92,110)(21,78,60,159)(22,160,41,79)(23,80,42,141)(24,142,43,61)(25,62,44,143)(26,144,45,63)(27,64,46,145)(28,146,47,65)(29,66,48,147)(30,148,49,67)(31,68,50,149)(32,150,51,69)(33,70,52,151)(34,152,53,71)(35,72,54,153)(36,154,55,73)(37,74,56,155)(38,156,57,75)(39,76,58,157)(40,158,59,77), (1,101)(2,127)(3,103)(4,129)(5,105)(6,131)(7,107)(8,133)(9,109)(10,135)(11,111)(12,137)(13,113)(14,139)(15,115)(16,121)(17,117)(18,123)(19,119)(20,125)(21,68)(22,150)(23,70)(24,152)(25,72)(26,154)(27,74)(28,156)(29,76)(30,158)(31,78)(32,160)(33,80)(34,142)(35,62)(36,144)(37,64)(38,146)(39,66)(40,148)(41,69)(42,151)(43,71)(44,153)(45,73)(46,155)(47,75)(48,157)(49,77)(50,159)(51,79)(52,141)(53,61)(54,143)(55,63)(56,145)(57,65)(58,147)(59,67)(60,149)(81,134)(82,110)(83,136)(84,112)(85,138)(86,114)(87,140)(88,116)(89,122)(90,118)(91,124)(92,120)(93,126)(94,102)(95,128)(96,104)(97,130)(98,106)(99,132)(100,108), (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), (1,156,11,146)(2,155,12,145)(3,154,13,144)(4,153,14,143)(5,152,15,142)(6,151,16,141)(7,150,17,160)(8,149,18,159)(9,148,19,158)(10,147,20,157)(21,123,31,133)(22,122,32,132)(23,121,33,131)(24,140,34,130)(25,139,35,129)(26,138,36,128)(27,137,37,127)(28,136,38,126)(29,135,39,125)(30,134,40,124)(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)(61,97,71,87)(62,96,72,86)(63,95,73,85)(64,94,74,84)(65,93,75,83)(66,92,76,82)(67,91,77,81)(68,90,78,100)(69,89,79,99)(70,88,80,98) );
G=PermutationGroup([[(1,111,93,136),(2,137,94,112),(3,113,95,138),(4,139,96,114),(5,115,97,140),(6,121,98,116),(7,117,99,122),(8,123,100,118),(9,119,81,124),(10,125,82,120),(11,101,83,126),(12,127,84,102),(13,103,85,128),(14,129,86,104),(15,105,87,130),(16,131,88,106),(17,107,89,132),(18,133,90,108),(19,109,91,134),(20,135,92,110),(21,78,60,159),(22,160,41,79),(23,80,42,141),(24,142,43,61),(25,62,44,143),(26,144,45,63),(27,64,46,145),(28,146,47,65),(29,66,48,147),(30,148,49,67),(31,68,50,149),(32,150,51,69),(33,70,52,151),(34,152,53,71),(35,72,54,153),(36,154,55,73),(37,74,56,155),(38,156,57,75),(39,76,58,157),(40,158,59,77)], [(1,101),(2,127),(3,103),(4,129),(5,105),(6,131),(7,107),(8,133),(9,109),(10,135),(11,111),(12,137),(13,113),(14,139),(15,115),(16,121),(17,117),(18,123),(19,119),(20,125),(21,68),(22,150),(23,70),(24,152),(25,72),(26,154),(27,74),(28,156),(29,76),(30,158),(31,78),(32,160),(33,80),(34,142),(35,62),(36,144),(37,64),(38,146),(39,66),(40,148),(41,69),(42,151),(43,71),(44,153),(45,73),(46,155),(47,75),(48,157),(49,77),(50,159),(51,79),(52,141),(53,61),(54,143),(55,63),(56,145),(57,65),(58,147),(59,67),(60,149),(81,134),(82,110),(83,136),(84,112),(85,138),(86,114),(87,140),(88,116),(89,122),(90,118),(91,124),(92,120),(93,126),(94,102),(95,128),(96,104),(97,130),(98,106),(99,132),(100,108)], [(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)], [(1,156,11,146),(2,155,12,145),(3,154,13,144),(4,153,14,143),(5,152,15,142),(6,151,16,141),(7,150,17,160),(8,149,18,159),(9,148,19,158),(10,147,20,157),(21,123,31,133),(22,122,32,132),(23,121,33,131),(24,140,34,130),(25,139,35,129),(26,138,36,128),(27,137,37,127),(28,136,38,126),(29,135,39,125),(30,134,40,124),(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),(61,97,71,87),(62,96,72,86),(63,95,73,85),(64,94,74,84),(65,93,75,83),(66,92,76,82),(67,91,77,81),(68,90,78,100),(69,89,79,99),(70,88,80,98)]])
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 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | - | + | + | + | + | + | + | - | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | Q8 | D5 | C4○D4 | D10 | D10 | D10 | D10 | D10 | Dic10 | 2+ 1+4 | D4⋊6D10 | D5×C4○D4 |
| kernel | D4⋊5Dic10 | C4×Dic10 | C20.6Q8 | Dic5.14D4 | C20⋊Q8 | C4.Dic10 | C2×C10.D4 | C20.48D4 | D4×Dic5 | D4×C20 | C5×D4 | C4×D4 | Dic5 | C42 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×D4 | D4 | C10 | C2 | C2 |
| # reps | 1 | 1 | 1 | 4 | 1 | 1 | 2 | 2 | 2 | 1 | 4 | 2 | 4 | 2 | 4 | 2 | 4 | 2 | 16 | 1 | 4 | 4 |
Matrix representation of D4⋊5Dic10 ►in GL6(𝔽41)
| 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 | 21 |
| 0 | 0 | 0 | 0 | 37 | 40 |
| 1 | 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 | 0 | 1 | 21 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 34 | 31 | 0 | 0 | 0 | 0 |
| 5 | 7 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 40 | 0 | 0 |
| 0 | 0 | 8 | 34 | 0 | 0 |
| 0 | 0 | 0 | 0 | 32 | 0 |
| 0 | 0 | 0 | 0 | 36 | 9 |
| 40 | 11 | 0 | 0 | 0 | 0 |
| 11 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 5 | 0 | 0 |
| 0 | 0 | 23 | 38 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 21 |
| 0 | 0 | 0 | 0 | 37 | 40 |
G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,37,0,0,0,0,21,40],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,21,40],[34,5,0,0,0,0,31,7,0,0,0,0,0,0,1,8,0,0,0,0,40,34,0,0,0,0,0,0,32,36,0,0,0,0,0,9],[40,11,0,0,0,0,11,1,0,0,0,0,0,0,3,23,0,0,0,0,5,38,0,0,0,0,0,0,1,37,0,0,0,0,21,40] >;
D4⋊5Dic10 in GAP, Magma, Sage, TeX
D_4\rtimes_5{\rm Dic}_{10} % in TeX
G:=Group("D4:5Dic10"); // GroupNames label
G:=SmallGroup(320,1211);
// by ID
G=gap.SmallGroup(320,1211);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,387,675,192,12550]);
// 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=d*b*d^-1=a^2*b,d*c*d^-1=c^-1>;
// generators/relations