metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Dic5⋊M4(2), C5⋊C8⋊1D4, C2.8(D4×F5), C5⋊1(C8⋊6D4), C10.4(C4×D4), D10⋊C8⋊8C2, C22⋊C4.4F5, C23.8(C2×F5), C10.5(C8○D4), C2.8(D4.F5), D10⋊C4.6C4, Dic5.68(C2×D4), C10.D4.2C4, C23.2F5⋊6C2, Dic5⋊C8⋊13C2, Dic5⋊4D4.9C2, C10.12(C2×M4(2)), Dic5.53(C4○D4), C22.71(C22×F5), C2.12(D5⋊M4(2)), (C2×Dic5).325C23, (C4×Dic5).248C22, (C22×Dic5).180C22, (C4×C5⋊C8)⋊12C2, (C2×C5⋊D4).6C4, (C2×C4.F5)⋊10C2, (C2×C4).22(C2×F5), (C2×C20).80(C2×C4), (C5×C22⋊C4).5C4, (C2×C5⋊C8).23C22, (C2×C22.F5)⋊2C2, (C2×C4×D5).274C22, (C2×C10).33(C22×C4), (C22×C10).16(C2×C4), (C2×Dic5).50(C2×C4), (C22×D5).42(C2×C4), SmallGroup(320,1033)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 442 in 122 conjugacy classes, 46 normal (42 characteristic)
C1, C2 [×3], C2 [×2], C4 [×7], C22, C22 [×6], C5, C8 [×5], C2×C4 [×2], C2×C4 [×7], D4 [×2], C23, C23, D5, C10 [×3], C10, C42, C22⋊C4, C22⋊C4, C4⋊C4, C2×C8 [×4], M4(2) [×4], C22×C4 [×2], C2×D4, Dic5 [×2], Dic5 [×2], Dic5, C20 [×2], D10 [×3], C2×C10, C2×C10 [×3], C4×C8, C22⋊C8 [×2], C4⋊C8, C4×D4, C2×M4(2) [×2], C5⋊C8 [×2], C5⋊C8 [×3], C4×D5 [×2], C2×Dic5 [×3], C2×Dic5 [×2], C5⋊D4 [×2], C2×C20 [×2], C22×D5, C22×C10, C8⋊6D4, C4×Dic5, C10.D4, D10⋊C4, C5×C22⋊C4, C4.F5 [×2], C2×C5⋊C8 [×4], C22.F5 [×2], C2×C4×D5, C22×Dic5, C2×C5⋊D4, C4×C5⋊C8, D10⋊C8, Dic5⋊C8, C23.2F5, Dic5⋊4D4, C2×C4.F5, C2×C22.F5, Dic5⋊M4(2)
Quotients:
C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×2], C23, M4(2) [×2], C22×C4, C2×D4, C4○D4, F5, C4×D4, C2×M4(2), C8○D4, C2×F5 [×3], C8⋊6D4, C22×F5, D5⋊M4(2), D4.F5, D4×F5, Dic5⋊M4(2)
Generators and relations
G = < a,b,c,d | a10=c8=d2=1, b2=a5, bab-1=a-1, cac-1=a7, ad=da, cbc-1=dbd=a5b, dcd=c5 >
►On 160 points
Generators in S160
(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 52 6 57)(2 51 7 56)(3 60 8 55)(4 59 9 54)(5 58 10 53)(11 123 16 128)(12 122 17 127)(13 121 18 126)(14 130 19 125)(15 129 20 124)(21 69 26 64)(22 68 27 63)(23 67 28 62)(24 66 29 61)(25 65 30 70)(31 75 36 80)(32 74 37 79)(33 73 38 78)(34 72 39 77)(35 71 40 76)(41 88 46 83)(42 87 47 82)(43 86 48 81)(44 85 49 90)(45 84 50 89)(91 131 96 136)(92 140 97 135)(93 139 98 134)(94 138 99 133)(95 137 100 132)(101 147 106 142)(102 146 107 141)(103 145 108 150)(104 144 109 149)(105 143 110 148)(111 157 116 152)(112 156 117 151)(113 155 118 160)(114 154 119 159)(115 153 120 158)
(1 119 33 104 25 126 45 96)(2 112 32 101 26 129 44 93)(3 115 31 108 27 122 43 100)(4 118 40 105 28 125 42 97)(5 111 39 102 29 128 41 94)(6 114 38 109 30 121 50 91)(7 117 37 106 21 124 49 98)(8 120 36 103 22 127 48 95)(9 113 35 110 23 130 47 92)(10 116 34 107 24 123 46 99)(11 83 138 53 157 72 146 66)(12 86 137 60 158 75 145 63)(13 89 136 57 159 78 144 70)(14 82 135 54 160 71 143 67)(15 85 134 51 151 74 142 64)(16 88 133 58 152 77 141 61)(17 81 132 55 153 80 150 68)(18 84 131 52 154 73 149 65)(19 87 140 59 155 76 148 62)(20 90 139 56 156 79 147 69)
(1 52)(2 53)(3 54)(4 55)(5 56)(6 57)(7 58)(8 59)(9 60)(10 51)(11 112)(12 113)(13 114)(14 115)(15 116)(16 117)(17 118)(18 119)(19 120)(20 111)(21 61)(22 62)(23 63)(24 64)(25 65)(26 66)(27 67)(28 68)(29 69)(30 70)(31 71)(32 72)(33 73)(34 74)(35 75)(36 76)(37 77)(38 78)(39 79)(40 80)(41 90)(42 81)(43 82)(44 83)(45 84)(46 85)(47 86)(48 87)(49 88)(50 89)(91 144)(92 145)(93 146)(94 147)(95 148)(96 149)(97 150)(98 141)(99 142)(100 143)(101 138)(102 139)(103 140)(104 131)(105 132)(106 133)(107 134)(108 135)(109 136)(110 137)(121 159)(122 160)(123 151)(124 152)(125 153)(126 154)(127 155)(128 156)(129 157)(130 158)
G:=sub<Sym(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), (1,52,6,57)(2,51,7,56)(3,60,8,55)(4,59,9,54)(5,58,10,53)(11,123,16,128)(12,122,17,127)(13,121,18,126)(14,130,19,125)(15,129,20,124)(21,69,26,64)(22,68,27,63)(23,67,28,62)(24,66,29,61)(25,65,30,70)(31,75,36,80)(32,74,37,79)(33,73,38,78)(34,72,39,77)(35,71,40,76)(41,88,46,83)(42,87,47,82)(43,86,48,81)(44,85,49,90)(45,84,50,89)(91,131,96,136)(92,140,97,135)(93,139,98,134)(94,138,99,133)(95,137,100,132)(101,147,106,142)(102,146,107,141)(103,145,108,150)(104,144,109,149)(105,143,110,148)(111,157,116,152)(112,156,117,151)(113,155,118,160)(114,154,119,159)(115,153,120,158), (1,119,33,104,25,126,45,96)(2,112,32,101,26,129,44,93)(3,115,31,108,27,122,43,100)(4,118,40,105,28,125,42,97)(5,111,39,102,29,128,41,94)(6,114,38,109,30,121,50,91)(7,117,37,106,21,124,49,98)(8,120,36,103,22,127,48,95)(9,113,35,110,23,130,47,92)(10,116,34,107,24,123,46,99)(11,83,138,53,157,72,146,66)(12,86,137,60,158,75,145,63)(13,89,136,57,159,78,144,70)(14,82,135,54,160,71,143,67)(15,85,134,51,151,74,142,64)(16,88,133,58,152,77,141,61)(17,81,132,55,153,80,150,68)(18,84,131,52,154,73,149,65)(19,87,140,59,155,76,148,62)(20,90,139,56,156,79,147,69), (1,52)(2,53)(3,54)(4,55)(5,56)(6,57)(7,58)(8,59)(9,60)(10,51)(11,112)(12,113)(13,114)(14,115)(15,116)(16,117)(17,118)(18,119)(19,120)(20,111)(21,61)(22,62)(23,63)(24,64)(25,65)(26,66)(27,67)(28,68)(29,69)(30,70)(31,71)(32,72)(33,73)(34,74)(35,75)(36,76)(37,77)(38,78)(39,79)(40,80)(41,90)(42,81)(43,82)(44,83)(45,84)(46,85)(47,86)(48,87)(49,88)(50,89)(91,144)(92,145)(93,146)(94,147)(95,148)(96,149)(97,150)(98,141)(99,142)(100,143)(101,138)(102,139)(103,140)(104,131)(105,132)(106,133)(107,134)(108,135)(109,136)(110,137)(121,159)(122,160)(123,151)(124,152)(125,153)(126,154)(127,155)(128,156)(129,157)(130,158)>;
G:=Group( (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), (1,52,6,57)(2,51,7,56)(3,60,8,55)(4,59,9,54)(5,58,10,53)(11,123,16,128)(12,122,17,127)(13,121,18,126)(14,130,19,125)(15,129,20,124)(21,69,26,64)(22,68,27,63)(23,67,28,62)(24,66,29,61)(25,65,30,70)(31,75,36,80)(32,74,37,79)(33,73,38,78)(34,72,39,77)(35,71,40,76)(41,88,46,83)(42,87,47,82)(43,86,48,81)(44,85,49,90)(45,84,50,89)(91,131,96,136)(92,140,97,135)(93,139,98,134)(94,138,99,133)(95,137,100,132)(101,147,106,142)(102,146,107,141)(103,145,108,150)(104,144,109,149)(105,143,110,148)(111,157,116,152)(112,156,117,151)(113,155,118,160)(114,154,119,159)(115,153,120,158), (1,119,33,104,25,126,45,96)(2,112,32,101,26,129,44,93)(3,115,31,108,27,122,43,100)(4,118,40,105,28,125,42,97)(5,111,39,102,29,128,41,94)(6,114,38,109,30,121,50,91)(7,117,37,106,21,124,49,98)(8,120,36,103,22,127,48,95)(9,113,35,110,23,130,47,92)(10,116,34,107,24,123,46,99)(11,83,138,53,157,72,146,66)(12,86,137,60,158,75,145,63)(13,89,136,57,159,78,144,70)(14,82,135,54,160,71,143,67)(15,85,134,51,151,74,142,64)(16,88,133,58,152,77,141,61)(17,81,132,55,153,80,150,68)(18,84,131,52,154,73,149,65)(19,87,140,59,155,76,148,62)(20,90,139,56,156,79,147,69), (1,52)(2,53)(3,54)(4,55)(5,56)(6,57)(7,58)(8,59)(9,60)(10,51)(11,112)(12,113)(13,114)(14,115)(15,116)(16,117)(17,118)(18,119)(19,120)(20,111)(21,61)(22,62)(23,63)(24,64)(25,65)(26,66)(27,67)(28,68)(29,69)(30,70)(31,71)(32,72)(33,73)(34,74)(35,75)(36,76)(37,77)(38,78)(39,79)(40,80)(41,90)(42,81)(43,82)(44,83)(45,84)(46,85)(47,86)(48,87)(49,88)(50,89)(91,144)(92,145)(93,146)(94,147)(95,148)(96,149)(97,150)(98,141)(99,142)(100,143)(101,138)(102,139)(103,140)(104,131)(105,132)(106,133)(107,134)(108,135)(109,136)(110,137)(121,159)(122,160)(123,151)(124,152)(125,153)(126,154)(127,155)(128,156)(129,157)(130,158) );
G=PermutationGroup([(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)], [(1,52,6,57),(2,51,7,56),(3,60,8,55),(4,59,9,54),(5,58,10,53),(11,123,16,128),(12,122,17,127),(13,121,18,126),(14,130,19,125),(15,129,20,124),(21,69,26,64),(22,68,27,63),(23,67,28,62),(24,66,29,61),(25,65,30,70),(31,75,36,80),(32,74,37,79),(33,73,38,78),(34,72,39,77),(35,71,40,76),(41,88,46,83),(42,87,47,82),(43,86,48,81),(44,85,49,90),(45,84,50,89),(91,131,96,136),(92,140,97,135),(93,139,98,134),(94,138,99,133),(95,137,100,132),(101,147,106,142),(102,146,107,141),(103,145,108,150),(104,144,109,149),(105,143,110,148),(111,157,116,152),(112,156,117,151),(113,155,118,160),(114,154,119,159),(115,153,120,158)], [(1,119,33,104,25,126,45,96),(2,112,32,101,26,129,44,93),(3,115,31,108,27,122,43,100),(4,118,40,105,28,125,42,97),(5,111,39,102,29,128,41,94),(6,114,38,109,30,121,50,91),(7,117,37,106,21,124,49,98),(8,120,36,103,22,127,48,95),(9,113,35,110,23,130,47,92),(10,116,34,107,24,123,46,99),(11,83,138,53,157,72,146,66),(12,86,137,60,158,75,145,63),(13,89,136,57,159,78,144,70),(14,82,135,54,160,71,143,67),(15,85,134,51,151,74,142,64),(16,88,133,58,152,77,141,61),(17,81,132,55,153,80,150,68),(18,84,131,52,154,73,149,65),(19,87,140,59,155,76,148,62),(20,90,139,56,156,79,147,69)], [(1,52),(2,53),(3,54),(4,55),(5,56),(6,57),(7,58),(8,59),(9,60),(10,51),(11,112),(12,113),(13,114),(14,115),(15,116),(16,117),(17,118),(18,119),(19,120),(20,111),(21,61),(22,62),(23,63),(24,64),(25,65),(26,66),(27,67),(28,68),(29,69),(30,70),(31,71),(32,72),(33,73),(34,74),(35,75),(36,76),(37,77),(38,78),(39,79),(40,80),(41,90),(42,81),(43,82),(44,83),(45,84),(46,85),(47,86),(48,87),(49,88),(50,89),(91,144),(92,145),(93,146),(94,147),(95,148),(96,149),(97,150),(98,141),(99,142),(100,143),(101,138),(102,139),(103,140),(104,131),(105,132),(106,133),(107,134),(108,135),(109,136),(110,137),(121,159),(122,160),(123,151),(124,152),(125,153),(126,154),(127,155),(128,156),(129,157),(130,158)])
Matrix representation ►G ⊆ GL6(𝔽41)
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 40 | 40 | 40 | 40 |
| 1 | 2 | 0 | 0 | 0 | 0 |
| 40 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 22 | 0 | 3 | 3 |
| 0 | 0 | 19 | 22 | 22 | 19 |
| 0 | 0 | 3 | 3 | 0 | 22 |
| 0 | 0 | 0 | 38 | 19 | 38 |
| 40 | 39 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 25 | 9 | 1 | 8 |
| 0 | 0 | 7 | 40 | 24 | 8 |
| 0 | 0 | 25 | 17 | 24 | 16 |
| 0 | 0 | 33 | 17 | 1 | 34 |
| 1 | 2 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 22 | 0 | 3 | 3 |
| 0 | 0 | 38 | 19 | 38 | 0 |
| 0 | 0 | 0 | 38 | 19 | 38 |
| 0 | 0 | 3 | 3 | 0 | 22 |
G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,40,0,0,1,0,0,40,0,0,0,1,0,40,0,0,0,0,1,40],[1,40,0,0,0,0,2,40,0,0,0,0,0,0,22,19,3,0,0,0,0,22,3,38,0,0,3,22,0,19,0,0,3,19,22,38],[40,0,0,0,0,0,39,1,0,0,0,0,0,0,25,7,25,33,0,0,9,40,17,17,0,0,1,24,24,1,0,0,8,8,16,34],[1,0,0,0,0,0,2,40,0,0,0,0,0,0,22,38,0,3,0,0,0,19,38,3,0,0,3,38,19,0,0,0,3,0,38,22] >;
38 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 5 | 8A | ··· | 8H | 8I | 8J | 8K | 8L | 10A | 10B | 10C | 10D | 10E | 20A | 20B | 20C | 20D |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 8 | ··· | 8 | 8 | 8 | 8 | 8 | 10 | 10 | 10 | 10 | 10 | 20 | 20 | 20 | 20 |
| size | 1 | 1 | 1 | 1 | 4 | 20 | 2 | 2 | 4 | 5 | 5 | 5 | 5 | 10 | 10 | 20 | 4 | 10 | ··· | 10 | 20 | 20 | 20 | 20 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 8 | 8 |
38 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 8 | 8 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | - | + | ||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | C4 | D4 | M4(2) | C4○D4 | C8○D4 | F5 | C2×F5 | C2×F5 | D5⋊M4(2) | D4.F5 | D4×F5 |
| kernel | Dic5⋊M4(2) | C4×C5⋊C8 | D10⋊C8 | Dic5⋊C8 | C23.2F5 | Dic5⋊4D4 | C2×C4.F5 | C2×C22.F5 | C10.D4 | D10⋊C4 | C5×C22⋊C4 | C2×C5⋊D4 | C5⋊C8 | Dic5 | Dic5 | C10 | C22⋊C4 | C2×C4 | C23 | C2 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 2 | 4 | 1 | 2 | 1 | 4 | 1 | 1 |
In GAP, Magma, Sage, TeX
Dic_5\rtimes M_{4(2)} % in TeX
G:=Group("Dic5:M4(2)"); // GroupNames label
G:=SmallGroup(320,1033);
// by ID
G=gap.SmallGroup(320,1033);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,344,1094,219,136,6278,1595]);
// Polycyclic
G:=Group<a,b,c,d|a^10=c^8=d^2=1,b^2=a^5,b*a*b^-1=a^-1,c*a*c^-1=a^7,a*d=d*a,c*b*c^-1=d*b*d=a^5*b,d*c*d=c^5>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀