metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Dic5⋊2M4(2), C5⋊2C8⋊16D4, C5⋊5(C8⋊6D4), C22⋊C8⋊14D5, C4.198(D4×D5), C10.58(C4×D4), (C8×Dic5)⋊16C2, (C2×C8).196D10, C20.357(C2×D4), D10⋊1C8⋊20C2, C23.14(C4×D5), C10.34(C8○D4), C20.8Q8⋊20C2, (C22×C4).80D10, C2.14(D5×M4(2)), C23.D5.13C4, D10⋊C4.16C4, C20.299(C4○D4), (C2×C40).173C22, (C2×C20).824C23, C10.D4.16C4, C10.57(C2×M4(2)), C4.125(D4⋊2D5), (C22×C20).95C22, C2.12(Dic5⋊4D4), C2.12(D20.3C4), (C4×Dic5).303C22, (C2×C4).33(C4×D5), (C4×C5⋊D4).2C2, (C2×C8⋊D5)⋊13C2, (C5×C22⋊C8)⋊18C2, (C2×C5⋊D4).15C4, (C2×C4.Dic5)⋊2C2, C22.106(C2×C4×D5), (C2×C20).214(C2×C4), (C2×C4×D5).230C22, (C2×Dic5).96(C2×C4), (C22×D5).18(C2×C4), (C2×C4).766(C22×D5), (C22×C10).110(C2×C4), (C2×C10).180(C22×C4), (C2×C5⋊2C8).196C22, SmallGroup(320,356)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic5⋊2M4(2)
G = < a,b,c,d | a10=c8=d2=1, b2=a5, bab-1=cac-1=a-1, ad=da, cbc-1=dbd=a5b, dcd=c5 >
Subgroups: 398 in 122 conjugacy classes, 51 normal (47 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, C23, C23, D5, C10, C10, C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, M4(2), C22×C4, C22×C4, C2×D4, Dic5, Dic5, C20, C20, D10, C2×C10, C2×C10, C4×C8, C22⋊C8, C22⋊C8, C4⋊C8, C4×D4, C2×M4(2), C5⋊2C8, C5⋊2C8, C40, C4×D5, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C22×D5, C22×C10, C8⋊6D4, C8⋊D5, C2×C5⋊2C8, C4.Dic5, C4×Dic5, C10.D4, D10⋊C4, C23.D5, C2×C40, C2×C4×D5, C2×C5⋊D4, C22×C20, C8×Dic5, C20.8Q8, D10⋊1C8, C5×C22⋊C8, C2×C8⋊D5, C2×C4.Dic5, C4×C5⋊D4, Dic5⋊2M4(2)
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D5, M4(2), C22×C4, C2×D4, C4○D4, D10, C4×D4, C2×M4(2), C8○D4, C4×D5, C22×D5, C8⋊6D4, C2×C4×D5, D4×D5, D4⋊2D5, Dic5⋊4D4, D20.3C4, D5×M4(2), Dic5⋊2M4(2)
►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 94 6 99)(2 93 7 98)(3 92 8 97)(4 91 9 96)(5 100 10 95)(11 81 16 86)(12 90 17 85)(13 89 18 84)(14 88 19 83)(15 87 20 82)(21 109 26 104)(22 108 27 103)(23 107 28 102)(24 106 29 101)(25 105 30 110)(31 126 36 121)(32 125 37 130)(33 124 38 129)(34 123 39 128)(35 122 40 127)(41 116 46 111)(42 115 47 120)(43 114 48 119)(44 113 49 118)(45 112 50 117)(51 132 56 137)(52 131 57 136)(53 140 58 135)(54 139 59 134)(55 138 60 133)(61 150 66 145)(62 149 67 144)(63 148 68 143)(64 147 69 142)(65 146 70 141)(71 152 76 157)(72 151 77 156)(73 160 78 155)(74 159 79 154)(75 158 80 153)
(1 79 48 63 27 88 39 54)(2 78 49 62 28 87 40 53)(3 77 50 61 29 86 31 52)(4 76 41 70 30 85 32 51)(5 75 42 69 21 84 33 60)(6 74 43 68 22 83 34 59)(7 73 44 67 23 82 35 58)(8 72 45 66 24 81 36 57)(9 71 46 65 25 90 37 56)(10 80 47 64 26 89 38 55)(11 121 131 97 156 112 150 106)(12 130 132 96 157 111 141 105)(13 129 133 95 158 120 142 104)(14 128 134 94 159 119 143 103)(15 127 135 93 160 118 144 102)(16 126 136 92 151 117 145 101)(17 125 137 91 152 116 146 110)(18 124 138 100 153 115 147 109)(19 123 139 99 154 114 148 108)(20 122 140 98 155 113 149 107)
(1 103)(2 104)(3 105)(4 106)(5 107)(6 108)(7 109)(8 110)(9 101)(10 102)(11 85)(12 86)(13 87)(14 88)(15 89)(16 90)(17 81)(18 82)(19 83)(20 84)(21 98)(22 99)(23 100)(24 91)(25 92)(26 93)(27 94)(28 95)(29 96)(30 97)(31 111)(32 112)(33 113)(34 114)(35 115)(36 116)(37 117)(38 118)(39 119)(40 120)(41 121)(42 122)(43 123)(44 124)(45 125)(46 126)(47 127)(48 128)(49 129)(50 130)(51 131)(52 132)(53 133)(54 134)(55 135)(56 136)(57 137)(58 138)(59 139)(60 140)(61 141)(62 142)(63 143)(64 144)(65 145)(66 146)(67 147)(68 148)(69 149)(70 150)(71 151)(72 152)(73 153)(74 154)(75 155)(76 156)(77 157)(78 158)(79 159)(80 160)
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), (1,94,6,99)(2,93,7,98)(3,92,8,97)(4,91,9,96)(5,100,10,95)(11,81,16,86)(12,90,17,85)(13,89,18,84)(14,88,19,83)(15,87,20,82)(21,109,26,104)(22,108,27,103)(23,107,28,102)(24,106,29,101)(25,105,30,110)(31,126,36,121)(32,125,37,130)(33,124,38,129)(34,123,39,128)(35,122,40,127)(41,116,46,111)(42,115,47,120)(43,114,48,119)(44,113,49,118)(45,112,50,117)(51,132,56,137)(52,131,57,136)(53,140,58,135)(54,139,59,134)(55,138,60,133)(61,150,66,145)(62,149,67,144)(63,148,68,143)(64,147,69,142)(65,146,70,141)(71,152,76,157)(72,151,77,156)(73,160,78,155)(74,159,79,154)(75,158,80,153), (1,79,48,63,27,88,39,54)(2,78,49,62,28,87,40,53)(3,77,50,61,29,86,31,52)(4,76,41,70,30,85,32,51)(5,75,42,69,21,84,33,60)(6,74,43,68,22,83,34,59)(7,73,44,67,23,82,35,58)(8,72,45,66,24,81,36,57)(9,71,46,65,25,90,37,56)(10,80,47,64,26,89,38,55)(11,121,131,97,156,112,150,106)(12,130,132,96,157,111,141,105)(13,129,133,95,158,120,142,104)(14,128,134,94,159,119,143,103)(15,127,135,93,160,118,144,102)(16,126,136,92,151,117,145,101)(17,125,137,91,152,116,146,110)(18,124,138,100,153,115,147,109)(19,123,139,99,154,114,148,108)(20,122,140,98,155,113,149,107), (1,103)(2,104)(3,105)(4,106)(5,107)(6,108)(7,109)(8,110)(9,101)(10,102)(11,85)(12,86)(13,87)(14,88)(15,89)(16,90)(17,81)(18,82)(19,83)(20,84)(21,98)(22,99)(23,100)(24,91)(25,92)(26,93)(27,94)(28,95)(29,96)(30,97)(31,111)(32,112)(33,113)(34,114)(35,115)(36,116)(37,117)(38,118)(39,119)(40,120)(41,121)(42,122)(43,123)(44,124)(45,125)(46,126)(47,127)(48,128)(49,129)(50,130)(51,131)(52,132)(53,133)(54,134)(55,135)(56,136)(57,137)(58,138)(59,139)(60,140)(61,141)(62,142)(63,143)(64,144)(65,145)(66,146)(67,147)(68,148)(69,149)(70,150)(71,151)(72,152)(73,153)(74,154)(75,155)(76,156)(77,157)(78,158)(79,159)(80,160)>;
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), (1,94,6,99)(2,93,7,98)(3,92,8,97)(4,91,9,96)(5,100,10,95)(11,81,16,86)(12,90,17,85)(13,89,18,84)(14,88,19,83)(15,87,20,82)(21,109,26,104)(22,108,27,103)(23,107,28,102)(24,106,29,101)(25,105,30,110)(31,126,36,121)(32,125,37,130)(33,124,38,129)(34,123,39,128)(35,122,40,127)(41,116,46,111)(42,115,47,120)(43,114,48,119)(44,113,49,118)(45,112,50,117)(51,132,56,137)(52,131,57,136)(53,140,58,135)(54,139,59,134)(55,138,60,133)(61,150,66,145)(62,149,67,144)(63,148,68,143)(64,147,69,142)(65,146,70,141)(71,152,76,157)(72,151,77,156)(73,160,78,155)(74,159,79,154)(75,158,80,153), (1,79,48,63,27,88,39,54)(2,78,49,62,28,87,40,53)(3,77,50,61,29,86,31,52)(4,76,41,70,30,85,32,51)(5,75,42,69,21,84,33,60)(6,74,43,68,22,83,34,59)(7,73,44,67,23,82,35,58)(8,72,45,66,24,81,36,57)(9,71,46,65,25,90,37,56)(10,80,47,64,26,89,38,55)(11,121,131,97,156,112,150,106)(12,130,132,96,157,111,141,105)(13,129,133,95,158,120,142,104)(14,128,134,94,159,119,143,103)(15,127,135,93,160,118,144,102)(16,126,136,92,151,117,145,101)(17,125,137,91,152,116,146,110)(18,124,138,100,153,115,147,109)(19,123,139,99,154,114,148,108)(20,122,140,98,155,113,149,107), (1,103)(2,104)(3,105)(4,106)(5,107)(6,108)(7,109)(8,110)(9,101)(10,102)(11,85)(12,86)(13,87)(14,88)(15,89)(16,90)(17,81)(18,82)(19,83)(20,84)(21,98)(22,99)(23,100)(24,91)(25,92)(26,93)(27,94)(28,95)(29,96)(30,97)(31,111)(32,112)(33,113)(34,114)(35,115)(36,116)(37,117)(38,118)(39,119)(40,120)(41,121)(42,122)(43,123)(44,124)(45,125)(46,126)(47,127)(48,128)(49,129)(50,130)(51,131)(52,132)(53,133)(54,134)(55,135)(56,136)(57,137)(58,138)(59,139)(60,140)(61,141)(62,142)(63,143)(64,144)(65,145)(66,146)(67,147)(68,148)(69,149)(70,150)(71,151)(72,152)(73,153)(74,154)(75,155)(76,156)(77,157)(78,158)(79,159)(80,160) );
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)], [(1,94,6,99),(2,93,7,98),(3,92,8,97),(4,91,9,96),(5,100,10,95),(11,81,16,86),(12,90,17,85),(13,89,18,84),(14,88,19,83),(15,87,20,82),(21,109,26,104),(22,108,27,103),(23,107,28,102),(24,106,29,101),(25,105,30,110),(31,126,36,121),(32,125,37,130),(33,124,38,129),(34,123,39,128),(35,122,40,127),(41,116,46,111),(42,115,47,120),(43,114,48,119),(44,113,49,118),(45,112,50,117),(51,132,56,137),(52,131,57,136),(53,140,58,135),(54,139,59,134),(55,138,60,133),(61,150,66,145),(62,149,67,144),(63,148,68,143),(64,147,69,142),(65,146,70,141),(71,152,76,157),(72,151,77,156),(73,160,78,155),(74,159,79,154),(75,158,80,153)], [(1,79,48,63,27,88,39,54),(2,78,49,62,28,87,40,53),(3,77,50,61,29,86,31,52),(4,76,41,70,30,85,32,51),(5,75,42,69,21,84,33,60),(6,74,43,68,22,83,34,59),(7,73,44,67,23,82,35,58),(8,72,45,66,24,81,36,57),(9,71,46,65,25,90,37,56),(10,80,47,64,26,89,38,55),(11,121,131,97,156,112,150,106),(12,130,132,96,157,111,141,105),(13,129,133,95,158,120,142,104),(14,128,134,94,159,119,143,103),(15,127,135,93,160,118,144,102),(16,126,136,92,151,117,145,101),(17,125,137,91,152,116,146,110),(18,124,138,100,153,115,147,109),(19,123,139,99,154,114,148,108),(20,122,140,98,155,113,149,107)], [(1,103),(2,104),(3,105),(4,106),(5,107),(6,108),(7,109),(8,110),(9,101),(10,102),(11,85),(12,86),(13,87),(14,88),(15,89),(16,90),(17,81),(18,82),(19,83),(20,84),(21,98),(22,99),(23,100),(24,91),(25,92),(26,93),(27,94),(28,95),(29,96),(30,97),(31,111),(32,112),(33,113),(34,114),(35,115),(36,116),(37,117),(38,118),(39,119),(40,120),(41,121),(42,122),(43,123),(44,124),(45,125),(46,126),(47,127),(48,128),(49,129),(50,130),(51,131),(52,132),(53,133),(54,134),(55,135),(56,136),(57,137),(58,138),(59,139),(60,140),(61,141),(62,142),(63,143),(64,144),(65,145),(66,146),(67,147),(68,148),(69,149),(70,150),(71,151),(72,152),(73,153),(74,154),(75,155),(76,156),(77,157),(78,158),(79,159),(80,160)]])
68 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 5A | 5B | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 8I | 8J | 8K | 8L | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 20A | ··· | 20H | 20I | 20J | 20K | 20L | 40A | ··· | 40P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 20 | ··· | 20 | 20 | 20 | 20 | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 4 | 20 | 1 | 1 | 1 | 1 | 4 | 10 | 10 | 10 | 10 | 20 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 10 | 10 | 10 | 10 | 20 | 20 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
68 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | - | |||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | C4 | D4 | D5 | M4(2) | C4○D4 | D10 | D10 | C8○D4 | C4×D5 | C4×D5 | D20.3C4 | D4×D5 | D4⋊2D5 | D5×M4(2) |
| kernel | Dic5⋊2M4(2) | C8×Dic5 | C20.8Q8 | D10⋊1C8 | C5×C22⋊C8 | C2×C8⋊D5 | C2×C4.Dic5 | C4×C5⋊D4 | C10.D4 | D10⋊C4 | C23.D5 | C2×C5⋊D4 | C5⋊2C8 | C22⋊C8 | Dic5 | C20 | C2×C8 | C22×C4 | C10 | C2×C4 | C23 | C2 | C4 | C4 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 2 | 4 | 2 | 4 | 4 | 4 | 16 | 2 | 2 | 4 |
Matrix representation of Dic5⋊2M4(2) ►in GL6(𝔽41)
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 40 | 34 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 9 | 5 | 0 | 0 | 0 | 0 |
| 0 | 32 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 7 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 8 |
| 0 | 0 | 0 | 0 | 10 | 1 |
| 1 | 37 | 0 | 0 | 0 | 0 |
| 21 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 7 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 14 | 0 |
| 0 | 0 | 0 | 0 | 24 | 27 |
| 32 | 36 | 0 | 0 | 0 | 0 |
| 16 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 8 |
| 0 | 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,40,0,0,0,0,1,34,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[9,0,0,0,0,0,5,32,0,0,0,0,0,0,40,7,0,0,0,0,0,1,0,0,0,0,0,0,40,10,0,0,0,0,8,1],[1,21,0,0,0,0,37,40,0,0,0,0,0,0,40,7,0,0,0,0,0,1,0,0,0,0,0,0,14,24,0,0,0,0,0,27],[32,16,0,0,0,0,36,9,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,8,1] >;
Dic5⋊2M4(2) in GAP, Magma, Sage, TeX
{\rm Dic}_5\rtimes_2M_4(2) % in TeX
G:=Group("Dic5:2M4(2)"); // GroupNames label
G:=SmallGroup(320,356);
// by ID
G=gap.SmallGroup(320,356);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,701,422,219,58,136,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^10=c^8=d^2=1,b^2=a^5,b*a*b^-1=c*a*c^-1=a^-1,a*d=d*a,c*b*c^-1=d*b*d=a^5*b,d*c*d=c^5>;
// generators/relations