metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D10⋊4M4(2), C5⋊2C8⋊25D4, C5⋊6(C8⋊9D4), C22⋊C8⋊13D5, C40⋊8C4⋊13C2, C10.57(C4×D4), C4.197(D4×D5), (C2×C8).195D10, C20.356(C2×D4), D10⋊1C8⋊19C2, C23.13(C4×D5), C10.33(C8○D4), C20.8Q8⋊19C2, (C22×C4).79D10, C2.13(D5×M4(2)), C23.D5.12C4, D10⋊C4.15C4, C20.298(C4○D4), (C2×C20).823C23, (C2×C40).172C22, C10.D4.15C4, C10.56(C2×M4(2)), C4.124(D4⋊2D5), (C22×C20).94C22, C2.11(Dic5⋊4D4), C2.11(D20.3C4), (C4×Dic5).202C22, (D5×C2×C8)⋊15C2, (C2×C4).32(C4×D5), (C4×C5⋊D4).1C2, (C5×C22⋊C8)⋊17C2, (C2×C5⋊D4).14C4, (C2×C4.Dic5)⋊1C2, C22.105(C2×C4×D5), (C2×C20).213(C2×C4), (C2×C4×D5).345C22, (C2×Dic5).19(C2×C4), (C22×D5).73(C2×C4), (C2×C4).765(C22×D5), (C2×C10).179(C22×C4), (C22×C10).109(C2×C4), (C2×C5⋊2C8).195C22, SmallGroup(320,355)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D10⋊4M4(2)
G = < a,b,c,d | a10=b2=c8=d2=1, bab=cac-1=a-1, ad=da, cbc-1=a3b, dbd=a5b, dcd=c5 >
Subgroups: 398 in 124 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, C20, C20, D10, D10, C2×C10, C2×C10, C8⋊C4, C22⋊C8, C22⋊C8, C4⋊C8, C4×D4, C22×C8, C2×M4(2), C5⋊2C8, C5⋊2C8, C40, C4×D5, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C22×D5, C22×C10, C8⋊9D4, 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, C20.8Q8, C40⋊8C4, D10⋊1C8, C5×C22⋊C8, D5×C2×C8, C2×C4.Dic5, C4×C5⋊D4, D10⋊4M4(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⋊9D4, C2×C4×D5, D4×D5, D4⋊2D5, Dic5⋊4D4, D20.3C4, D5×M4(2), D10⋊4M4(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 22)(2 21)(3 30)(4 29)(5 28)(6 27)(7 26)(8 25)(9 24)(10 23)(11 152)(12 151)(13 160)(14 159)(15 158)(16 157)(17 156)(18 155)(19 154)(20 153)(31 41)(32 50)(33 49)(34 48)(35 47)(36 46)(37 45)(38 44)(39 43)(40 42)(51 61)(52 70)(53 69)(54 68)(55 67)(56 66)(57 65)(58 64)(59 63)(60 62)(71 81)(72 90)(73 89)(74 88)(75 87)(76 86)(77 85)(78 84)(79 83)(80 82)(91 106)(92 105)(93 104)(94 103)(95 102)(96 101)(97 110)(98 109)(99 108)(100 107)(111 126)(112 125)(113 124)(114 123)(115 122)(116 121)(117 130)(118 129)(119 128)(120 127)(131 146)(132 145)(133 144)(134 143)(135 142)(136 141)(137 150)(138 149)(139 148)(140 147)
(1 11 45 138 23 158 38 145)(2 20 46 137 24 157 39 144)(3 19 47 136 25 156 40 143)(4 18 48 135 26 155 31 142)(5 17 49 134 27 154 32 141)(6 16 50 133 28 153 33 150)(7 15 41 132 29 152 34 149)(8 14 42 131 30 151 35 148)(9 13 43 140 21 160 36 147)(10 12 44 139 22 159 37 146)(51 102 71 122 68 95 88 115)(52 101 72 121 69 94 89 114)(53 110 73 130 70 93 90 113)(54 109 74 129 61 92 81 112)(55 108 75 128 62 91 82 111)(56 107 76 127 63 100 83 120)(57 106 77 126 64 99 84 119)(58 105 78 125 65 98 85 118)(59 104 79 124 66 97 86 117)(60 103 80 123 67 96 87 116)
(1 105)(2 106)(3 107)(4 108)(5 109)(6 110)(7 101)(8 102)(9 103)(10 104)(11 85)(12 86)(13 87)(14 88)(15 89)(16 90)(17 81)(18 82)(19 83)(20 84)(21 96)(22 97)(23 98)(24 99)(25 100)(26 91)(27 92)(28 93)(29 94)(30 95)(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), (1,22)(2,21)(3,30)(4,29)(5,28)(6,27)(7,26)(8,25)(9,24)(10,23)(11,152)(12,151)(13,160)(14,159)(15,158)(16,157)(17,156)(18,155)(19,154)(20,153)(31,41)(32,50)(33,49)(34,48)(35,47)(36,46)(37,45)(38,44)(39,43)(40,42)(51,61)(52,70)(53,69)(54,68)(55,67)(56,66)(57,65)(58,64)(59,63)(60,62)(71,81)(72,90)(73,89)(74,88)(75,87)(76,86)(77,85)(78,84)(79,83)(80,82)(91,106)(92,105)(93,104)(94,103)(95,102)(96,101)(97,110)(98,109)(99,108)(100,107)(111,126)(112,125)(113,124)(114,123)(115,122)(116,121)(117,130)(118,129)(119,128)(120,127)(131,146)(132,145)(133,144)(134,143)(135,142)(136,141)(137,150)(138,149)(139,148)(140,147), (1,11,45,138,23,158,38,145)(2,20,46,137,24,157,39,144)(3,19,47,136,25,156,40,143)(4,18,48,135,26,155,31,142)(5,17,49,134,27,154,32,141)(6,16,50,133,28,153,33,150)(7,15,41,132,29,152,34,149)(8,14,42,131,30,151,35,148)(9,13,43,140,21,160,36,147)(10,12,44,139,22,159,37,146)(51,102,71,122,68,95,88,115)(52,101,72,121,69,94,89,114)(53,110,73,130,70,93,90,113)(54,109,74,129,61,92,81,112)(55,108,75,128,62,91,82,111)(56,107,76,127,63,100,83,120)(57,106,77,126,64,99,84,119)(58,105,78,125,65,98,85,118)(59,104,79,124,66,97,86,117)(60,103,80,123,67,96,87,116), (1,105)(2,106)(3,107)(4,108)(5,109)(6,110)(7,101)(8,102)(9,103)(10,104)(11,85)(12,86)(13,87)(14,88)(15,89)(16,90)(17,81)(18,82)(19,83)(20,84)(21,96)(22,97)(23,98)(24,99)(25,100)(26,91)(27,92)(28,93)(29,94)(30,95)(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), (1,22)(2,21)(3,30)(4,29)(5,28)(6,27)(7,26)(8,25)(9,24)(10,23)(11,152)(12,151)(13,160)(14,159)(15,158)(16,157)(17,156)(18,155)(19,154)(20,153)(31,41)(32,50)(33,49)(34,48)(35,47)(36,46)(37,45)(38,44)(39,43)(40,42)(51,61)(52,70)(53,69)(54,68)(55,67)(56,66)(57,65)(58,64)(59,63)(60,62)(71,81)(72,90)(73,89)(74,88)(75,87)(76,86)(77,85)(78,84)(79,83)(80,82)(91,106)(92,105)(93,104)(94,103)(95,102)(96,101)(97,110)(98,109)(99,108)(100,107)(111,126)(112,125)(113,124)(114,123)(115,122)(116,121)(117,130)(118,129)(119,128)(120,127)(131,146)(132,145)(133,144)(134,143)(135,142)(136,141)(137,150)(138,149)(139,148)(140,147), (1,11,45,138,23,158,38,145)(2,20,46,137,24,157,39,144)(3,19,47,136,25,156,40,143)(4,18,48,135,26,155,31,142)(5,17,49,134,27,154,32,141)(6,16,50,133,28,153,33,150)(7,15,41,132,29,152,34,149)(8,14,42,131,30,151,35,148)(9,13,43,140,21,160,36,147)(10,12,44,139,22,159,37,146)(51,102,71,122,68,95,88,115)(52,101,72,121,69,94,89,114)(53,110,73,130,70,93,90,113)(54,109,74,129,61,92,81,112)(55,108,75,128,62,91,82,111)(56,107,76,127,63,100,83,120)(57,106,77,126,64,99,84,119)(58,105,78,125,65,98,85,118)(59,104,79,124,66,97,86,117)(60,103,80,123,67,96,87,116), (1,105)(2,106)(3,107)(4,108)(5,109)(6,110)(7,101)(8,102)(9,103)(10,104)(11,85)(12,86)(13,87)(14,88)(15,89)(16,90)(17,81)(18,82)(19,83)(20,84)(21,96)(22,97)(23,98)(24,99)(25,100)(26,91)(27,92)(28,93)(29,94)(30,95)(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)], [(1,22),(2,21),(3,30),(4,29),(5,28),(6,27),(7,26),(8,25),(9,24),(10,23),(11,152),(12,151),(13,160),(14,159),(15,158),(16,157),(17,156),(18,155),(19,154),(20,153),(31,41),(32,50),(33,49),(34,48),(35,47),(36,46),(37,45),(38,44),(39,43),(40,42),(51,61),(52,70),(53,69),(54,68),(55,67),(56,66),(57,65),(58,64),(59,63),(60,62),(71,81),(72,90),(73,89),(74,88),(75,87),(76,86),(77,85),(78,84),(79,83),(80,82),(91,106),(92,105),(93,104),(94,103),(95,102),(96,101),(97,110),(98,109),(99,108),(100,107),(111,126),(112,125),(113,124),(114,123),(115,122),(116,121),(117,130),(118,129),(119,128),(120,127),(131,146),(132,145),(133,144),(134,143),(135,142),(136,141),(137,150),(138,149),(139,148),(140,147)], [(1,11,45,138,23,158,38,145),(2,20,46,137,24,157,39,144),(3,19,47,136,25,156,40,143),(4,18,48,135,26,155,31,142),(5,17,49,134,27,154,32,141),(6,16,50,133,28,153,33,150),(7,15,41,132,29,152,34,149),(8,14,42,131,30,151,35,148),(9,13,43,140,21,160,36,147),(10,12,44,139,22,159,37,146),(51,102,71,122,68,95,88,115),(52,101,72,121,69,94,89,114),(53,110,73,130,70,93,90,113),(54,109,74,129,61,92,81,112),(55,108,75,128,62,91,82,111),(56,107,76,127,63,100,83,120),(57,106,77,126,64,99,84,119),(58,105,78,125,65,98,85,118),(59,104,79,124,66,97,86,117),(60,103,80,123,67,96,87,116)], [(1,105),(2,106),(3,107),(4,108),(5,109),(6,110),(7,101),(8,102),(9,103),(10,104),(11,85),(12,86),(13,87),(14,88),(15,89),(16,90),(17,81),(18,82),(19,83),(20,84),(21,96),(22,97),(23,98),(24,99),(25,100),(26,91),(27,92),(28,93),(29,94),(30,95),(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 | 2F | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 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 | 2 | 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 | 10 | 10 | 1 | 1 | 1 | 1 | 4 | 10 | 10 | 20 | 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 | C4○D4 | M4(2) | D10 | D10 | C8○D4 | C4×D5 | C4×D5 | D20.3C4 | D4×D5 | D4⋊2D5 | D5×M4(2) |
| kernel | D10⋊4M4(2) | C20.8Q8 | C40⋊8C4 | D10⋊1C8 | C5×C22⋊C8 | D5×C2×C8 | C2×C4.Dic5 | C4×C5⋊D4 | C10.D4 | D10⋊C4 | C23.D5 | C2×C5⋊D4 | C5⋊2C8 | C22⋊C8 | C20 | D10 | 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 | 2 | 4 | 4 | 2 | 4 | 4 | 4 | 16 | 2 | 2 | 4 |
Matrix representation of D10⋊4M4(2) ►in GL4(𝔽41) generated by
| 0 | 7 | 0 | 0 |
| 35 | 7 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 40 |
| 7 | 34 | 0 | 0 |
| 1 | 34 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 22 | 1 |
| 1 | 7 | 0 | 0 |
| 7 | 40 | 0 | 0 |
| 0 | 0 | 34 | 18 |
| 0 | 0 | 20 | 7 |
| 17 | 7 | 0 | 0 |
| 35 | 24 | 0 | 0 |
| 0 | 0 | 19 | 39 |
| 0 | 0 | 16 | 22 |
G:=sub<GL(4,GF(41))| [0,35,0,0,7,7,0,0,0,0,40,0,0,0,0,40],[7,1,0,0,34,34,0,0,0,0,40,22,0,0,0,1],[1,7,0,0,7,40,0,0,0,0,34,20,0,0,18,7],[17,35,0,0,7,24,0,0,0,0,19,16,0,0,39,22] >;
D10⋊4M4(2) in GAP, Magma, Sage, TeX
D_{10}\rtimes_4M_4(2) % in TeX
G:=Group("D10:4M4(2)"); // GroupNames label
G:=SmallGroup(320,355);
// by ID
G=gap.SmallGroup(320,355);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,701,219,58,136,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^10=b^2=c^8=d^2=1,b*a*b=c*a*c^-1=a^-1,a*d=d*a,c*b*c^-1=a^3*b,d*b*d=a^5*b,d*c*d=c^5>;
// generators/relations