metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D10⋊1M4(2), C5⋊C8⋊5D4, C2.7(D4×F5), C5⋊2(C8⋊9D4), C10.3(C4×D4), C20⋊C8⋊10C2, C22⋊C4.3F5, C23.7(C2×F5), C10.4(C8○D4), D10⋊C8⋊12C2, C2.7(D4.F5), D10⋊C4.2C4, Dic5.67(C2×D4), C10.D4.7C4, C23.2F5⋊5C2, Dic5⋊4D4.8C2, C10.11(C2×M4(2)), C10.C42⋊10C2, Dic5.52(C4○D4), C22.70(C22×F5), C2.11(D5⋊M4(2)), (C2×Dic5).324C23, (C4×Dic5).241C22, (C22×Dic5).179C22, (C2×D5⋊C8)⋊9C2, (C2×C5⋊D4).5C4, (C2×C4).21(C2×F5), (C2×C20).79(C2×C4), (C5×C22⋊C4).4C4, (C2×C5⋊C8).22C22, (C2×C22.F5)⋊1C2, (C2×C4×D5).287C22, (C22×C10).15(C2×C4), (C2×C10).32(C22×C4), (C2×Dic5).49(C2×C4), (C22×D5).41(C2×C4), SmallGroup(320,1032)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D10⋊M4(2)
G = < a,b,c,d | a10=b2=c8=d2=1, bab=a-1, cac-1=a3, ad=da, cbc-1=a7b, dbd=a5b, dcd=c5 >
Subgroups: 442 in 124 conjugacy classes, 46 normal (42 characteristic)
C1, C2, C2, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, C23, C23, D5, C10, C10, C42, C22⋊C4, C22⋊C4, C4⋊C4, C2×C8, M4(2), C22×C4, C2×D4, Dic5, Dic5, C20, D10, D10, C2×C10, C2×C10, C8⋊C4, C22⋊C8, C4⋊C8, C4×D4, C22×C8, C2×M4(2), C5⋊C8, C5⋊C8, C4×D5, C2×Dic5, C2×Dic5, C5⋊D4, C2×C20, C22×D5, C22×C10, C8⋊9D4, C4×Dic5, C10.D4, D10⋊C4, C5×C22⋊C4, D5⋊C8, C2×C5⋊C8, C22.F5, C2×C4×D5, C22×Dic5, C2×C5⋊D4, C20⋊C8, C10.C42, D10⋊C8, C23.2F5, Dic5⋊4D4, C2×D5⋊C8, C2×C22.F5, D10⋊M4(2)
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, M4(2), C22×C4, C2×D4, C4○D4, F5, C4×D4, C2×M4(2), C8○D4, C2×F5, C8⋊9D4, C22×F5, D5⋊M4(2), D4.F5, D4×F5, D10⋊M4(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 10)(2 9)(3 8)(4 7)(5 6)(12 20)(13 19)(14 18)(15 17)(21 30)(22 29)(23 28)(24 27)(25 26)(31 38)(32 37)(33 36)(34 35)(39 40)(41 46)(42 45)(43 44)(47 50)(48 49)(51 53)(54 60)(55 59)(56 58)(61 65)(62 64)(66 70)(67 69)(71 73)(74 80)(75 79)(76 78)(81 85)(82 84)(86 90)(87 89)(91 100)(92 99)(93 98)(94 97)(95 96)(101 108)(102 107)(103 106)(104 105)(109 110)(111 116)(112 115)(113 114)(117 120)(118 119)(121 128)(122 127)(123 126)(124 125)(129 130)(131 135)(132 134)(136 140)(137 139)(141 143)(144 150)(145 149)(146 148)(152 160)(153 159)(154 158)(155 157)
(1 154 40 145 21 14 44 136)(2 151 39 148 22 11 43 139)(3 158 38 141 23 18 42 132)(4 155 37 144 24 15 41 135)(5 152 36 147 25 12 50 138)(6 159 35 150 26 19 49 131)(7 156 34 143 27 16 48 134)(8 153 33 146 28 13 47 137)(9 160 32 149 29 20 46 140)(10 157 31 142 30 17 45 133)(51 111 79 108 62 127 85 99)(52 118 78 101 63 124 84 92)(53 115 77 104 64 121 83 95)(54 112 76 107 65 128 82 98)(55 119 75 110 66 125 81 91)(56 116 74 103 67 122 90 94)(57 113 73 106 68 129 89 97)(58 120 72 109 69 126 88 100)(59 117 71 102 70 123 87 93)(60 114 80 105 61 130 86 96)
(1 60)(2 51)(3 52)(4 53)(5 54)(6 55)(7 56)(8 57)(9 58)(10 59)(11 111)(12 112)(13 113)(14 114)(15 115)(16 116)(17 117)(18 118)(19 119)(20 120)(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 83)(42 84)(43 85)(44 86)(45 87)(46 88)(47 89)(48 90)(49 81)(50 82)(91 150)(92 141)(93 142)(94 143)(95 144)(96 145)(97 146)(98 147)(99 148)(100 149)(101 132)(102 133)(103 134)(104 135)(105 136)(106 137)(107 138)(108 139)(109 140)(110 131)(121 155)(122 156)(123 157)(124 158)(125 159)(126 160)(127 151)(128 152)(129 153)(130 154)
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), (1,10)(2,9)(3,8)(4,7)(5,6)(12,20)(13,19)(14,18)(15,17)(21,30)(22,29)(23,28)(24,27)(25,26)(31,38)(32,37)(33,36)(34,35)(39,40)(41,46)(42,45)(43,44)(47,50)(48,49)(51,53)(54,60)(55,59)(56,58)(61,65)(62,64)(66,70)(67,69)(71,73)(74,80)(75,79)(76,78)(81,85)(82,84)(86,90)(87,89)(91,100)(92,99)(93,98)(94,97)(95,96)(101,108)(102,107)(103,106)(104,105)(109,110)(111,116)(112,115)(113,114)(117,120)(118,119)(121,128)(122,127)(123,126)(124,125)(129,130)(131,135)(132,134)(136,140)(137,139)(141,143)(144,150)(145,149)(146,148)(152,160)(153,159)(154,158)(155,157), (1,154,40,145,21,14,44,136)(2,151,39,148,22,11,43,139)(3,158,38,141,23,18,42,132)(4,155,37,144,24,15,41,135)(5,152,36,147,25,12,50,138)(6,159,35,150,26,19,49,131)(7,156,34,143,27,16,48,134)(8,153,33,146,28,13,47,137)(9,160,32,149,29,20,46,140)(10,157,31,142,30,17,45,133)(51,111,79,108,62,127,85,99)(52,118,78,101,63,124,84,92)(53,115,77,104,64,121,83,95)(54,112,76,107,65,128,82,98)(55,119,75,110,66,125,81,91)(56,116,74,103,67,122,90,94)(57,113,73,106,68,129,89,97)(58,120,72,109,69,126,88,100)(59,117,71,102,70,123,87,93)(60,114,80,105,61,130,86,96), (1,60)(2,51)(3,52)(4,53)(5,54)(6,55)(7,56)(8,57)(9,58)(10,59)(11,111)(12,112)(13,113)(14,114)(15,115)(16,116)(17,117)(18,118)(19,119)(20,120)(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)(41,83)(42,84)(43,85)(44,86)(45,87)(46,88)(47,89)(48,90)(49,81)(50,82)(91,150)(92,141)(93,142)(94,143)(95,144)(96,145)(97,146)(98,147)(99,148)(100,149)(101,132)(102,133)(103,134)(104,135)(105,136)(106,137)(107,138)(108,139)(109,140)(110,131)(121,155)(122,156)(123,157)(124,158)(125,159)(126,160)(127,151)(128,152)(129,153)(130,154)>;
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), (1,10)(2,9)(3,8)(4,7)(5,6)(12,20)(13,19)(14,18)(15,17)(21,30)(22,29)(23,28)(24,27)(25,26)(31,38)(32,37)(33,36)(34,35)(39,40)(41,46)(42,45)(43,44)(47,50)(48,49)(51,53)(54,60)(55,59)(56,58)(61,65)(62,64)(66,70)(67,69)(71,73)(74,80)(75,79)(76,78)(81,85)(82,84)(86,90)(87,89)(91,100)(92,99)(93,98)(94,97)(95,96)(101,108)(102,107)(103,106)(104,105)(109,110)(111,116)(112,115)(113,114)(117,120)(118,119)(121,128)(122,127)(123,126)(124,125)(129,130)(131,135)(132,134)(136,140)(137,139)(141,143)(144,150)(145,149)(146,148)(152,160)(153,159)(154,158)(155,157), (1,154,40,145,21,14,44,136)(2,151,39,148,22,11,43,139)(3,158,38,141,23,18,42,132)(4,155,37,144,24,15,41,135)(5,152,36,147,25,12,50,138)(6,159,35,150,26,19,49,131)(7,156,34,143,27,16,48,134)(8,153,33,146,28,13,47,137)(9,160,32,149,29,20,46,140)(10,157,31,142,30,17,45,133)(51,111,79,108,62,127,85,99)(52,118,78,101,63,124,84,92)(53,115,77,104,64,121,83,95)(54,112,76,107,65,128,82,98)(55,119,75,110,66,125,81,91)(56,116,74,103,67,122,90,94)(57,113,73,106,68,129,89,97)(58,120,72,109,69,126,88,100)(59,117,71,102,70,123,87,93)(60,114,80,105,61,130,86,96), (1,60)(2,51)(3,52)(4,53)(5,54)(6,55)(7,56)(8,57)(9,58)(10,59)(11,111)(12,112)(13,113)(14,114)(15,115)(16,116)(17,117)(18,118)(19,119)(20,120)(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)(41,83)(42,84)(43,85)(44,86)(45,87)(46,88)(47,89)(48,90)(49,81)(50,82)(91,150)(92,141)(93,142)(94,143)(95,144)(96,145)(97,146)(98,147)(99,148)(100,149)(101,132)(102,133)(103,134)(104,135)(105,136)(106,137)(107,138)(108,139)(109,140)(110,131)(121,155)(122,156)(123,157)(124,158)(125,159)(126,160)(127,151)(128,152)(129,153)(130,154) );
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)], [(1,10),(2,9),(3,8),(4,7),(5,6),(12,20),(13,19),(14,18),(15,17),(21,30),(22,29),(23,28),(24,27),(25,26),(31,38),(32,37),(33,36),(34,35),(39,40),(41,46),(42,45),(43,44),(47,50),(48,49),(51,53),(54,60),(55,59),(56,58),(61,65),(62,64),(66,70),(67,69),(71,73),(74,80),(75,79),(76,78),(81,85),(82,84),(86,90),(87,89),(91,100),(92,99),(93,98),(94,97),(95,96),(101,108),(102,107),(103,106),(104,105),(109,110),(111,116),(112,115),(113,114),(117,120),(118,119),(121,128),(122,127),(123,126),(124,125),(129,130),(131,135),(132,134),(136,140),(137,139),(141,143),(144,150),(145,149),(146,148),(152,160),(153,159),(154,158),(155,157)], [(1,154,40,145,21,14,44,136),(2,151,39,148,22,11,43,139),(3,158,38,141,23,18,42,132),(4,155,37,144,24,15,41,135),(5,152,36,147,25,12,50,138),(6,159,35,150,26,19,49,131),(7,156,34,143,27,16,48,134),(8,153,33,146,28,13,47,137),(9,160,32,149,29,20,46,140),(10,157,31,142,30,17,45,133),(51,111,79,108,62,127,85,99),(52,118,78,101,63,124,84,92),(53,115,77,104,64,121,83,95),(54,112,76,107,65,128,82,98),(55,119,75,110,66,125,81,91),(56,116,74,103,67,122,90,94),(57,113,73,106,68,129,89,97),(58,120,72,109,69,126,88,100),(59,117,71,102,70,123,87,93),(60,114,80,105,61,130,86,96)], [(1,60),(2,51),(3,52),(4,53),(5,54),(6,55),(7,56),(8,57),(9,58),(10,59),(11,111),(12,112),(13,113),(14,114),(15,115),(16,116),(17,117),(18,118),(19,119),(20,120),(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),(41,83),(42,84),(43,85),(44,86),(45,87),(46,88),(47,89),(48,90),(49,81),(50,82),(91,150),(92,141),(93,142),(94,143),(95,144),(96,145),(97,146),(98,147),(99,148),(100,149),(101,132),(102,133),(103,134),(104,135),(105,136),(106,137),(107,138),(108,139),(109,140),(110,131),(121,155),(122,156),(123,157),(124,158),(125,159),(126,160),(127,151),(128,152),(129,153),(130,154)]])
38 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 5 | 8A | ··· | 8H | 8I | 8J | 8K | 8L | 10A | 10B | 10C | 10D | 10E | 20A | 20B | 20C | 20D |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 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 | 10 | 10 | 2 | 2 | 4 | 5 | 5 | 5 | 5 | 20 | 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 | C4○D4 | M4(2) | C8○D4 | F5 | C2×F5 | C2×F5 | D5⋊M4(2) | D4.F5 | D4×F5 |
| kernel | D10⋊M4(2) | C20⋊C8 | C10.C42 | D10⋊C8 | C23.2F5 | Dic5⋊4D4 | C2×D5⋊C8 | C2×C22.F5 | C10.D4 | D10⋊C4 | C5×C22⋊C4 | C2×C5⋊D4 | C5⋊C8 | Dic5 | D10 | C10 | C22⋊C4 | C2×C4 | C23 | C2 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 1 | 2 | 1 | 4 | 1 | 1 |
Matrix representation of D10⋊M4(2) ►in GL6(𝔽41)
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 1 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 1 | 0 | 40 | 0 |
| 0 | 0 | 0 | 1 | 40 | 0 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 40 | 0 |
| 0 | 0 | 1 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 40 | 1 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 24 | 17 | 2 | 0 |
| 0 | 0 | 26 | 17 | 0 | 24 |
| 0 | 0 | 24 | 0 | 17 | 26 |
| 0 | 0 | 0 | 2 | 17 | 24 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 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 |
G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,40,40,40,40,0,0,1,0,0,0],[40,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,40,40,40,40,0,0,0,0,0,1],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,24,26,24,0,0,0,17,17,0,2,0,0,2,0,17,17,0,0,0,24,26,24],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,22,0,3,3,0,0,38,19,38,0,0,0,0,38,19,38,0,0,3,3,0,22] >;
D10⋊M4(2) in GAP, Magma, Sage, TeX
D_{10}\rtimes M_4(2) % in TeX
G:=Group("D10:M4(2)"); // GroupNames label
G:=SmallGroup(320,1032);
// by ID
G=gap.SmallGroup(320,1032);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,232,758,219,136,6278,1595]);
// Polycyclic
G:=Group<a,b,c,d|a^10=b^2=c^8=d^2=1,b*a*b=a^-1,c*a*c^-1=a^3,a*d=d*a,c*b*c^-1=a^7*b,d*b*d=a^5*b,d*c*d=c^5>;
// generators/relations