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)), (C4×Dic5).241C22, (C2×Dic5).324C23, (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, (C2×C10).32(C22×C4), (C22×C10).15(C2×C4), (C2×Dic5).49(C2×C4), (C22×D5).41(C2×C4), SmallGroup(320,1032)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 442 in 124 conjugacy classes, 46 normal (42 characteristic)
C1, C2 [×3], C2 [×3], C4 [×6], C22, C22 [×7], C5, C8 [×5], C2×C4 [×2], C2×C4 [×7], D4 [×2], C23, C23, D5 [×2], C10 [×3], C10, C42, C22⋊C4, C22⋊C4, C4⋊C4, C2×C8 [×6], M4(2) [×2], C22×C4 [×2], C2×D4, Dic5 [×2], Dic5 [×2], C20 [×2], D10 [×2], D10 [×2], C2×C10, C2×C10 [×3], C8⋊C4, C22⋊C8 [×2], C4⋊C8, C4×D4, C22×C8, C2×M4(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⋊9D4, C4×Dic5, C10.D4, D10⋊C4, C5×C22⋊C4, D5⋊C8 [×2], C2×C5⋊C8 [×4], C22.F5 [×2], 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 [×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⋊9D4, C22×F5, D5⋊M4(2), D4.F5, D4×F5, D10⋊M4(2)
Generators and relations
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 >
►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 98)(92 97)(93 96)(94 95)(99 100)(101 108)(102 107)(103 106)(104 105)(109 110)(111 118)(112 117)(113 116)(114 115)(119 120)(121 128)(122 127)(123 126)(124 125)(129 130)(131 133)(134 140)(135 139)(136 138)(141 143)(144 150)(145 149)(146 148)(151 153)(154 160)(155 159)(156 158)
(1 160 40 145 21 14 44 140)(2 157 39 148 22 11 43 133)(3 154 38 141 23 18 42 136)(4 151 37 144 24 15 41 139)(5 158 36 147 25 12 50 132)(6 155 35 150 26 19 49 135)(7 152 34 143 27 16 48 138)(8 159 33 146 28 13 47 131)(9 156 32 149 29 20 46 134)(10 153 31 142 30 17 45 137)(51 117 79 108 62 127 85 93)(52 114 78 101 63 124 84 96)(53 111 77 104 64 121 83 99)(54 118 76 107 65 128 82 92)(55 115 75 110 66 125 81 95)(56 112 74 103 67 122 90 98)(57 119 73 106 68 129 89 91)(58 116 72 109 69 126 88 94)(59 113 71 102 70 123 87 97)(60 120 80 105 61 130 86 100)
(1 60)(2 51)(3 52)(4 53)(5 54)(6 55)(7 56)(8 57)(9 58)(10 59)(11 117)(12 118)(13 119)(14 120)(15 111)(16 112)(17 113)(18 114)(19 115)(20 116)(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)(91 146)(92 147)(93 148)(94 149)(95 150)(96 141)(97 142)(98 143)(99 144)(100 145)(101 136)(102 137)(103 138)(104 139)(105 140)(106 131)(107 132)(108 133)(109 134)(110 135)(121 151)(122 152)(123 153)(124 154)(125 155)(126 156)(127 157)(128 158)(129 159)(130 160)
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), (1,10)(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)(91,98)(92,97)(93,96)(94,95)(99,100)(101,108)(102,107)(103,106)(104,105)(109,110)(111,118)(112,117)(113,116)(114,115)(119,120)(121,128)(122,127)(123,126)(124,125)(129,130)(131,133)(134,140)(135,139)(136,138)(141,143)(144,150)(145,149)(146,148)(151,153)(154,160)(155,159)(156,158), (1,160,40,145,21,14,44,140)(2,157,39,148,22,11,43,133)(3,154,38,141,23,18,42,136)(4,151,37,144,24,15,41,139)(5,158,36,147,25,12,50,132)(6,155,35,150,26,19,49,135)(7,152,34,143,27,16,48,138)(8,159,33,146,28,13,47,131)(9,156,32,149,29,20,46,134)(10,153,31,142,30,17,45,137)(51,117,79,108,62,127,85,93)(52,114,78,101,63,124,84,96)(53,111,77,104,64,121,83,99)(54,118,76,107,65,128,82,92)(55,115,75,110,66,125,81,95)(56,112,74,103,67,122,90,98)(57,119,73,106,68,129,89,91)(58,116,72,109,69,126,88,94)(59,113,71,102,70,123,87,97)(60,120,80,105,61,130,86,100), (1,60)(2,51)(3,52)(4,53)(5,54)(6,55)(7,56)(8,57)(9,58)(10,59)(11,117)(12,118)(13,119)(14,120)(15,111)(16,112)(17,113)(18,114)(19,115)(20,116)(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)(91,146)(92,147)(93,148)(94,149)(95,150)(96,141)(97,142)(98,143)(99,144)(100,145)(101,136)(102,137)(103,138)(104,139)(105,140)(106,131)(107,132)(108,133)(109,134)(110,135)(121,151)(122,152)(123,153)(124,154)(125,155)(126,156)(127,157)(128,158)(129,159)(130,160)>;
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), (1,10)(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)(91,98)(92,97)(93,96)(94,95)(99,100)(101,108)(102,107)(103,106)(104,105)(109,110)(111,118)(112,117)(113,116)(114,115)(119,120)(121,128)(122,127)(123,126)(124,125)(129,130)(131,133)(134,140)(135,139)(136,138)(141,143)(144,150)(145,149)(146,148)(151,153)(154,160)(155,159)(156,158), (1,160,40,145,21,14,44,140)(2,157,39,148,22,11,43,133)(3,154,38,141,23,18,42,136)(4,151,37,144,24,15,41,139)(5,158,36,147,25,12,50,132)(6,155,35,150,26,19,49,135)(7,152,34,143,27,16,48,138)(8,159,33,146,28,13,47,131)(9,156,32,149,29,20,46,134)(10,153,31,142,30,17,45,137)(51,117,79,108,62,127,85,93)(52,114,78,101,63,124,84,96)(53,111,77,104,64,121,83,99)(54,118,76,107,65,128,82,92)(55,115,75,110,66,125,81,95)(56,112,74,103,67,122,90,98)(57,119,73,106,68,129,89,91)(58,116,72,109,69,126,88,94)(59,113,71,102,70,123,87,97)(60,120,80,105,61,130,86,100), (1,60)(2,51)(3,52)(4,53)(5,54)(6,55)(7,56)(8,57)(9,58)(10,59)(11,117)(12,118)(13,119)(14,120)(15,111)(16,112)(17,113)(18,114)(19,115)(20,116)(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)(91,146)(92,147)(93,148)(94,149)(95,150)(96,141)(97,142)(98,143)(99,144)(100,145)(101,136)(102,137)(103,138)(104,139)(105,140)(106,131)(107,132)(108,133)(109,134)(110,135)(121,151)(122,152)(123,153)(124,154)(125,155)(126,156)(127,157)(128,158)(129,159)(130,160) );
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)], [(1,10),(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),(91,98),(92,97),(93,96),(94,95),(99,100),(101,108),(102,107),(103,106),(104,105),(109,110),(111,118),(112,117),(113,116),(114,115),(119,120),(121,128),(122,127),(123,126),(124,125),(129,130),(131,133),(134,140),(135,139),(136,138),(141,143),(144,150),(145,149),(146,148),(151,153),(154,160),(155,159),(156,158)], [(1,160,40,145,21,14,44,140),(2,157,39,148,22,11,43,133),(3,154,38,141,23,18,42,136),(4,151,37,144,24,15,41,139),(5,158,36,147,25,12,50,132),(6,155,35,150,26,19,49,135),(7,152,34,143,27,16,48,138),(8,159,33,146,28,13,47,131),(9,156,32,149,29,20,46,134),(10,153,31,142,30,17,45,137),(51,117,79,108,62,127,85,93),(52,114,78,101,63,124,84,96),(53,111,77,104,64,121,83,99),(54,118,76,107,65,128,82,92),(55,115,75,110,66,125,81,95),(56,112,74,103,67,122,90,98),(57,119,73,106,68,129,89,91),(58,116,72,109,69,126,88,94),(59,113,71,102,70,123,87,97),(60,120,80,105,61,130,86,100)], [(1,60),(2,51),(3,52),(4,53),(5,54),(6,55),(7,56),(8,57),(9,58),(10,59),(11,117),(12,118),(13,119),(14,120),(15,111),(16,112),(17,113),(18,114),(19,115),(20,116),(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),(91,146),(92,147),(93,148),(94,149),(95,150),(96,141),(97,142),(98,143),(99,144),(100,145),(101,136),(102,137),(103,138),(104,139),(105,140),(106,131),(107,132),(108,133),(109,134),(110,135),(121,151),(122,152),(123,153),(124,154),(125,155),(126,156),(127,157),(128,158),(129,159),(130,160)])
Matrix representation ►G ⊆ 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] >;
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 |
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
×
⋊
ℤ
𝔽
○
≀