metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D10⋊4(C4⋊C4), C20⋊4(C22⋊C4), (C2×C4).143D20, (C2×C20).141D4, C4⋊3(D10⋊C4), C22.26(Q8×D5), C2.4(C20⋊2D4), C2.4(C4⋊D20), (C22×D5).88D4, C22.110(D4×D5), C22.47(C2×D20), (C22×D5).15Q8, C10.53(C4⋊D4), C5⋊4(C23.7Q8), C2.5(D10⋊2Q8), C2.3(D10⋊3Q8), (C22×C4).334D10, C10.72(C22⋊Q8), C23.295(C22×D5), C10.10C42⋊19C2, C10.56(C42⋊C2), C22.59(D4⋊2D5), (C22×C10).351C23, (C22×C20).143C22, C22.27(Q8⋊2D5), (C23×D5).101C22, (C22×Dic5).58C22, (C2×C4×D5)⋊7C4, (C2×C4⋊C4)⋊5D5, (C10×C4⋊C4)⋊5C2, C2.21(D5×C4⋊C4), C10.43(C2×C4⋊C4), (D5×C22×C4).1C2, (C2×C4⋊Dic5)⋊30C2, (C2×C4).154(C4×D5), (C2×C10).83(C2×Q8), C22.136(C2×C4×D5), (C2×C20).258(C2×C4), (C2×C10).451(C2×D4), C10.83(C2×C22⋊C4), C22.66(C2×C5⋊D4), C2.13(C4⋊C4⋊7D5), C2.15(C2×D10⋊C4), (C2×C4).184(C5⋊D4), (C2×D10⋊C4).13C2, (C2×C10).156(C4○D4), (C2×C10).220(C22×C4), (C2×Dic5).151(C2×C4), (C22×D5).105(C2×C4), SmallGroup(320,614)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D10⋊4(C4⋊C4)
G = < a,b,c,d | a10=b2=c4=d4=1, bab=a-1, ac=ca, ad=da, bc=cb, dbd-1=a5b, dcd-1=c-1 >
Subgroups: 894 in 234 conjugacy classes, 87 normal (41 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C2×C4, C2×C4, C23, C23, D5, C10, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C22×C4, C24, Dic5, C20, C20, D10, D10, C2×C10, C2.C42, C2×C22⋊C4, C2×C4⋊C4, C2×C4⋊C4, C23×C4, C4×D5, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C22×D5, C22×D5, C22×C10, C23.7Q8, C4⋊Dic5, D10⋊C4, C5×C4⋊C4, C2×C4×D5, C2×C4×D5, C22×Dic5, C22×Dic5, C22×C20, C22×C20, C23×D5, C10.10C42, C2×C4⋊Dic5, C2×D10⋊C4, C10×C4⋊C4, D5×C22×C4, D10⋊4(C4⋊C4)
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, D5, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C2×Q8, C4○D4, D10, C2×C22⋊C4, C2×C4⋊C4, C42⋊C2, C4⋊D4, C22⋊Q8, C4×D5, D20, C5⋊D4, C22×D5, C23.7Q8, D10⋊C4, C2×C4×D5, C2×D20, D4×D5, D4⋊2D5, Q8×D5, Q8⋊2D5, C2×C5⋊D4, D5×C4⋊C4, C4⋊C4⋊7D5, C4⋊D20, D10⋊2Q8, C2×D10⋊C4, C20⋊2D4, D10⋊3Q8, D10⋊4(C4⋊C4)
►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 33)(2 32)(3 31)(4 40)(5 39)(6 38)(7 37)(8 36)(9 35)(10 34)(11 149)(12 148)(13 147)(14 146)(15 145)(16 144)(17 143)(18 142)(19 141)(20 150)(21 48)(22 47)(23 46)(24 45)(25 44)(26 43)(27 42)(28 41)(29 50)(30 49)(51 71)(52 80)(53 79)(54 78)(55 77)(56 76)(57 75)(58 74)(59 73)(60 72)(61 83)(62 82)(63 81)(64 90)(65 89)(66 88)(67 87)(68 86)(69 85)(70 84)(91 116)(92 115)(93 114)(94 113)(95 112)(96 111)(97 120)(98 119)(99 118)(100 117)(101 128)(102 127)(103 126)(104 125)(105 124)(106 123)(107 122)(108 121)(109 130)(110 129)(131 151)(132 160)(133 159)(134 158)(135 157)(136 156)(137 155)(138 154)(139 153)(140 152)
(1 130 30 114)(2 121 21 115)(3 122 22 116)(4 123 23 117)(5 124 24 118)(6 125 25 119)(7 126 26 120)(8 127 27 111)(9 128 28 112)(10 129 29 113)(11 65 159 59)(12 66 160 60)(13 67 151 51)(14 68 152 52)(15 69 153 53)(16 70 154 54)(17 61 155 55)(18 62 156 56)(19 63 157 57)(20 64 158 58)(31 107 47 91)(32 108 48 92)(33 109 49 93)(34 110 50 94)(35 101 41 95)(36 102 42 96)(37 103 43 97)(38 104 44 98)(39 105 45 99)(40 106 46 100)(71 147 87 131)(72 148 88 132)(73 149 89 133)(74 150 90 134)(75 141 81 135)(76 142 82 136)(77 143 83 137)(78 144 84 138)(79 145 85 139)(80 146 86 140)
(1 74 34 54)(2 75 35 55)(3 76 36 56)(4 77 37 57)(5 78 38 58)(6 79 39 59)(7 80 40 60)(8 71 31 51)(9 72 32 52)(10 73 33 53)(11 119 145 99)(12 120 146 100)(13 111 147 91)(14 112 148 92)(15 113 149 93)(16 114 150 94)(17 115 141 95)(18 116 142 96)(19 117 143 97)(20 118 144 98)(21 81 41 61)(22 82 42 62)(23 83 43 63)(24 84 44 64)(25 85 45 65)(26 86 46 66)(27 87 47 67)(28 88 48 68)(29 89 49 69)(30 90 50 70)(101 155 121 135)(102 156 122 136)(103 157 123 137)(104 158 124 138)(105 159 125 139)(106 160 126 140)(107 151 127 131)(108 152 128 132)(109 153 129 133)(110 154 130 134)
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), (1,33)(2,32)(3,31)(4,40)(5,39)(6,38)(7,37)(8,36)(9,35)(10,34)(11,149)(12,148)(13,147)(14,146)(15,145)(16,144)(17,143)(18,142)(19,141)(20,150)(21,48)(22,47)(23,46)(24,45)(25,44)(26,43)(27,42)(28,41)(29,50)(30,49)(51,71)(52,80)(53,79)(54,78)(55,77)(56,76)(57,75)(58,74)(59,73)(60,72)(61,83)(62,82)(63,81)(64,90)(65,89)(66,88)(67,87)(68,86)(69,85)(70,84)(91,116)(92,115)(93,114)(94,113)(95,112)(96,111)(97,120)(98,119)(99,118)(100,117)(101,128)(102,127)(103,126)(104,125)(105,124)(106,123)(107,122)(108,121)(109,130)(110,129)(131,151)(132,160)(133,159)(134,158)(135,157)(136,156)(137,155)(138,154)(139,153)(140,152), (1,130,30,114)(2,121,21,115)(3,122,22,116)(4,123,23,117)(5,124,24,118)(6,125,25,119)(7,126,26,120)(8,127,27,111)(9,128,28,112)(10,129,29,113)(11,65,159,59)(12,66,160,60)(13,67,151,51)(14,68,152,52)(15,69,153,53)(16,70,154,54)(17,61,155,55)(18,62,156,56)(19,63,157,57)(20,64,158,58)(31,107,47,91)(32,108,48,92)(33,109,49,93)(34,110,50,94)(35,101,41,95)(36,102,42,96)(37,103,43,97)(38,104,44,98)(39,105,45,99)(40,106,46,100)(71,147,87,131)(72,148,88,132)(73,149,89,133)(74,150,90,134)(75,141,81,135)(76,142,82,136)(77,143,83,137)(78,144,84,138)(79,145,85,139)(80,146,86,140), (1,74,34,54)(2,75,35,55)(3,76,36,56)(4,77,37,57)(5,78,38,58)(6,79,39,59)(7,80,40,60)(8,71,31,51)(9,72,32,52)(10,73,33,53)(11,119,145,99)(12,120,146,100)(13,111,147,91)(14,112,148,92)(15,113,149,93)(16,114,150,94)(17,115,141,95)(18,116,142,96)(19,117,143,97)(20,118,144,98)(21,81,41,61)(22,82,42,62)(23,83,43,63)(24,84,44,64)(25,85,45,65)(26,86,46,66)(27,87,47,67)(28,88,48,68)(29,89,49,69)(30,90,50,70)(101,155,121,135)(102,156,122,136)(103,157,123,137)(104,158,124,138)(105,159,125,139)(106,160,126,140)(107,151,127,131)(108,152,128,132)(109,153,129,133)(110,154,130,134)>;
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), (1,33)(2,32)(3,31)(4,40)(5,39)(6,38)(7,37)(8,36)(9,35)(10,34)(11,149)(12,148)(13,147)(14,146)(15,145)(16,144)(17,143)(18,142)(19,141)(20,150)(21,48)(22,47)(23,46)(24,45)(25,44)(26,43)(27,42)(28,41)(29,50)(30,49)(51,71)(52,80)(53,79)(54,78)(55,77)(56,76)(57,75)(58,74)(59,73)(60,72)(61,83)(62,82)(63,81)(64,90)(65,89)(66,88)(67,87)(68,86)(69,85)(70,84)(91,116)(92,115)(93,114)(94,113)(95,112)(96,111)(97,120)(98,119)(99,118)(100,117)(101,128)(102,127)(103,126)(104,125)(105,124)(106,123)(107,122)(108,121)(109,130)(110,129)(131,151)(132,160)(133,159)(134,158)(135,157)(136,156)(137,155)(138,154)(139,153)(140,152), (1,130,30,114)(2,121,21,115)(3,122,22,116)(4,123,23,117)(5,124,24,118)(6,125,25,119)(7,126,26,120)(8,127,27,111)(9,128,28,112)(10,129,29,113)(11,65,159,59)(12,66,160,60)(13,67,151,51)(14,68,152,52)(15,69,153,53)(16,70,154,54)(17,61,155,55)(18,62,156,56)(19,63,157,57)(20,64,158,58)(31,107,47,91)(32,108,48,92)(33,109,49,93)(34,110,50,94)(35,101,41,95)(36,102,42,96)(37,103,43,97)(38,104,44,98)(39,105,45,99)(40,106,46,100)(71,147,87,131)(72,148,88,132)(73,149,89,133)(74,150,90,134)(75,141,81,135)(76,142,82,136)(77,143,83,137)(78,144,84,138)(79,145,85,139)(80,146,86,140), (1,74,34,54)(2,75,35,55)(3,76,36,56)(4,77,37,57)(5,78,38,58)(6,79,39,59)(7,80,40,60)(8,71,31,51)(9,72,32,52)(10,73,33,53)(11,119,145,99)(12,120,146,100)(13,111,147,91)(14,112,148,92)(15,113,149,93)(16,114,150,94)(17,115,141,95)(18,116,142,96)(19,117,143,97)(20,118,144,98)(21,81,41,61)(22,82,42,62)(23,83,43,63)(24,84,44,64)(25,85,45,65)(26,86,46,66)(27,87,47,67)(28,88,48,68)(29,89,49,69)(30,90,50,70)(101,155,121,135)(102,156,122,136)(103,157,123,137)(104,158,124,138)(105,159,125,139)(106,160,126,140)(107,151,127,131)(108,152,128,132)(109,153,129,133)(110,154,130,134) );
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)], [(1,33),(2,32),(3,31),(4,40),(5,39),(6,38),(7,37),(8,36),(9,35),(10,34),(11,149),(12,148),(13,147),(14,146),(15,145),(16,144),(17,143),(18,142),(19,141),(20,150),(21,48),(22,47),(23,46),(24,45),(25,44),(26,43),(27,42),(28,41),(29,50),(30,49),(51,71),(52,80),(53,79),(54,78),(55,77),(56,76),(57,75),(58,74),(59,73),(60,72),(61,83),(62,82),(63,81),(64,90),(65,89),(66,88),(67,87),(68,86),(69,85),(70,84),(91,116),(92,115),(93,114),(94,113),(95,112),(96,111),(97,120),(98,119),(99,118),(100,117),(101,128),(102,127),(103,126),(104,125),(105,124),(106,123),(107,122),(108,121),(109,130),(110,129),(131,151),(132,160),(133,159),(134,158),(135,157),(136,156),(137,155),(138,154),(139,153),(140,152)], [(1,130,30,114),(2,121,21,115),(3,122,22,116),(4,123,23,117),(5,124,24,118),(6,125,25,119),(7,126,26,120),(8,127,27,111),(9,128,28,112),(10,129,29,113),(11,65,159,59),(12,66,160,60),(13,67,151,51),(14,68,152,52),(15,69,153,53),(16,70,154,54),(17,61,155,55),(18,62,156,56),(19,63,157,57),(20,64,158,58),(31,107,47,91),(32,108,48,92),(33,109,49,93),(34,110,50,94),(35,101,41,95),(36,102,42,96),(37,103,43,97),(38,104,44,98),(39,105,45,99),(40,106,46,100),(71,147,87,131),(72,148,88,132),(73,149,89,133),(74,150,90,134),(75,141,81,135),(76,142,82,136),(77,143,83,137),(78,144,84,138),(79,145,85,139),(80,146,86,140)], [(1,74,34,54),(2,75,35,55),(3,76,36,56),(4,77,37,57),(5,78,38,58),(6,79,39,59),(7,80,40,60),(8,71,31,51),(9,72,32,52),(10,73,33,53),(11,119,145,99),(12,120,146,100),(13,111,147,91),(14,112,148,92),(15,113,149,93),(16,114,150,94),(17,115,141,95),(18,116,142,96),(19,117,143,97),(20,118,144,98),(21,81,41,61),(22,82,42,62),(23,83,43,63),(24,84,44,64),(25,85,45,65),(26,86,46,66),(27,87,47,67),(28,88,48,68),(29,89,49,69),(30,90,50,70),(101,155,121,135),(102,156,122,136),(103,157,123,137),(104,158,124,138),(105,159,125,139),(106,160,126,140),(107,151,127,131),(108,152,128,132),(109,153,129,133),(110,154,130,134)]])
68 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 4O | 4P | 5A | 5B | 10A | ··· | 10N | 20A | ··· | 20X |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 10 | ··· | 10 | 20 | ··· | 20 |
| size | 1 | 1 | ··· | 1 | 10 | 10 | 10 | 10 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 10 | 10 | 10 | 10 | 20 | 20 | 20 | 20 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 |
68 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | - | + | + | + | + | - | - | + | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | D4 | D4 | Q8 | D5 | C4○D4 | D10 | C4×D5 | D20 | C5⋊D4 | D4×D5 | D4⋊2D5 | Q8×D5 | Q8⋊2D5 |
| kernel | D10⋊4(C4⋊C4) | C10.10C42 | C2×C4⋊Dic5 | C2×D10⋊C4 | C10×C4⋊C4 | D5×C22×C4 | C2×C4×D5 | C2×C20 | C22×D5 | C22×D5 | C2×C4⋊C4 | C2×C10 | C22×C4 | C2×C4 | C2×C4 | C2×C4 | C22 | C22 | C22 | C22 |
| # reps | 1 | 2 | 1 | 2 | 1 | 1 | 8 | 4 | 2 | 2 | 2 | 4 | 6 | 8 | 8 | 8 | 2 | 2 | 2 | 2 |
Matrix representation of D10⋊4(C4⋊C4) ►in GL6(𝔽41)
| 34 | 34 | 0 | 0 | 0 | 0 |
| 7 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 7 | 7 | 0 | 0 |
| 0 | 0 | 34 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 7 | 7 | 0 | 0 | 0 | 0 |
| 40 | 34 | 0 | 0 | 0 | 0 |
| 0 | 0 | 34 | 34 | 0 | 0 |
| 0 | 0 | 1 | 7 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 9 | 40 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 32 | 0 |
| 0 | 0 | 0 | 0 | 1 | 9 |
| 30 | 32 | 0 | 0 | 0 | 0 |
| 9 | 11 | 0 | 0 | 0 | 0 |
| 0 | 0 | 32 | 0 | 0 | 0 |
| 0 | 0 | 0 | 32 | 0 | 0 |
| 0 | 0 | 0 | 0 | 21 | 9 |
| 0 | 0 | 0 | 0 | 1 | 20 |
G:=sub<GL(6,GF(41))| [34,7,0,0,0,0,34,1,0,0,0,0,0,0,7,34,0,0,0,0,7,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[7,40,0,0,0,0,7,34,0,0,0,0,0,0,34,1,0,0,0,0,34,7,0,0,0,0,0,0,1,9,0,0,0,0,0,40],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,32,1,0,0,0,0,0,9],[30,9,0,0,0,0,32,11,0,0,0,0,0,0,32,0,0,0,0,0,0,32,0,0,0,0,0,0,21,1,0,0,0,0,9,20] >;
D10⋊4(C4⋊C4) in GAP, Magma, Sage, TeX
D_{10}\rtimes_4(C_4\rtimes C_4) % in TeX
G:=Group("D10:4(C4:C4)"); // GroupNames label
G:=SmallGroup(320,614);
// by ID
G=gap.SmallGroup(320,614);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,232,422,387,58,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^10=b^2=c^4=d^4=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=a^5*b,d*c*d^-1=c^-1>;
// generators/relations