metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2.6(C4×D20), D10⋊3(C4⋊C4), C10.33(C4×D4), D10⋊C4⋊5C4, C10.3C22≀C2, (C2×C4).112D20, (C2×C20).234D4, C22.62(D4×D5), C22.17(Q8×D5), C2.C42⋊8D5, (C22×C4).17D10, C22.25(C2×D20), (C22×D5).13Q8, C2.2(C22⋊D20), C5⋊2(C23.8Q8), C2.4(D10⋊Q8), C2.2(D10⋊2Q8), (C2×Dic5).187D4, (C22×D5).104D4, C10.25(C22⋊Q8), C10.10C42⋊2C2, C2.8(Dic5⋊4D4), C22.36(C4○D20), (C23×D5).97C22, C23.258(C22×D5), C2.4(D10.12D4), C22.37(D4⋊2D5), (C22×C20).333C22, (C22×C10).293C23, C10.9(C22.D4), (C22×Dic5).16C22, (C2×C4)⋊3(C4×D5), C2.8(D5×C4⋊C4), (C2×C20)⋊18(C2×C4), C10.29(C2×C4⋊C4), (C2×C4⋊Dic5)⋊1C2, C22.91(C2×C4×D5), (C2×Dic5)⋊5(C2×C4), (C2×C10).68(C2×Q8), (D5×C22×C4).15C2, (C2×C10).202(C2×D4), (C2×D10⋊C4).5C2, (C2×C10.D4)⋊31C2, (C22×D5).67(C2×C4), (C2×C10).132(C4○D4), (C2×C10).153(C22×C4), (C5×C2.C42)⋊15C2, SmallGroup(320,295)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D10⋊3(C4⋊C4)
G = < a,b,c,d | a10=b2=c4=d4=1, bab=cac-1=a-1, ad=da, cbc-1=a3b, bd=db, dcd-1=c-1 >
Subgroups: 910 in 234 conjugacy classes, 77 normal (51 characteristic)
C1, C2 [×7], C2 [×4], C4 [×10], C22 [×7], C22 [×16], C5, C2×C4 [×4], C2×C4 [×26], C23, C23 [×10], D5 [×4], C10 [×7], C22⋊C4 [×6], C4⋊C4 [×4], C22×C4 [×3], C22×C4 [×9], C24, Dic5 [×5], C20 [×5], D10 [×4], D10 [×12], C2×C10 [×7], C2.C42, C2.C42, C2×C22⋊C4 [×2], C2×C4⋊C4 [×2], C23×C4, C4×D5 [×8], C2×Dic5 [×4], C2×Dic5 [×7], C2×C20 [×4], C2×C20 [×7], C22×D5 [×6], C22×D5 [×4], C22×C10, C23.8Q8, C10.D4 [×2], C4⋊Dic5 [×2], D10⋊C4 [×4], D10⋊C4 [×2], C2×C4×D5 [×6], C22×Dic5 [×3], C22×C20 [×3], C23×D5, C10.10C42, C5×C2.C42, C2×C10.D4, C2×C4⋊Dic5, C2×D10⋊C4 [×2], D5×C22×C4, D10⋊3(C4⋊C4)
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×6], Q8 [×2], C23, D5, C4⋊C4 [×4], C22×C4, C2×D4 [×3], C2×Q8, C4○D4 [×2], D10 [×3], C2×C4⋊C4, C4×D4 [×2], C22≀C2, C22⋊Q8 [×2], C22.D4, C4×D5 [×2], D20 [×2], C22×D5, C23.8Q8, C2×C4×D5, C2×D20, C4○D20, D4×D5 [×2], D4⋊2D5, Q8×D5, C4×D20, Dic5⋊4D4, C22⋊D20, D10.12D4, D5×C4⋊C4, D10⋊Q8, D10⋊2Q8, D10⋊3(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 35)(2 34)(3 33)(4 32)(5 31)(6 40)(7 39)(8 38)(9 37)(10 36)(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 80)(52 79)(53 78)(54 77)(55 76)(56 75)(57 74)(58 73)(59 72)(60 71)(61 88)(62 87)(63 86)(64 85)(65 84)(66 83)(67 82)(68 81)(69 90)(70 89)(91 115)(92 114)(93 113)(94 112)(95 111)(96 120)(97 119)(98 118)(99 117)(100 116)(101 123)(102 122)(103 121)(104 130)(105 129)(106 128)(107 127)(108 126)(109 125)(110 124)(131 155)(132 154)(133 153)(134 152)(135 151)(136 160)(137 159)(138 158)(139 157)(140 156)
(1 105 30 91)(2 104 21 100)(3 103 22 99)(4 102 23 98)(5 101 24 97)(6 110 25 96)(7 109 26 95)(8 108 27 94)(9 107 28 93)(10 106 29 92)(11 90 151 76)(12 89 152 75)(13 88 153 74)(14 87 154 73)(15 86 155 72)(16 85 156 71)(17 84 157 80)(18 83 158 79)(19 82 159 78)(20 81 160 77)(31 130 45 116)(32 129 46 115)(33 128 47 114)(34 127 48 113)(35 126 49 112)(36 125 50 111)(37 124 41 120)(38 123 42 119)(39 122 43 118)(40 121 44 117)(51 150 65 136)(52 149 66 135)(53 148 67 134)(54 147 68 133)(55 146 69 132)(56 145 70 131)(57 144 61 140)(58 143 62 139)(59 142 63 138)(60 141 64 137)
(1 76 36 56)(2 77 37 57)(3 78 38 58)(4 79 39 59)(5 80 40 60)(6 71 31 51)(7 72 32 52)(8 73 33 53)(9 74 34 54)(10 75 35 55)(11 111 145 91)(12 112 146 92)(13 113 147 93)(14 114 148 94)(15 115 149 95)(16 116 150 96)(17 117 141 97)(18 118 142 98)(19 119 143 99)(20 120 144 100)(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)(101 157 121 137)(102 158 122 138)(103 159 123 139)(104 160 124 140)(105 151 125 131)(106 152 126 132)(107 153 127 133)(108 154 128 134)(109 155 129 135)(110 156 130 136)
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), (1,35)(2,34)(3,33)(4,32)(5,31)(6,40)(7,39)(8,38)(9,37)(10,36)(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)(51,80)(52,79)(53,78)(54,77)(55,76)(56,75)(57,74)(58,73)(59,72)(60,71)(61,88)(62,87)(63,86)(64,85)(65,84)(66,83)(67,82)(68,81)(69,90)(70,89)(91,115)(92,114)(93,113)(94,112)(95,111)(96,120)(97,119)(98,118)(99,117)(100,116)(101,123)(102,122)(103,121)(104,130)(105,129)(106,128)(107,127)(108,126)(109,125)(110,124)(131,155)(132,154)(133,153)(134,152)(135,151)(136,160)(137,159)(138,158)(139,157)(140,156), (1,105,30,91)(2,104,21,100)(3,103,22,99)(4,102,23,98)(5,101,24,97)(6,110,25,96)(7,109,26,95)(8,108,27,94)(9,107,28,93)(10,106,29,92)(11,90,151,76)(12,89,152,75)(13,88,153,74)(14,87,154,73)(15,86,155,72)(16,85,156,71)(17,84,157,80)(18,83,158,79)(19,82,159,78)(20,81,160,77)(31,130,45,116)(32,129,46,115)(33,128,47,114)(34,127,48,113)(35,126,49,112)(36,125,50,111)(37,124,41,120)(38,123,42,119)(39,122,43,118)(40,121,44,117)(51,150,65,136)(52,149,66,135)(53,148,67,134)(54,147,68,133)(55,146,69,132)(56,145,70,131)(57,144,61,140)(58,143,62,139)(59,142,63,138)(60,141,64,137), (1,76,36,56)(2,77,37,57)(3,78,38,58)(4,79,39,59)(5,80,40,60)(6,71,31,51)(7,72,32,52)(8,73,33,53)(9,74,34,54)(10,75,35,55)(11,111,145,91)(12,112,146,92)(13,113,147,93)(14,114,148,94)(15,115,149,95)(16,116,150,96)(17,117,141,97)(18,118,142,98)(19,119,143,99)(20,120,144,100)(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)(101,157,121,137)(102,158,122,138)(103,159,123,139)(104,160,124,140)(105,151,125,131)(106,152,126,132)(107,153,127,133)(108,154,128,134)(109,155,129,135)(110,156,130,136)>;
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), (1,35)(2,34)(3,33)(4,32)(5,31)(6,40)(7,39)(8,38)(9,37)(10,36)(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)(51,80)(52,79)(53,78)(54,77)(55,76)(56,75)(57,74)(58,73)(59,72)(60,71)(61,88)(62,87)(63,86)(64,85)(65,84)(66,83)(67,82)(68,81)(69,90)(70,89)(91,115)(92,114)(93,113)(94,112)(95,111)(96,120)(97,119)(98,118)(99,117)(100,116)(101,123)(102,122)(103,121)(104,130)(105,129)(106,128)(107,127)(108,126)(109,125)(110,124)(131,155)(132,154)(133,153)(134,152)(135,151)(136,160)(137,159)(138,158)(139,157)(140,156), (1,105,30,91)(2,104,21,100)(3,103,22,99)(4,102,23,98)(5,101,24,97)(6,110,25,96)(7,109,26,95)(8,108,27,94)(9,107,28,93)(10,106,29,92)(11,90,151,76)(12,89,152,75)(13,88,153,74)(14,87,154,73)(15,86,155,72)(16,85,156,71)(17,84,157,80)(18,83,158,79)(19,82,159,78)(20,81,160,77)(31,130,45,116)(32,129,46,115)(33,128,47,114)(34,127,48,113)(35,126,49,112)(36,125,50,111)(37,124,41,120)(38,123,42,119)(39,122,43,118)(40,121,44,117)(51,150,65,136)(52,149,66,135)(53,148,67,134)(54,147,68,133)(55,146,69,132)(56,145,70,131)(57,144,61,140)(58,143,62,139)(59,142,63,138)(60,141,64,137), (1,76,36,56)(2,77,37,57)(3,78,38,58)(4,79,39,59)(5,80,40,60)(6,71,31,51)(7,72,32,52)(8,73,33,53)(9,74,34,54)(10,75,35,55)(11,111,145,91)(12,112,146,92)(13,113,147,93)(14,114,148,94)(15,115,149,95)(16,116,150,96)(17,117,141,97)(18,118,142,98)(19,119,143,99)(20,120,144,100)(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)(101,157,121,137)(102,158,122,138)(103,159,123,139)(104,160,124,140)(105,151,125,131)(106,152,126,132)(107,153,127,133)(108,154,128,134)(109,155,129,135)(110,156,130,136) );
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)], [(1,35),(2,34),(3,33),(4,32),(5,31),(6,40),(7,39),(8,38),(9,37),(10,36),(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),(51,80),(52,79),(53,78),(54,77),(55,76),(56,75),(57,74),(58,73),(59,72),(60,71),(61,88),(62,87),(63,86),(64,85),(65,84),(66,83),(67,82),(68,81),(69,90),(70,89),(91,115),(92,114),(93,113),(94,112),(95,111),(96,120),(97,119),(98,118),(99,117),(100,116),(101,123),(102,122),(103,121),(104,130),(105,129),(106,128),(107,127),(108,126),(109,125),(110,124),(131,155),(132,154),(133,153),(134,152),(135,151),(136,160),(137,159),(138,158),(139,157),(140,156)], [(1,105,30,91),(2,104,21,100),(3,103,22,99),(4,102,23,98),(5,101,24,97),(6,110,25,96),(7,109,26,95),(8,108,27,94),(9,107,28,93),(10,106,29,92),(11,90,151,76),(12,89,152,75),(13,88,153,74),(14,87,154,73),(15,86,155,72),(16,85,156,71),(17,84,157,80),(18,83,158,79),(19,82,159,78),(20,81,160,77),(31,130,45,116),(32,129,46,115),(33,128,47,114),(34,127,48,113),(35,126,49,112),(36,125,50,111),(37,124,41,120),(38,123,42,119),(39,122,43,118),(40,121,44,117),(51,150,65,136),(52,149,66,135),(53,148,67,134),(54,147,68,133),(55,146,69,132),(56,145,70,131),(57,144,61,140),(58,143,62,139),(59,142,63,138),(60,141,64,137)], [(1,76,36,56),(2,77,37,57),(3,78,38,58),(4,79,39,59),(5,80,40,60),(6,71,31,51),(7,72,32,52),(8,73,33,53),(9,74,34,54),(10,75,35,55),(11,111,145,91),(12,112,146,92),(13,113,147,93),(14,114,148,94),(15,115,149,95),(16,116,150,96),(17,117,141,97),(18,118,142,98),(19,119,143,99),(20,120,144,100),(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),(101,157,121,137),(102,158,122,138),(103,159,123,139),(104,160,124,140),(105,151,125,131),(106,152,126,132),(107,153,127,133),(108,154,128,134),(109,155,129,135),(110,156,130,136)])
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 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | - | + | + | + | + | - | - | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | D4 | D4 | D4 | Q8 | D5 | C4○D4 | D10 | C4×D5 | D20 | C4○D20 | D4×D5 | D4⋊2D5 | Q8×D5 |
| kernel | D10⋊3(C4⋊C4) | C10.10C42 | C5×C2.C42 | C2×C10.D4 | C2×C4⋊Dic5 | C2×D10⋊C4 | D5×C22×C4 | D10⋊C4 | C2×Dic5 | C2×C20 | C22×D5 | C22×D5 | C2.C42 | C2×C10 | C22×C4 | C2×C4 | C2×C4 | C22 | C22 | C22 | C22 |
| # reps | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 8 | 2 | 2 | 2 | 2 | 2 | 4 | 6 | 8 | 8 | 8 | 4 | 2 | 2 |
Matrix representation of D10⋊3(C4⋊C4) ►in GL6(𝔽41)
| 34 | 34 | 0 | 0 | 0 | 0 |
| 6 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 34 | 33 | 0 | 0 | 0 | 0 |
| 6 | 7 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 22 | 40 |
| 7 | 7 | 0 | 0 | 0 | 0 |
| 40 | 34 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 39 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 14 | 36 |
| 0 | 0 | 0 | 0 | 23 | 27 |
| 32 | 0 | 0 | 0 | 0 | 0 |
| 0 | 32 | 0 | 0 | 0 | 0 |
| 0 | 0 | 32 | 0 | 0 | 0 |
| 0 | 0 | 9 | 9 | 0 | 0 |
| 0 | 0 | 0 | 0 | 32 | 0 |
| 0 | 0 | 0 | 0 | 7 | 9 |
G:=sub<GL(6,GF(41))| [34,6,0,0,0,0,34,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[34,6,0,0,0,0,33,7,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,22,0,0,0,0,0,40],[7,40,0,0,0,0,7,34,0,0,0,0,0,0,40,1,0,0,0,0,39,1,0,0,0,0,0,0,14,23,0,0,0,0,36,27],[32,0,0,0,0,0,0,32,0,0,0,0,0,0,32,9,0,0,0,0,0,9,0,0,0,0,0,0,32,7,0,0,0,0,0,9] >;
D10⋊3(C4⋊C4) in GAP, Magma, Sage, TeX
D_{10}\rtimes_3(C_4\rtimes C_4) % in TeX
G:=Group("D10:3(C4:C4)"); // GroupNames label
G:=SmallGroup(320,295);
// by ID
G=gap.SmallGroup(320,295);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,253,344,387,58,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^10=b^2=c^4=d^4=1,b*a*b=c*a*c^-1=a^-1,a*d=d*a,c*b*c^-1=a^3*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations