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, C2, C4, C22, C22, C5, C2×C4, C2×C4, C23, C23, D5, C10, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C24, Dic5, C20, D10, D10, C2×C10, C2.C42, C2.C42, C2×C22⋊C4, C2×C4⋊C4, C23×C4, C4×D5, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C22×D5, C22×D5, C22×C10, C23.8Q8, C10.D4, C4⋊Dic5, D10⋊C4, D10⋊C4, C2×C4×D5, C22×Dic5, C22×C20, C23×D5, C10.10C42, C5×C2.C42, C2×C10.D4, C2×C4⋊Dic5, C2×D10⋊C4, D5×C22×C4, D10⋊3(C4⋊C4)
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, D5, C4⋊C4, C22×C4, C2×D4, C2×Q8, C4○D4, D10, C2×C4⋊C4, C4×D4, C22≀C2, C22⋊Q8, C22.D4, C4×D5, D20, C22×D5, C23.8Q8, C2×C4×D5, C2×D20, C4○D20, D4×D5, 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 39)(2 38)(3 37)(4 36)(5 35)(6 34)(7 33)(8 32)(9 31)(10 40)(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 78)(52 77)(53 76)(54 75)(55 74)(56 73)(57 72)(58 71)(59 80)(60 79)(61 88)(62 87)(63 86)(64 85)(65 84)(66 83)(67 82)(68 81)(69 90)(70 89)(91 113)(92 112)(93 111)(94 120)(95 119)(96 118)(97 117)(98 116)(99 115)(100 114)(101 123)(102 122)(103 121)(104 130)(105 129)(106 128)(107 127)(108 126)(109 125)(110 124)(131 153)(132 152)(133 151)(134 160)(135 159)(136 158)(137 157)(138 156)(139 155)(140 154)
(1 105 30 95)(2 104 21 94)(3 103 22 93)(4 102 23 92)(5 101 24 91)(6 110 25 100)(7 109 26 99)(8 108 27 98)(9 107 28 97)(10 106 29 96)(11 90 155 80)(12 89 156 79)(13 88 157 78)(14 87 158 77)(15 86 159 76)(16 85 160 75)(17 84 151 74)(18 83 152 73)(19 82 153 72)(20 81 154 71)(31 124 41 114)(32 123 42 113)(33 122 43 112)(34 121 44 111)(35 130 45 120)(36 129 46 119)(37 128 47 118)(38 127 48 117)(39 126 49 116)(40 125 50 115)(51 144 61 134)(52 143 62 133)(53 142 63 132)(54 141 64 131)(55 150 65 140)(56 149 66 139)(57 148 67 138)(58 147 68 137)(59 146 69 136)(60 145 70 135)
(1 80 40 60)(2 71 31 51)(3 72 32 52)(4 73 33 53)(5 74 34 54)(6 75 35 55)(7 76 36 56)(8 77 37 57)(9 78 38 58)(10 79 39 59)(11 115 145 95)(12 116 146 96)(13 117 147 97)(14 118 148 98)(15 119 149 99)(16 120 150 100)(17 111 141 91)(18 112 142 92)(19 113 143 93)(20 114 144 94)(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)(101 151 121 131)(102 152 122 132)(103 153 123 133)(104 154 124 134)(105 155 125 135)(106 156 126 136)(107 157 127 137)(108 158 128 138)(109 159 129 139)(110 160 130 140)
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), (1,39)(2,38)(3,37)(4,36)(5,35)(6,34)(7,33)(8,32)(9,31)(10,40)(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)(51,78)(52,77)(53,76)(54,75)(55,74)(56,73)(57,72)(58,71)(59,80)(60,79)(61,88)(62,87)(63,86)(64,85)(65,84)(66,83)(67,82)(68,81)(69,90)(70,89)(91,113)(92,112)(93,111)(94,120)(95,119)(96,118)(97,117)(98,116)(99,115)(100,114)(101,123)(102,122)(103,121)(104,130)(105,129)(106,128)(107,127)(108,126)(109,125)(110,124)(131,153)(132,152)(133,151)(134,160)(135,159)(136,158)(137,157)(138,156)(139,155)(140,154), (1,105,30,95)(2,104,21,94)(3,103,22,93)(4,102,23,92)(5,101,24,91)(6,110,25,100)(7,109,26,99)(8,108,27,98)(9,107,28,97)(10,106,29,96)(11,90,155,80)(12,89,156,79)(13,88,157,78)(14,87,158,77)(15,86,159,76)(16,85,160,75)(17,84,151,74)(18,83,152,73)(19,82,153,72)(20,81,154,71)(31,124,41,114)(32,123,42,113)(33,122,43,112)(34,121,44,111)(35,130,45,120)(36,129,46,119)(37,128,47,118)(38,127,48,117)(39,126,49,116)(40,125,50,115)(51,144,61,134)(52,143,62,133)(53,142,63,132)(54,141,64,131)(55,150,65,140)(56,149,66,139)(57,148,67,138)(58,147,68,137)(59,146,69,136)(60,145,70,135), (1,80,40,60)(2,71,31,51)(3,72,32,52)(4,73,33,53)(5,74,34,54)(6,75,35,55)(7,76,36,56)(8,77,37,57)(9,78,38,58)(10,79,39,59)(11,115,145,95)(12,116,146,96)(13,117,147,97)(14,118,148,98)(15,119,149,99)(16,120,150,100)(17,111,141,91)(18,112,142,92)(19,113,143,93)(20,114,144,94)(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)(101,151,121,131)(102,152,122,132)(103,153,123,133)(104,154,124,134)(105,155,125,135)(106,156,126,136)(107,157,127,137)(108,158,128,138)(109,159,129,139)(110,160,130,140)>;
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), (1,39)(2,38)(3,37)(4,36)(5,35)(6,34)(7,33)(8,32)(9,31)(10,40)(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)(51,78)(52,77)(53,76)(54,75)(55,74)(56,73)(57,72)(58,71)(59,80)(60,79)(61,88)(62,87)(63,86)(64,85)(65,84)(66,83)(67,82)(68,81)(69,90)(70,89)(91,113)(92,112)(93,111)(94,120)(95,119)(96,118)(97,117)(98,116)(99,115)(100,114)(101,123)(102,122)(103,121)(104,130)(105,129)(106,128)(107,127)(108,126)(109,125)(110,124)(131,153)(132,152)(133,151)(134,160)(135,159)(136,158)(137,157)(138,156)(139,155)(140,154), (1,105,30,95)(2,104,21,94)(3,103,22,93)(4,102,23,92)(5,101,24,91)(6,110,25,100)(7,109,26,99)(8,108,27,98)(9,107,28,97)(10,106,29,96)(11,90,155,80)(12,89,156,79)(13,88,157,78)(14,87,158,77)(15,86,159,76)(16,85,160,75)(17,84,151,74)(18,83,152,73)(19,82,153,72)(20,81,154,71)(31,124,41,114)(32,123,42,113)(33,122,43,112)(34,121,44,111)(35,130,45,120)(36,129,46,119)(37,128,47,118)(38,127,48,117)(39,126,49,116)(40,125,50,115)(51,144,61,134)(52,143,62,133)(53,142,63,132)(54,141,64,131)(55,150,65,140)(56,149,66,139)(57,148,67,138)(58,147,68,137)(59,146,69,136)(60,145,70,135), (1,80,40,60)(2,71,31,51)(3,72,32,52)(4,73,33,53)(5,74,34,54)(6,75,35,55)(7,76,36,56)(8,77,37,57)(9,78,38,58)(10,79,39,59)(11,115,145,95)(12,116,146,96)(13,117,147,97)(14,118,148,98)(15,119,149,99)(16,120,150,100)(17,111,141,91)(18,112,142,92)(19,113,143,93)(20,114,144,94)(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)(101,151,121,131)(102,152,122,132)(103,153,123,133)(104,154,124,134)(105,155,125,135)(106,156,126,136)(107,157,127,137)(108,158,128,138)(109,159,129,139)(110,160,130,140) );
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)], [(1,39),(2,38),(3,37),(4,36),(5,35),(6,34),(7,33),(8,32),(9,31),(10,40),(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),(51,78),(52,77),(53,76),(54,75),(55,74),(56,73),(57,72),(58,71),(59,80),(60,79),(61,88),(62,87),(63,86),(64,85),(65,84),(66,83),(67,82),(68,81),(69,90),(70,89),(91,113),(92,112),(93,111),(94,120),(95,119),(96,118),(97,117),(98,116),(99,115),(100,114),(101,123),(102,122),(103,121),(104,130),(105,129),(106,128),(107,127),(108,126),(109,125),(110,124),(131,153),(132,152),(133,151),(134,160),(135,159),(136,158),(137,157),(138,156),(139,155),(140,154)], [(1,105,30,95),(2,104,21,94),(3,103,22,93),(4,102,23,92),(5,101,24,91),(6,110,25,100),(7,109,26,99),(8,108,27,98),(9,107,28,97),(10,106,29,96),(11,90,155,80),(12,89,156,79),(13,88,157,78),(14,87,158,77),(15,86,159,76),(16,85,160,75),(17,84,151,74),(18,83,152,73),(19,82,153,72),(20,81,154,71),(31,124,41,114),(32,123,42,113),(33,122,43,112),(34,121,44,111),(35,130,45,120),(36,129,46,119),(37,128,47,118),(38,127,48,117),(39,126,49,116),(40,125,50,115),(51,144,61,134),(52,143,62,133),(53,142,63,132),(54,141,64,131),(55,150,65,140),(56,149,66,139),(57,148,67,138),(58,147,68,137),(59,146,69,136),(60,145,70,135)], [(1,80,40,60),(2,71,31,51),(3,72,32,52),(4,73,33,53),(5,74,34,54),(6,75,35,55),(7,76,36,56),(8,77,37,57),(9,78,38,58),(10,79,39,59),(11,115,145,95),(12,116,146,96),(13,117,147,97),(14,118,148,98),(15,119,149,99),(16,120,150,100),(17,111,141,91),(18,112,142,92),(19,113,143,93),(20,114,144,94),(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),(101,151,121,131),(102,152,122,132),(103,153,123,133),(104,154,124,134),(105,155,125,135),(106,156,126,136),(107,157,127,137),(108,158,128,138),(109,159,129,139),(110,160,130,140)]])
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