metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D10⋊5(C4⋊C4), C10.98(C4×D4), D10⋊C4⋊9C4, (C2×C20).251D4, C10.41C22≀C2, C22.27(Q8×D5), C22.112(D4×D5), (C22×C4).42D10, (C22×D5).16Q8, C2.4(C23⋊D10), C5⋊5(C23.8Q8), C2.7(D10⋊Q8), C2.4(D10⋊3Q8), (C2×Dic5).233D4, (C22×D5).126D4, C10.49(C22⋊Q8), C2.19(D20⋊8C4), C22.58(C4○D20), C23.297(C22×D5), C10.10C42⋊40C2, C2.6(D10.13D4), (C22×C20).347C22, (C22×C10).353C23, C22.29(Q8⋊2D5), (C23×D5).102C22, C10.51(C22.D4), (C22×Dic5).59C22, (C2×C4⋊C4)⋊7D5, (C2×C4)⋊4(C4×D5), (C10×C4⋊C4)⋊24C2, C2.22(D5×C4⋊C4), (C2×C20)⋊34(C2×C4), C10.44(C2×C4⋊C4), C2.13(C4×C5⋊D4), (C2×Dic5)⋊9(C2×C4), (C2×C10).84(C2×Q8), (D5×C22×C4).19C2, C22.138(C2×C4×D5), (C2×C10).334(C2×D4), C22.68(C2×C5⋊D4), (C2×C10.D4)⋊13C2, (C2×C4).169(C5⋊D4), (C22×D5).79(C2×C4), (C2×D10⋊C4).14C2, (C2×C10).190(C4○D4), (C2×C10).222(C22×C4), SmallGroup(320,616)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D10⋊5(C4⋊C4)
G = < a,b,c,d | a10=b2=c4=d4=1, bab=a-1, ac=ca, ad=da, cbc-1=a5b, bd=db, dcd-1=c-1 >
Subgroups: 894 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 [×2], C2×C22⋊C4 [×2], C2×C4⋊C4, C2×C4⋊C4, 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], D10⋊C4 [×4], D10⋊C4 [×2], C5×C4⋊C4 [×2], C2×C4×D5 [×6], C22×Dic5 [×3], C22×C20 [×3], C23×D5, C10.10C42 [×2], C2×C10.D4, C2×D10⋊C4 [×2], C10×C4⋊C4, D5×C22×C4, D10⋊5(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], C5⋊D4 [×2], C22×D5, C23.8Q8, C2×C4×D5, C4○D20, D4×D5 [×2], Q8×D5, Q8⋊2D5, C2×C5⋊D4, D5×C4⋊C4, D20⋊8C4, D10.13D4, D10⋊Q8, C4×C5⋊D4, C23⋊D10, D10⋊3Q8, D10⋊5(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)
(1 130 30 120)(2 121 21 111)(3 122 22 112)(4 123 23 113)(5 124 24 114)(6 125 25 115)(7 126 26 116)(8 127 27 117)(9 128 28 118)(10 129 29 119)(11 65 155 55)(12 66 156 56)(13 67 157 57)(14 68 158 58)(15 69 159 59)(16 70 160 60)(17 61 151 51)(18 62 152 52)(19 63 153 53)(20 64 154 54)(31 101 41 91)(32 102 42 92)(33 103 43 93)(34 104 44 94)(35 105 45 95)(36 106 46 96)(37 107 47 97)(38 108 48 98)(39 109 49 99)(40 110 50 100)(71 141 81 131)(72 142 82 132)(73 143 83 133)(74 144 84 134)(75 145 85 135)(76 146 86 136)(77 147 87 137)(78 148 88 138)(79 149 89 139)(80 150 90 140)
(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), (1,130,30,120)(2,121,21,111)(3,122,22,112)(4,123,23,113)(5,124,24,114)(6,125,25,115)(7,126,26,116)(8,127,27,117)(9,128,28,118)(10,129,29,119)(11,65,155,55)(12,66,156,56)(13,67,157,57)(14,68,158,58)(15,69,159,59)(16,70,160,60)(17,61,151,51)(18,62,152,52)(19,63,153,53)(20,64,154,54)(31,101,41,91)(32,102,42,92)(33,103,43,93)(34,104,44,94)(35,105,45,95)(36,106,46,96)(37,107,47,97)(38,108,48,98)(39,109,49,99)(40,110,50,100)(71,141,81,131)(72,142,82,132)(73,143,83,133)(74,144,84,134)(75,145,85,135)(76,146,86,136)(77,147,87,137)(78,148,88,138)(79,149,89,139)(80,150,90,140), (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), (1,130,30,120)(2,121,21,111)(3,122,22,112)(4,123,23,113)(5,124,24,114)(6,125,25,115)(7,126,26,116)(8,127,27,117)(9,128,28,118)(10,129,29,119)(11,65,155,55)(12,66,156,56)(13,67,157,57)(14,68,158,58)(15,69,159,59)(16,70,160,60)(17,61,151,51)(18,62,152,52)(19,63,153,53)(20,64,154,54)(31,101,41,91)(32,102,42,92)(33,103,43,93)(34,104,44,94)(35,105,45,95)(36,106,46,96)(37,107,47,97)(38,108,48,98)(39,109,49,99)(40,110,50,100)(71,141,81,131)(72,142,82,132)(73,143,83,133)(74,144,84,134)(75,145,85,135)(76,146,86,136)(77,147,87,137)(78,148,88,138)(79,149,89,139)(80,150,90,140), (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)], [(1,130,30,120),(2,121,21,111),(3,122,22,112),(4,123,23,113),(5,124,24,114),(6,125,25,115),(7,126,26,116),(8,127,27,117),(9,128,28,118),(10,129,29,119),(11,65,155,55),(12,66,156,56),(13,67,157,57),(14,68,158,58),(15,69,159,59),(16,70,160,60),(17,61,151,51),(18,62,152,52),(19,63,153,53),(20,64,154,54),(31,101,41,91),(32,102,42,92),(33,103,43,93),(34,104,44,94),(35,105,45,95),(36,106,46,96),(37,107,47,97),(38,108,48,98),(39,109,49,99),(40,110,50,100),(71,141,81,131),(72,142,82,132),(73,143,83,133),(74,144,84,134),(75,145,85,135),(76,146,86,136),(77,147,87,137),(78,148,88,138),(79,149,89,139),(80,150,90,140)], [(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 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | - | + | + | + | - | + | |||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | D4 | D4 | D4 | Q8 | D5 | C4○D4 | D10 | C4×D5 | C5⋊D4 | C4○D20 | D4×D5 | Q8×D5 | Q8⋊2D5 |
| kernel | D10⋊5(C4⋊C4) | C10.10C42 | C2×C10.D4 | C2×D10⋊C4 | C10×C4⋊C4 | D5×C22×C4 | D10⋊C4 | C2×Dic5 | C2×C20 | C22×D5 | C22×D5 | C2×C4⋊C4 | C2×C10 | C22×C4 | C2×C4 | C2×C4 | C22 | C22 | C22 | C22 |
| # reps | 1 | 2 | 1 | 2 | 1 | 1 | 8 | 2 | 2 | 2 | 2 | 2 | 4 | 6 | 8 | 8 | 8 | 4 | 2 | 2 |
Matrix representation of D10⋊5(C4⋊C4) ►in GL6(𝔽41)
| 34 | 34 | 0 | 0 | 0 | 0 |
| 7 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 7 | 0 | 0 |
| 0 | 0 | 34 | 7 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 34 | 34 | 0 | 0 | 0 | 0 |
| 1 | 7 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 7 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 17 | 40 | 0 | 0 | 0 | 0 |
| 1 | 24 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 33 | 13 |
| 0 | 0 | 0 | 0 | 36 | 8 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 32 | 0 | 0 | 0 |
| 0 | 0 | 0 | 32 | 0 | 0 |
| 0 | 0 | 0 | 0 | 32 | 37 |
| 0 | 0 | 0 | 0 | 0 | 9 |
G:=sub<GL(6,GF(41))| [34,7,0,0,0,0,34,1,0,0,0,0,0,0,40,34,0,0,0,0,7,7,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[34,1,0,0,0,0,34,7,0,0,0,0,0,0,1,7,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[17,1,0,0,0,0,40,24,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,33,36,0,0,0,0,13,8],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,32,0,0,0,0,0,0,32,0,0,0,0,0,0,32,0,0,0,0,0,37,9] >;
D10⋊5(C4⋊C4) in GAP, Magma, Sage, TeX
D_{10}\rtimes_5(C_4\rtimes C_4) % in TeX
G:=Group("D10:5(C4:C4)"); // GroupNames label
G:=SmallGroup(320,616);
// by ID
G=gap.SmallGroup(320,616);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,253,232,387,58,12550]);
// 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,c*b*c^-1=a^5*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations