metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D10⋊2(C4⋊C4), C22.61(D4×D5), C22.16(Q8×D5), C10.4(C4⋊D4), (C22×D5).76D4, C2.C42⋊2D5, (C22×C4).16D10, (C22×D5).12Q8, C2.3(D10⋊D4), C5⋊2(C23.7Q8), Dic5⋊3(C22⋊C4), C2.3(D10⋊Q8), (C2×Dic5).186D4, C2.9(C42⋊D5), C10.24(C22⋊Q8), C22.35(C4○D20), (C22×C20).13C22, (C23×D5).96C22, C23.257(C22×D5), C10.10C42⋊36C2, C10.24(C42⋊C2), (C22×C10).292C23, (C22×Dic5).15C22, (C2×C4×D5)⋊14C4, C2.7(D5×C4⋊C4), C10.28(C2×C4⋊C4), C2.8(D5×C22⋊C4), C22.90(C2×C4×D5), (C2×C4).125(C4×D5), (C2×C10).67(C2×Q8), (D5×C22×C4).14C2, (C2×C20).316(C2×C4), (C2×C10.D4)⋊1C2, (C2×C10).201(C2×D4), C10.47(C2×C22⋊C4), (C2×D10⋊C4).4C2, (C2×C10).58(C4○D4), (C22×D5).93(C2×C4), (C5×C2.C42)⋊20C2, (C2×C10).152(C22×C4), (C2×Dic5).133(C2×C4), SmallGroup(320,294)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D10⋊2(C4⋊C4)
G = < a,b,c,d | a10=b2=c4=d4=1, bab=cac-1=a-1, ad=da, cbc-1=a8b, dbd-1=a5b, dcd-1=c-1 >
Subgroups: 910 in 234 conjugacy classes, 79 normal (25 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C5, C2×C4, C2×C4, C23, C23, D5, C10, C10, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C22×C4, C24, Dic5, Dic5, C20, D10, D10, C2×C10, 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, C23.7Q8, C10.D4, D10⋊C4, C2×C4×D5, C2×C4×D5, C22×Dic5, C22×Dic5, C22×C20, C22×C20, C23×D5, C10.10C42, C5×C2.C42, C2×C10.D4, C2×D10⋊C4, D5×C22×C4, D10⋊2(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, C22×D5, C23.7Q8, C2×C4×D5, C4○D20, D4×D5, Q8×D5, C42⋊D5, D5×C22⋊C4, D10⋊D4, D5×C4⋊C4, D10⋊Q8, D10⋊2(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 37)(2 36)(3 35)(4 34)(5 33)(6 32)(7 31)(8 40)(9 39)(10 38)(11 132)(12 131)(13 140)(14 139)(15 138)(16 137)(17 136)(18 135)(19 134)(20 133)(21 46)(22 45)(23 44)(24 43)(25 42)(26 41)(27 50)(28 49)(29 48)(30 47)(51 86)(52 85)(53 84)(54 83)(55 82)(56 81)(57 90)(58 89)(59 88)(60 87)(61 76)(62 75)(63 74)(64 73)(65 72)(66 71)(67 80)(68 79)(69 78)(70 77)(91 121)(92 130)(93 129)(94 128)(95 127)(96 126)(97 125)(98 124)(99 123)(100 122)(101 111)(102 120)(103 119)(104 118)(105 117)(106 116)(107 115)(108 114)(109 113)(110 112)(141 156)(142 155)(143 154)(144 153)(145 152)(146 151)(147 160)(148 159)(149 158)(150 157)
(1 110 28 98)(2 109 29 97)(3 108 30 96)(4 107 21 95)(5 106 22 94)(6 105 23 93)(7 104 24 92)(8 103 25 91)(9 102 26 100)(10 101 27 99)(11 73 153 85)(12 72 154 84)(13 71 155 83)(14 80 156 82)(15 79 157 81)(16 78 158 90)(17 77 159 89)(18 76 160 88)(19 75 151 87)(20 74 152 86)(31 120 43 122)(32 119 44 121)(33 118 45 130)(34 117 46 129)(35 116 47 128)(36 115 48 127)(37 114 49 126)(38 113 50 125)(39 112 41 124)(40 111 42 123)(51 135 63 147)(52 134 64 146)(53 133 65 145)(54 132 66 144)(55 131 67 143)(56 140 68 142)(57 139 69 141)(58 138 70 150)(59 137 61 149)(60 136 62 148)
(1 85 45 65)(2 86 46 66)(3 87 47 67)(4 88 48 68)(5 89 49 69)(6 90 50 70)(7 81 41 61)(8 82 42 62)(9 83 43 63)(10 84 44 64)(11 118 145 98)(12 119 146 99)(13 120 147 100)(14 111 148 91)(15 112 149 92)(16 113 150 93)(17 114 141 94)(18 115 142 95)(19 116 143 96)(20 117 144 97)(21 76 36 56)(22 77 37 57)(23 78 38 58)(24 79 39 59)(25 80 40 60)(26 71 31 51)(27 72 32 52)(28 73 33 53)(29 74 34 54)(30 75 35 55)(101 154 121 134)(102 155 122 135)(103 156 123 136)(104 157 124 137)(105 158 125 138)(106 159 126 139)(107 160 127 140)(108 151 128 131)(109 152 129 132)(110 153 130 133)
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), (1,37)(2,36)(3,35)(4,34)(5,33)(6,32)(7,31)(8,40)(9,39)(10,38)(11,132)(12,131)(13,140)(14,139)(15,138)(16,137)(17,136)(18,135)(19,134)(20,133)(21,46)(22,45)(23,44)(24,43)(25,42)(26,41)(27,50)(28,49)(29,48)(30,47)(51,86)(52,85)(53,84)(54,83)(55,82)(56,81)(57,90)(58,89)(59,88)(60,87)(61,76)(62,75)(63,74)(64,73)(65,72)(66,71)(67,80)(68,79)(69,78)(70,77)(91,121)(92,130)(93,129)(94,128)(95,127)(96,126)(97,125)(98,124)(99,123)(100,122)(101,111)(102,120)(103,119)(104,118)(105,117)(106,116)(107,115)(108,114)(109,113)(110,112)(141,156)(142,155)(143,154)(144,153)(145,152)(146,151)(147,160)(148,159)(149,158)(150,157), (1,110,28,98)(2,109,29,97)(3,108,30,96)(4,107,21,95)(5,106,22,94)(6,105,23,93)(7,104,24,92)(8,103,25,91)(9,102,26,100)(10,101,27,99)(11,73,153,85)(12,72,154,84)(13,71,155,83)(14,80,156,82)(15,79,157,81)(16,78,158,90)(17,77,159,89)(18,76,160,88)(19,75,151,87)(20,74,152,86)(31,120,43,122)(32,119,44,121)(33,118,45,130)(34,117,46,129)(35,116,47,128)(36,115,48,127)(37,114,49,126)(38,113,50,125)(39,112,41,124)(40,111,42,123)(51,135,63,147)(52,134,64,146)(53,133,65,145)(54,132,66,144)(55,131,67,143)(56,140,68,142)(57,139,69,141)(58,138,70,150)(59,137,61,149)(60,136,62,148), (1,85,45,65)(2,86,46,66)(3,87,47,67)(4,88,48,68)(5,89,49,69)(6,90,50,70)(7,81,41,61)(8,82,42,62)(9,83,43,63)(10,84,44,64)(11,118,145,98)(12,119,146,99)(13,120,147,100)(14,111,148,91)(15,112,149,92)(16,113,150,93)(17,114,141,94)(18,115,142,95)(19,116,143,96)(20,117,144,97)(21,76,36,56)(22,77,37,57)(23,78,38,58)(24,79,39,59)(25,80,40,60)(26,71,31,51)(27,72,32,52)(28,73,33,53)(29,74,34,54)(30,75,35,55)(101,154,121,134)(102,155,122,135)(103,156,123,136)(104,157,124,137)(105,158,125,138)(106,159,126,139)(107,160,127,140)(108,151,128,131)(109,152,129,132)(110,153,130,133)>;
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), (1,37)(2,36)(3,35)(4,34)(5,33)(6,32)(7,31)(8,40)(9,39)(10,38)(11,132)(12,131)(13,140)(14,139)(15,138)(16,137)(17,136)(18,135)(19,134)(20,133)(21,46)(22,45)(23,44)(24,43)(25,42)(26,41)(27,50)(28,49)(29,48)(30,47)(51,86)(52,85)(53,84)(54,83)(55,82)(56,81)(57,90)(58,89)(59,88)(60,87)(61,76)(62,75)(63,74)(64,73)(65,72)(66,71)(67,80)(68,79)(69,78)(70,77)(91,121)(92,130)(93,129)(94,128)(95,127)(96,126)(97,125)(98,124)(99,123)(100,122)(101,111)(102,120)(103,119)(104,118)(105,117)(106,116)(107,115)(108,114)(109,113)(110,112)(141,156)(142,155)(143,154)(144,153)(145,152)(146,151)(147,160)(148,159)(149,158)(150,157), (1,110,28,98)(2,109,29,97)(3,108,30,96)(4,107,21,95)(5,106,22,94)(6,105,23,93)(7,104,24,92)(8,103,25,91)(9,102,26,100)(10,101,27,99)(11,73,153,85)(12,72,154,84)(13,71,155,83)(14,80,156,82)(15,79,157,81)(16,78,158,90)(17,77,159,89)(18,76,160,88)(19,75,151,87)(20,74,152,86)(31,120,43,122)(32,119,44,121)(33,118,45,130)(34,117,46,129)(35,116,47,128)(36,115,48,127)(37,114,49,126)(38,113,50,125)(39,112,41,124)(40,111,42,123)(51,135,63,147)(52,134,64,146)(53,133,65,145)(54,132,66,144)(55,131,67,143)(56,140,68,142)(57,139,69,141)(58,138,70,150)(59,137,61,149)(60,136,62,148), (1,85,45,65)(2,86,46,66)(3,87,47,67)(4,88,48,68)(5,89,49,69)(6,90,50,70)(7,81,41,61)(8,82,42,62)(9,83,43,63)(10,84,44,64)(11,118,145,98)(12,119,146,99)(13,120,147,100)(14,111,148,91)(15,112,149,92)(16,113,150,93)(17,114,141,94)(18,115,142,95)(19,116,143,96)(20,117,144,97)(21,76,36,56)(22,77,37,57)(23,78,38,58)(24,79,39,59)(25,80,40,60)(26,71,31,51)(27,72,32,52)(28,73,33,53)(29,74,34,54)(30,75,35,55)(101,154,121,134)(102,155,122,135)(103,156,123,136)(104,157,124,137)(105,158,125,138)(106,159,126,139)(107,160,127,140)(108,151,128,131)(109,152,129,132)(110,153,130,133) );
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)], [(1,37),(2,36),(3,35),(4,34),(5,33),(6,32),(7,31),(8,40),(9,39),(10,38),(11,132),(12,131),(13,140),(14,139),(15,138),(16,137),(17,136),(18,135),(19,134),(20,133),(21,46),(22,45),(23,44),(24,43),(25,42),(26,41),(27,50),(28,49),(29,48),(30,47),(51,86),(52,85),(53,84),(54,83),(55,82),(56,81),(57,90),(58,89),(59,88),(60,87),(61,76),(62,75),(63,74),(64,73),(65,72),(66,71),(67,80),(68,79),(69,78),(70,77),(91,121),(92,130),(93,129),(94,128),(95,127),(96,126),(97,125),(98,124),(99,123),(100,122),(101,111),(102,120),(103,119),(104,118),(105,117),(106,116),(107,115),(108,114),(109,113),(110,112),(141,156),(142,155),(143,154),(144,153),(145,152),(146,151),(147,160),(148,159),(149,158),(150,157)], [(1,110,28,98),(2,109,29,97),(3,108,30,96),(4,107,21,95),(5,106,22,94),(6,105,23,93),(7,104,24,92),(8,103,25,91),(9,102,26,100),(10,101,27,99),(11,73,153,85),(12,72,154,84),(13,71,155,83),(14,80,156,82),(15,79,157,81),(16,78,158,90),(17,77,159,89),(18,76,160,88),(19,75,151,87),(20,74,152,86),(31,120,43,122),(32,119,44,121),(33,118,45,130),(34,117,46,129),(35,116,47,128),(36,115,48,127),(37,114,49,126),(38,113,50,125),(39,112,41,124),(40,111,42,123),(51,135,63,147),(52,134,64,146),(53,133,65,145),(54,132,66,144),(55,131,67,143),(56,140,68,142),(57,139,69,141),(58,138,70,150),(59,137,61,149),(60,136,62,148)], [(1,85,45,65),(2,86,46,66),(3,87,47,67),(4,88,48,68),(5,89,49,69),(6,90,50,70),(7,81,41,61),(8,82,42,62),(9,83,43,63),(10,84,44,64),(11,118,145,98),(12,119,146,99),(13,120,147,100),(14,111,148,91),(15,112,149,92),(16,113,150,93),(17,114,141,94),(18,115,142,95),(19,116,143,96),(20,117,144,97),(21,76,36,56),(22,77,37,57),(23,78,38,58),(24,79,39,59),(25,80,40,60),(26,71,31,51),(27,72,32,52),(28,73,33,53),(29,74,34,54),(30,75,35,55),(101,154,121,134),(102,155,122,135),(103,156,123,136),(104,157,124,137),(105,158,125,138),(106,159,126,139),(107,160,127,140),(108,151,128,131),(109,152,129,132),(110,153,130,133)]])
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 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | - | + | + | + | - | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | D4 | D4 | Q8 | D5 | C4○D4 | D10 | C4×D5 | C4○D20 | D4×D5 | Q8×D5 |
| kernel | D10⋊2(C4⋊C4) | C10.10C42 | C5×C2.C42 | C2×C10.D4 | C2×D10⋊C4 | D5×C22×C4 | C2×C4×D5 | C2×Dic5 | C22×D5 | C22×D5 | C2.C42 | C2×C10 | C22×C4 | C2×C4 | C22 | C22 | C22 |
| # reps | 1 | 1 | 1 | 2 | 2 | 1 | 8 | 4 | 2 | 2 | 2 | 4 | 6 | 8 | 16 | 6 | 2 |
Matrix representation of D10⋊2(C4⋊C4) ►in GL6(𝔽41)
| 1 | 6 | 0 | 0 | 0 | 0 |
| 35 | 6 | 0 | 0 | 0 | 0 |
| 0 | 0 | 34 | 34 | 0 | 0 |
| 0 | 0 | 7 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 35 | 40 | 0 | 0 | 0 | 0 |
| 35 | 6 | 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 | 1 | 40 |
| 9 | 13 | 0 | 0 | 0 | 0 |
| 0 | 32 | 0 | 0 | 0 | 0 |
| 0 | 0 | 32 | 0 | 0 | 0 |
| 0 | 0 | 22 | 9 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 2 | 28 | 0 | 0 | 0 | 0 |
| 13 | 39 | 0 | 0 | 0 | 0 |
| 0 | 0 | 17 | 40 | 0 | 0 |
| 0 | 0 | 1 | 24 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 39 |
| 0 | 0 | 0 | 0 | 0 | 40 |
G:=sub<GL(6,GF(41))| [1,35,0,0,0,0,6,6,0,0,0,0,0,0,34,7,0,0,0,0,34,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[35,35,0,0,0,0,40,6,0,0,0,0,0,0,34,1,0,0,0,0,34,7,0,0,0,0,0,0,1,1,0,0,0,0,0,40],[9,0,0,0,0,0,13,32,0,0,0,0,0,0,32,22,0,0,0,0,0,9,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[2,13,0,0,0,0,28,39,0,0,0,0,0,0,17,1,0,0,0,0,40,24,0,0,0,0,0,0,1,0,0,0,0,0,39,40] >;
D10⋊2(C4⋊C4) in GAP, Magma, Sage, TeX
D_{10}\rtimes_2(C_4\rtimes C_4) % in TeX
G:=Group("D10:2(C4:C4)"); // GroupNames label
G:=SmallGroup(320,294);
// by ID
G=gap.SmallGroup(320,294);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,120,1094,387,58,12550]);
// 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^8*b,d*b*d^-1=a^5*b,d*c*d^-1=c^-1>;
// generators/relations