metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D10⋊2C42, C10.53(C4×D4), D10⋊C4⋊7C4, C2.6(D5×C42), C22.60(D4×D5), C10.24(C2×C42), C2.1(D20⋊8C4), Dic5⋊7(C22⋊C4), (C2×Dic5).266D4, (C22×C4).298D10, C2.C42⋊15D5, C2.2(Dic5⋊4D4), (C23×D5).95C22, C23.256(C22×D5), C10.10C42⋊35C2, C10.48(C42⋊C2), C22.36(D4⋊2D5), (C22×C20).332C22, (C22×C10).291C23, C22.18(Q8⋊2D5), (C22×Dic5).199C22, (C2×C4×D5)⋊13C4, (C2×C4)⋊8(C4×D5), C5⋊3(C4×C22⋊C4), (C2×C20)⋊32(C2×C4), (C2×C4×Dic5)⋊16C2, C2.2(D5×C22⋊C4), C22.32(C2×C4×D5), (D5×C22×C4).13C2, C2.2(C4⋊C4⋊7D5), (C2×Dic5)⋊21(C2×C4), (C2×C10).200(C2×D4), C10.46(C2×C22⋊C4), (C22×D5).66(C2×C4), (C2×D10⋊C4).24C2, (C2×C10).131(C4○D4), (C5×C2.C42)⋊19C2, (C2×C10).151(C22×C4), SmallGroup(320,293)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D10⋊2C42
G = < a,b,c,d | a10=b2=c4=d4=1, bab=cac-1=a-1, ad=da, cbc-1=a8b, dbd-1=a5b, cd=dc >
Subgroups: 910 in 258 conjugacy classes, 99 normal (25 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C5, C2×C4, C2×C4, C23, C23, D5, C10, C10, C42, C22⋊C4, C22×C4, C22×C4, C22×C4, C24, Dic5, Dic5, C20, D10, D10, C2×C10, C2×C10, C2.C42, C2.C42, C2×C42, C2×C22⋊C4, C23×C4, C4×D5, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C22×D5, C22×D5, C22×C10, C4×C22⋊C4, C4×Dic5, D10⋊C4, C2×C4×D5, C2×C4×D5, C22×Dic5, C22×Dic5, C22×C20, C22×C20, C23×D5, C10.10C42, C5×C2.C42, C2×C4×Dic5, C2×D10⋊C4, D5×C22×C4, D10⋊2C42
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D5, C42, C22⋊C4, C22×C4, C2×D4, C4○D4, D10, C2×C42, C2×C22⋊C4, C42⋊C2, C4×D4, C4×D5, C22×D5, C4×C22⋊C4, C2×C4×D5, D4×D5, D4⋊2D5, Q8⋊2D5, D5×C42, D5×C22⋊C4, Dic5⋊4D4, C4⋊C4⋊7D5, D20⋊8C4, D10⋊2C42
►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 10)(2 9)(3 8)(4 7)(5 6)(11 14)(12 13)(15 20)(16 19)(17 18)(21 24)(22 23)(25 30)(26 29)(27 28)(31 34)(32 33)(35 40)(36 39)(37 38)(41 49)(42 48)(43 47)(44 46)(51 59)(52 58)(53 57)(54 56)(61 69)(62 68)(63 67)(64 66)(71 79)(72 78)(73 77)(74 76)(81 84)(82 83)(85 90)(86 89)(87 88)(91 94)(92 93)(95 100)(96 99)(97 98)(101 104)(102 103)(105 110)(106 109)(107 108)(111 114)(112 113)(115 120)(116 119)(117 118)(121 129)(122 128)(123 127)(124 126)(131 139)(132 138)(133 137)(134 136)(141 149)(142 148)(143 147)(144 146)(151 159)(152 158)(153 157)(154 156)
(1 93 13 83)(2 92 14 82)(3 91 15 81)(4 100 16 90)(5 99 17 89)(6 98 18 88)(7 97 19 87)(8 96 20 86)(9 95 11 85)(10 94 12 84)(21 115 31 105)(22 114 32 104)(23 113 33 103)(24 112 34 102)(25 111 35 101)(26 120 36 110)(27 119 37 109)(28 118 38 108)(29 117 39 107)(30 116 40 106)(41 140 51 130)(42 139 52 129)(43 138 53 128)(44 137 54 127)(45 136 55 126)(46 135 56 125)(47 134 57 124)(48 133 58 123)(49 132 59 122)(50 131 60 121)(61 160 71 150)(62 159 72 149)(63 158 73 148)(64 157 74 147)(65 156 75 146)(66 155 76 145)(67 154 77 144)(68 153 78 143)(69 152 79 142)(70 151 80 141)
(1 63 23 43)(2 64 24 44)(3 65 25 45)(4 66 26 46)(5 67 27 47)(6 68 28 48)(7 69 29 49)(8 70 30 50)(9 61 21 41)(10 62 22 42)(11 71 31 51)(12 72 32 52)(13 73 33 53)(14 74 34 54)(15 75 35 55)(16 76 36 56)(17 77 37 57)(18 78 38 58)(19 79 39 59)(20 80 40 60)(81 146 101 126)(82 147 102 127)(83 148 103 128)(84 149 104 129)(85 150 105 130)(86 141 106 121)(87 142 107 122)(88 143 108 123)(89 144 109 124)(90 145 110 125)(91 156 111 136)(92 157 112 137)(93 158 113 138)(94 159 114 139)(95 160 115 140)(96 151 116 131)(97 152 117 132)(98 153 118 133)(99 154 119 134)(100 155 120 135)
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), (1,10)(2,9)(3,8)(4,7)(5,6)(11,14)(12,13)(15,20)(16,19)(17,18)(21,24)(22,23)(25,30)(26,29)(27,28)(31,34)(32,33)(35,40)(36,39)(37,38)(41,49)(42,48)(43,47)(44,46)(51,59)(52,58)(53,57)(54,56)(61,69)(62,68)(63,67)(64,66)(71,79)(72,78)(73,77)(74,76)(81,84)(82,83)(85,90)(86,89)(87,88)(91,94)(92,93)(95,100)(96,99)(97,98)(101,104)(102,103)(105,110)(106,109)(107,108)(111,114)(112,113)(115,120)(116,119)(117,118)(121,129)(122,128)(123,127)(124,126)(131,139)(132,138)(133,137)(134,136)(141,149)(142,148)(143,147)(144,146)(151,159)(152,158)(153,157)(154,156), (1,93,13,83)(2,92,14,82)(3,91,15,81)(4,100,16,90)(5,99,17,89)(6,98,18,88)(7,97,19,87)(8,96,20,86)(9,95,11,85)(10,94,12,84)(21,115,31,105)(22,114,32,104)(23,113,33,103)(24,112,34,102)(25,111,35,101)(26,120,36,110)(27,119,37,109)(28,118,38,108)(29,117,39,107)(30,116,40,106)(41,140,51,130)(42,139,52,129)(43,138,53,128)(44,137,54,127)(45,136,55,126)(46,135,56,125)(47,134,57,124)(48,133,58,123)(49,132,59,122)(50,131,60,121)(61,160,71,150)(62,159,72,149)(63,158,73,148)(64,157,74,147)(65,156,75,146)(66,155,76,145)(67,154,77,144)(68,153,78,143)(69,152,79,142)(70,151,80,141), (1,63,23,43)(2,64,24,44)(3,65,25,45)(4,66,26,46)(5,67,27,47)(6,68,28,48)(7,69,29,49)(8,70,30,50)(9,61,21,41)(10,62,22,42)(11,71,31,51)(12,72,32,52)(13,73,33,53)(14,74,34,54)(15,75,35,55)(16,76,36,56)(17,77,37,57)(18,78,38,58)(19,79,39,59)(20,80,40,60)(81,146,101,126)(82,147,102,127)(83,148,103,128)(84,149,104,129)(85,150,105,130)(86,141,106,121)(87,142,107,122)(88,143,108,123)(89,144,109,124)(90,145,110,125)(91,156,111,136)(92,157,112,137)(93,158,113,138)(94,159,114,139)(95,160,115,140)(96,151,116,131)(97,152,117,132)(98,153,118,133)(99,154,119,134)(100,155,120,135)>;
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), (1,10)(2,9)(3,8)(4,7)(5,6)(11,14)(12,13)(15,20)(16,19)(17,18)(21,24)(22,23)(25,30)(26,29)(27,28)(31,34)(32,33)(35,40)(36,39)(37,38)(41,49)(42,48)(43,47)(44,46)(51,59)(52,58)(53,57)(54,56)(61,69)(62,68)(63,67)(64,66)(71,79)(72,78)(73,77)(74,76)(81,84)(82,83)(85,90)(86,89)(87,88)(91,94)(92,93)(95,100)(96,99)(97,98)(101,104)(102,103)(105,110)(106,109)(107,108)(111,114)(112,113)(115,120)(116,119)(117,118)(121,129)(122,128)(123,127)(124,126)(131,139)(132,138)(133,137)(134,136)(141,149)(142,148)(143,147)(144,146)(151,159)(152,158)(153,157)(154,156), (1,93,13,83)(2,92,14,82)(3,91,15,81)(4,100,16,90)(5,99,17,89)(6,98,18,88)(7,97,19,87)(8,96,20,86)(9,95,11,85)(10,94,12,84)(21,115,31,105)(22,114,32,104)(23,113,33,103)(24,112,34,102)(25,111,35,101)(26,120,36,110)(27,119,37,109)(28,118,38,108)(29,117,39,107)(30,116,40,106)(41,140,51,130)(42,139,52,129)(43,138,53,128)(44,137,54,127)(45,136,55,126)(46,135,56,125)(47,134,57,124)(48,133,58,123)(49,132,59,122)(50,131,60,121)(61,160,71,150)(62,159,72,149)(63,158,73,148)(64,157,74,147)(65,156,75,146)(66,155,76,145)(67,154,77,144)(68,153,78,143)(69,152,79,142)(70,151,80,141), (1,63,23,43)(2,64,24,44)(3,65,25,45)(4,66,26,46)(5,67,27,47)(6,68,28,48)(7,69,29,49)(8,70,30,50)(9,61,21,41)(10,62,22,42)(11,71,31,51)(12,72,32,52)(13,73,33,53)(14,74,34,54)(15,75,35,55)(16,76,36,56)(17,77,37,57)(18,78,38,58)(19,79,39,59)(20,80,40,60)(81,146,101,126)(82,147,102,127)(83,148,103,128)(84,149,104,129)(85,150,105,130)(86,141,106,121)(87,142,107,122)(88,143,108,123)(89,144,109,124)(90,145,110,125)(91,156,111,136)(92,157,112,137)(93,158,113,138)(94,159,114,139)(95,160,115,140)(96,151,116,131)(97,152,117,132)(98,153,118,133)(99,154,119,134)(100,155,120,135) );
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)], [(1,10),(2,9),(3,8),(4,7),(5,6),(11,14),(12,13),(15,20),(16,19),(17,18),(21,24),(22,23),(25,30),(26,29),(27,28),(31,34),(32,33),(35,40),(36,39),(37,38),(41,49),(42,48),(43,47),(44,46),(51,59),(52,58),(53,57),(54,56),(61,69),(62,68),(63,67),(64,66),(71,79),(72,78),(73,77),(74,76),(81,84),(82,83),(85,90),(86,89),(87,88),(91,94),(92,93),(95,100),(96,99),(97,98),(101,104),(102,103),(105,110),(106,109),(107,108),(111,114),(112,113),(115,120),(116,119),(117,118),(121,129),(122,128),(123,127),(124,126),(131,139),(132,138),(133,137),(134,136),(141,149),(142,148),(143,147),(144,146),(151,159),(152,158),(153,157),(154,156)], [(1,93,13,83),(2,92,14,82),(3,91,15,81),(4,100,16,90),(5,99,17,89),(6,98,18,88),(7,97,19,87),(8,96,20,86),(9,95,11,85),(10,94,12,84),(21,115,31,105),(22,114,32,104),(23,113,33,103),(24,112,34,102),(25,111,35,101),(26,120,36,110),(27,119,37,109),(28,118,38,108),(29,117,39,107),(30,116,40,106),(41,140,51,130),(42,139,52,129),(43,138,53,128),(44,137,54,127),(45,136,55,126),(46,135,56,125),(47,134,57,124),(48,133,58,123),(49,132,59,122),(50,131,60,121),(61,160,71,150),(62,159,72,149),(63,158,73,148),(64,157,74,147),(65,156,75,146),(66,155,76,145),(67,154,77,144),(68,153,78,143),(69,152,79,142),(70,151,80,141)], [(1,63,23,43),(2,64,24,44),(3,65,25,45),(4,66,26,46),(5,67,27,47),(6,68,28,48),(7,69,29,49),(8,70,30,50),(9,61,21,41),(10,62,22,42),(11,71,31,51),(12,72,32,52),(13,73,33,53),(14,74,34,54),(15,75,35,55),(16,76,36,56),(17,77,37,57),(18,78,38,58),(19,79,39,59),(20,80,40,60),(81,146,101,126),(82,147,102,127),(83,148,103,128),(84,149,104,129),(85,150,105,130),(86,141,106,121),(87,142,107,122),(88,143,108,123),(89,144,109,124),(90,145,110,125),(91,156,111,136),(92,157,112,137),(93,158,113,138),(94,159,114,139),(95,160,115,140),(96,151,116,131),(97,152,117,132),(98,153,118,133),(99,154,119,134),(100,155,120,135)]])
80 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 4A | ··· | 4L | 4M | ··· | 4T | 4U | ··· | 4AB | 5A | 5B | 10A | ··· | 10N | 20A | ··· | 20X |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 4 | ··· | 4 | 5 | 5 | 10 | ··· | 10 | 20 | ··· | 20 |
| size | 1 | 1 | ··· | 1 | 10 | 10 | 10 | 10 | 2 | ··· | 2 | 5 | ··· | 5 | 10 | ··· | 10 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 |
80 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | - | + | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | D4 | D5 | C4○D4 | D10 | C4×D5 | D4×D5 | D4⋊2D5 | Q8⋊2D5 |
| kernel | D10⋊2C42 | C10.10C42 | C5×C2.C42 | C2×C4×Dic5 | C2×D10⋊C4 | D5×C22×C4 | D10⋊C4 | C2×C4×D5 | C2×Dic5 | C2.C42 | C2×C10 | C22×C4 | C2×C4 | C22 | C22 | C22 |
| # reps | 1 | 1 | 1 | 2 | 2 | 1 | 16 | 8 | 4 | 2 | 4 | 6 | 24 | 4 | 2 | 2 |
Matrix representation of D10⋊2C42 ►in GL6(𝔽41)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 35 | 35 | 0 | 0 |
| 0 | 0 | 6 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 35 | 35 | 0 | 0 |
| 0 | 0 | 40 | 6 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 1 | 1 |
| 32 | 0 | 0 | 0 | 0 | 0 |
| 0 | 32 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 35 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 32 | 0 |
| 0 | 0 | 0 | 0 | 0 | 32 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 39 |
| 0 | 0 | 0 | 0 | 1 | 1 |
G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,35,6,0,0,0,0,35,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,35,40,0,0,0,0,35,6,0,0,0,0,0,0,40,1,0,0,0,0,0,1],[32,0,0,0,0,0,0,32,0,0,0,0,0,0,40,35,0,0,0,0,0,1,0,0,0,0,0,0,32,0,0,0,0,0,0,32],[1,0,0,0,0,0,0,9,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,1,0,0,0,0,39,1] >;
D10⋊2C42 in GAP, Magma, Sage, TeX
D_{10}\rtimes_2C_4^2 % in TeX
G:=Group("D10:2C4^2"); // GroupNames label
G:=SmallGroup(320,293);
// by ID
G=gap.SmallGroup(320,293);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,120,758,387,58,12550]);
// 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,c*d=d*c>;
// generators/relations