direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C10×D4⋊C4, D4⋊3(C2×C20), (C2×D4)⋊7C20, C2.1(C10×D8), (D4×C10)⋊31C4, (C22×C8)⋊3C10, (C22×C40)⋊7C2, C4.51(D4×C10), C10.73(C2×D8), (C2×C10).53D8, (C2×C40)⋊43C22, C20.458(C2×D4), (C2×C20).414D4, C4.1(C22×C20), C2.1(C10×SD16), C22.12(C5×D8), C23.54(C5×D4), (C2×C10).44SD16, C10.81(C2×SD16), (C22×D4).6C10, C22.41(D4×C10), (C2×C20).890C23, C20.205(C22×C4), (C22×C10).215D4, C22.10(C5×SD16), C20.127(C22⋊C4), (D4×C10).286C22, (C22×C20).581C22, (C2×C4⋊C4)⋊9C10, C4⋊C4⋊7(C2×C10), (C10×C4⋊C4)⋊36C2, (C2×C8)⋊11(C2×C10), (C5×D4)⋊33(C2×C4), (D4×C2×C10).18C2, (C2×C4).68(C5×D4), (C5×C4⋊C4)⋊63C22, (C2×C4).47(C2×C20), C4.12(C5×C22⋊C4), (C2×C20).441(C2×C4), (C2×D4).44(C2×C10), (C2×C10).617(C2×D4), C2.17(C10×C22⋊C4), C10.146(C2×C22⋊C4), (C2×C4).65(C22×C10), C22.33(C5×C22⋊C4), (C22×C4).110(C2×C10), (C2×C10).202(C22⋊C4), SmallGroup(320,915)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C10×D4⋊C4
G = < a,b,c,d | a10=b4=c2=d4=1, ab=ba, ac=ca, ad=da, cbc=dbd-1=b-1, dcd-1=bc >
Subgroups: 402 in 202 conjugacy classes, 98 normal (30 characteristic)
C1, C2 [×3], C2 [×4], C2 [×4], C4 [×2], C4 [×2], C4 [×2], C22, C22 [×6], C22 [×16], C5, C8 [×2], C2×C4 [×2], C2×C4 [×4], C2×C4 [×4], D4 [×4], D4 [×6], C23, C23 [×10], C10 [×3], C10 [×4], C10 [×4], C4⋊C4 [×2], C4⋊C4, C2×C8 [×2], C2×C8 [×2], C22×C4, C22×C4, C2×D4 [×6], C2×D4 [×3], C24, C20 [×2], C20 [×2], C20 [×2], C2×C10, C2×C10 [×6], C2×C10 [×16], D4⋊C4 [×4], C2×C4⋊C4, C22×C8, C22×D4, C40 [×2], C2×C20 [×2], C2×C20 [×4], C2×C20 [×4], C5×D4 [×4], C5×D4 [×6], C22×C10, C22×C10 [×10], C2×D4⋊C4, C5×C4⋊C4 [×2], C5×C4⋊C4, C2×C40 [×2], C2×C40 [×2], C22×C20, C22×C20, D4×C10 [×6], D4×C10 [×3], C23×C10, C5×D4⋊C4 [×4], C10×C4⋊C4, C22×C40, D4×C2×C10, C10×D4⋊C4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C5, C2×C4 [×6], D4 [×4], C23, C10 [×7], C22⋊C4 [×4], D8 [×2], SD16 [×2], C22×C4, C2×D4 [×2], C20 [×4], C2×C10 [×7], D4⋊C4 [×4], C2×C22⋊C4, C2×D8, C2×SD16, C2×C20 [×6], C5×D4 [×4], C22×C10, C2×D4⋊C4, C5×C22⋊C4 [×4], C5×D8 [×2], C5×SD16 [×2], C22×C20, D4×C10 [×2], C5×D4⋊C4 [×4], C10×C22⋊C4, C10×D8, C10×SD16, C10×D4⋊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)
(1 147 137 96)(2 148 138 97)(3 149 139 98)(4 150 140 99)(5 141 131 100)(6 142 132 91)(7 143 133 92)(8 144 134 93)(9 145 135 94)(10 146 136 95)(11 22 51 160)(12 23 52 151)(13 24 53 152)(14 25 54 153)(15 26 55 154)(16 27 56 155)(17 28 57 156)(18 29 58 157)(19 30 59 158)(20 21 60 159)(31 70 71 43)(32 61 72 44)(33 62 73 45)(34 63 74 46)(35 64 75 47)(36 65 76 48)(37 66 77 49)(38 67 78 50)(39 68 79 41)(40 69 80 42)(81 103 115 127)(82 104 116 128)(83 105 117 129)(84 106 118 130)(85 107 119 121)(86 108 120 122)(87 109 111 123)(88 110 112 124)(89 101 113 125)(90 102 114 126)
(1 102)(2 103)(3 104)(4 105)(5 106)(6 107)(7 108)(8 109)(9 110)(10 101)(11 50)(12 41)(13 42)(14 43)(15 44)(16 45)(17 46)(18 47)(19 48)(20 49)(21 77)(22 78)(23 79)(24 80)(25 71)(26 72)(27 73)(28 74)(29 75)(30 76)(31 153)(32 154)(33 155)(34 156)(35 157)(36 158)(37 159)(38 160)(39 151)(40 152)(51 67)(52 68)(53 69)(54 70)(55 61)(56 62)(57 63)(58 64)(59 65)(60 66)(81 148)(82 149)(83 150)(84 141)(85 142)(86 143)(87 144)(88 145)(89 146)(90 147)(91 119)(92 120)(93 111)(94 112)(95 113)(96 114)(97 115)(98 116)(99 117)(100 118)(121 132)(122 133)(123 134)(124 135)(125 136)(126 137)(127 138)(128 139)(129 140)(130 131)
(1 157 114 75)(2 158 115 76)(3 159 116 77)(4 160 117 78)(5 151 118 79)(6 152 119 80)(7 153 120 71)(8 154 111 72)(9 155 112 73)(10 156 113 74)(11 105 50 99)(12 106 41 100)(13 107 42 91)(14 108 43 92)(15 109 44 93)(16 110 45 94)(17 101 46 95)(18 102 47 96)(19 103 48 97)(20 104 49 98)(21 82 37 139)(22 83 38 140)(23 84 39 131)(24 85 40 132)(25 86 31 133)(26 87 32 134)(27 88 33 135)(28 89 34 136)(29 90 35 137)(30 81 36 138)(51 129 67 150)(52 130 68 141)(53 121 69 142)(54 122 70 143)(55 123 61 144)(56 124 62 145)(57 125 63 146)(58 126 64 147)(59 127 65 148)(60 128 66 149)
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), (1,147,137,96)(2,148,138,97)(3,149,139,98)(4,150,140,99)(5,141,131,100)(6,142,132,91)(7,143,133,92)(8,144,134,93)(9,145,135,94)(10,146,136,95)(11,22,51,160)(12,23,52,151)(13,24,53,152)(14,25,54,153)(15,26,55,154)(16,27,56,155)(17,28,57,156)(18,29,58,157)(19,30,59,158)(20,21,60,159)(31,70,71,43)(32,61,72,44)(33,62,73,45)(34,63,74,46)(35,64,75,47)(36,65,76,48)(37,66,77,49)(38,67,78,50)(39,68,79,41)(40,69,80,42)(81,103,115,127)(82,104,116,128)(83,105,117,129)(84,106,118,130)(85,107,119,121)(86,108,120,122)(87,109,111,123)(88,110,112,124)(89,101,113,125)(90,102,114,126), (1,102)(2,103)(3,104)(4,105)(5,106)(6,107)(7,108)(8,109)(9,110)(10,101)(11,50)(12,41)(13,42)(14,43)(15,44)(16,45)(17,46)(18,47)(19,48)(20,49)(21,77)(22,78)(23,79)(24,80)(25,71)(26,72)(27,73)(28,74)(29,75)(30,76)(31,153)(32,154)(33,155)(34,156)(35,157)(36,158)(37,159)(38,160)(39,151)(40,152)(51,67)(52,68)(53,69)(54,70)(55,61)(56,62)(57,63)(58,64)(59,65)(60,66)(81,148)(82,149)(83,150)(84,141)(85,142)(86,143)(87,144)(88,145)(89,146)(90,147)(91,119)(92,120)(93,111)(94,112)(95,113)(96,114)(97,115)(98,116)(99,117)(100,118)(121,132)(122,133)(123,134)(124,135)(125,136)(126,137)(127,138)(128,139)(129,140)(130,131), (1,157,114,75)(2,158,115,76)(3,159,116,77)(4,160,117,78)(5,151,118,79)(6,152,119,80)(7,153,120,71)(8,154,111,72)(9,155,112,73)(10,156,113,74)(11,105,50,99)(12,106,41,100)(13,107,42,91)(14,108,43,92)(15,109,44,93)(16,110,45,94)(17,101,46,95)(18,102,47,96)(19,103,48,97)(20,104,49,98)(21,82,37,139)(22,83,38,140)(23,84,39,131)(24,85,40,132)(25,86,31,133)(26,87,32,134)(27,88,33,135)(28,89,34,136)(29,90,35,137)(30,81,36,138)(51,129,67,150)(52,130,68,141)(53,121,69,142)(54,122,70,143)(55,123,61,144)(56,124,62,145)(57,125,63,146)(58,126,64,147)(59,127,65,148)(60,128,66,149)>;
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), (1,147,137,96)(2,148,138,97)(3,149,139,98)(4,150,140,99)(5,141,131,100)(6,142,132,91)(7,143,133,92)(8,144,134,93)(9,145,135,94)(10,146,136,95)(11,22,51,160)(12,23,52,151)(13,24,53,152)(14,25,54,153)(15,26,55,154)(16,27,56,155)(17,28,57,156)(18,29,58,157)(19,30,59,158)(20,21,60,159)(31,70,71,43)(32,61,72,44)(33,62,73,45)(34,63,74,46)(35,64,75,47)(36,65,76,48)(37,66,77,49)(38,67,78,50)(39,68,79,41)(40,69,80,42)(81,103,115,127)(82,104,116,128)(83,105,117,129)(84,106,118,130)(85,107,119,121)(86,108,120,122)(87,109,111,123)(88,110,112,124)(89,101,113,125)(90,102,114,126), (1,102)(2,103)(3,104)(4,105)(5,106)(6,107)(7,108)(8,109)(9,110)(10,101)(11,50)(12,41)(13,42)(14,43)(15,44)(16,45)(17,46)(18,47)(19,48)(20,49)(21,77)(22,78)(23,79)(24,80)(25,71)(26,72)(27,73)(28,74)(29,75)(30,76)(31,153)(32,154)(33,155)(34,156)(35,157)(36,158)(37,159)(38,160)(39,151)(40,152)(51,67)(52,68)(53,69)(54,70)(55,61)(56,62)(57,63)(58,64)(59,65)(60,66)(81,148)(82,149)(83,150)(84,141)(85,142)(86,143)(87,144)(88,145)(89,146)(90,147)(91,119)(92,120)(93,111)(94,112)(95,113)(96,114)(97,115)(98,116)(99,117)(100,118)(121,132)(122,133)(123,134)(124,135)(125,136)(126,137)(127,138)(128,139)(129,140)(130,131), (1,157,114,75)(2,158,115,76)(3,159,116,77)(4,160,117,78)(5,151,118,79)(6,152,119,80)(7,153,120,71)(8,154,111,72)(9,155,112,73)(10,156,113,74)(11,105,50,99)(12,106,41,100)(13,107,42,91)(14,108,43,92)(15,109,44,93)(16,110,45,94)(17,101,46,95)(18,102,47,96)(19,103,48,97)(20,104,49,98)(21,82,37,139)(22,83,38,140)(23,84,39,131)(24,85,40,132)(25,86,31,133)(26,87,32,134)(27,88,33,135)(28,89,34,136)(29,90,35,137)(30,81,36,138)(51,129,67,150)(52,130,68,141)(53,121,69,142)(54,122,70,143)(55,123,61,144)(56,124,62,145)(57,125,63,146)(58,126,64,147)(59,127,65,148)(60,128,66,149) );
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)], [(1,147,137,96),(2,148,138,97),(3,149,139,98),(4,150,140,99),(5,141,131,100),(6,142,132,91),(7,143,133,92),(8,144,134,93),(9,145,135,94),(10,146,136,95),(11,22,51,160),(12,23,52,151),(13,24,53,152),(14,25,54,153),(15,26,55,154),(16,27,56,155),(17,28,57,156),(18,29,58,157),(19,30,59,158),(20,21,60,159),(31,70,71,43),(32,61,72,44),(33,62,73,45),(34,63,74,46),(35,64,75,47),(36,65,76,48),(37,66,77,49),(38,67,78,50),(39,68,79,41),(40,69,80,42),(81,103,115,127),(82,104,116,128),(83,105,117,129),(84,106,118,130),(85,107,119,121),(86,108,120,122),(87,109,111,123),(88,110,112,124),(89,101,113,125),(90,102,114,126)], [(1,102),(2,103),(3,104),(4,105),(5,106),(6,107),(7,108),(8,109),(9,110),(10,101),(11,50),(12,41),(13,42),(14,43),(15,44),(16,45),(17,46),(18,47),(19,48),(20,49),(21,77),(22,78),(23,79),(24,80),(25,71),(26,72),(27,73),(28,74),(29,75),(30,76),(31,153),(32,154),(33,155),(34,156),(35,157),(36,158),(37,159),(38,160),(39,151),(40,152),(51,67),(52,68),(53,69),(54,70),(55,61),(56,62),(57,63),(58,64),(59,65),(60,66),(81,148),(82,149),(83,150),(84,141),(85,142),(86,143),(87,144),(88,145),(89,146),(90,147),(91,119),(92,120),(93,111),(94,112),(95,113),(96,114),(97,115),(98,116),(99,117),(100,118),(121,132),(122,133),(123,134),(124,135),(125,136),(126,137),(127,138),(128,139),(129,140),(130,131)], [(1,157,114,75),(2,158,115,76),(3,159,116,77),(4,160,117,78),(5,151,118,79),(6,152,119,80),(7,153,120,71),(8,154,111,72),(9,155,112,73),(10,156,113,74),(11,105,50,99),(12,106,41,100),(13,107,42,91),(14,108,43,92),(15,109,44,93),(16,110,45,94),(17,101,46,95),(18,102,47,96),(19,103,48,97),(20,104,49,98),(21,82,37,139),(22,83,38,140),(23,84,39,131),(24,85,40,132),(25,86,31,133),(26,87,32,134),(27,88,33,135),(28,89,34,136),(29,90,35,137),(30,81,36,138),(51,129,67,150),(52,130,68,141),(53,121,69,142),(54,122,70,143),(55,123,61,144),(56,124,62,145),(57,125,63,146),(58,126,64,147),(59,127,65,148),(60,128,66,149)])
140 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 5A | 5B | 5C | 5D | 8A | ··· | 8H | 10A | ··· | 10AB | 10AC | ··· | 10AR | 20A | ··· | 20P | 20Q | ··· | 20AF | 40A | ··· | 40AF |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 8 | ··· | 8 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 | 40 | ··· | 40 |
| size | 1 | 1 | ··· | 1 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 |
140 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | ||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | C5 | C10 | C10 | C10 | C10 | C20 | D4 | D4 | D8 | SD16 | C5×D4 | C5×D4 | C5×D8 | C5×SD16 |
| kernel | C10×D4⋊C4 | C5×D4⋊C4 | C10×C4⋊C4 | C22×C40 | D4×C2×C10 | D4×C10 | C2×D4⋊C4 | D4⋊C4 | C2×C4⋊C4 | C22×C8 | C22×D4 | C2×D4 | C2×C20 | C22×C10 | C2×C10 | C2×C10 | C2×C4 | C23 | C22 | C22 |
| # reps | 1 | 4 | 1 | 1 | 1 | 8 | 4 | 16 | 4 | 4 | 4 | 32 | 3 | 1 | 4 | 4 | 12 | 4 | 16 | 16 |
Matrix representation of C10×D4⋊C4 ►in GL4(𝔽41) generated by
| 40 | 0 | 0 | 0 |
| 0 | 25 | 0 | 0 |
| 0 | 0 | 25 | 0 |
| 0 | 0 | 0 | 25 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 39 |
| 0 | 0 | 1 | 40 |
| 40 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 39 |
| 0 | 0 | 0 | 40 |
| 1 | 0 | 0 | 0 |
| 0 | 32 | 0 | 0 |
| 0 | 0 | 0 | 17 |
| 0 | 0 | 29 | 0 |
G:=sub<GL(4,GF(41))| [40,0,0,0,0,25,0,0,0,0,25,0,0,0,0,25],[1,0,0,0,0,1,0,0,0,0,1,1,0,0,39,40],[40,0,0,0,0,1,0,0,0,0,1,0,0,0,39,40],[1,0,0,0,0,32,0,0,0,0,0,29,0,0,17,0] >;
C10×D4⋊C4 in GAP, Magma, Sage, TeX
C_{10}\times D_4\rtimes C_4 % in TeX
G:=Group("C10xD4:C4"); // GroupNames label
G:=SmallGroup(320,915);
// by ID
G=gap.SmallGroup(320,915);
# by ID
G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-2,560,589,7004,3511,172]);
// Polycyclic
G:=Group<a,b,c,d|a^10=b^4=c^2=d^4=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=d*b*d^-1=b^-1,d*c*d^-1=b*c>;
// generators/relations