direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C10×C4⋊1D4, C4⋊1(D4×C10), C20⋊12(C2×D4), (C2×C20)⋊33D4, C42⋊18(C2×C10), (C2×C42)⋊11C10, (C4×C20)⋊59C22, (C22×D4)⋊5C10, (D4×C10)⋊63C22, C24.15(C2×C10), C22.63(D4×C10), (C2×C10).350C24, (C2×C20).961C23, C10.186(C22×D4), C23.8(C22×C10), (C23×C10).15C22, (C22×C10).88C23, C22.24(C23×C10), (C22×C20).596C22, (C2×C4×C20)⋊24C2, (C2×C4)⋊7(C5×D4), (D4×C2×C10)⋊20C2, C2.10(D4×C2×C10), (C2×D4)⋊11(C2×C10), (C2×C10).684(C2×D4), (C22×C4).124(C2×C10), (C2×C4).136(C22×C10), SmallGroup(320,1532)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 882 in 498 conjugacy classes, 210 normal (10 characteristic)
C1, C2 [×7], C2 [×8], C4 [×12], C22, C22 [×6], C22 [×40], C5, C2×C4 [×18], D4 [×48], C23, C23 [×8], C23 [×24], C10 [×7], C10 [×8], C42 [×4], C22×C4 [×3], C2×D4 [×24], C2×D4 [×24], C24 [×4], C20 [×12], C2×C10, C2×C10 [×6], C2×C10 [×40], C2×C42, C4⋊1D4 [×8], C22×D4 [×6], C2×C20 [×18], C5×D4 [×48], C22×C10, C22×C10 [×8], C22×C10 [×24], C2×C4⋊1D4, C4×C20 [×4], C22×C20 [×3], D4×C10 [×24], D4×C10 [×24], C23×C10 [×4], C2×C4×C20, C5×C4⋊1D4 [×8], D4×C2×C10 [×6], C10×C4⋊1D4
Quotients:
C1, C2 [×15], C22 [×35], C5, D4 [×12], C23 [×15], C10 [×15], C2×D4 [×18], C24, C2×C10 [×35], C4⋊1D4 [×4], C22×D4 [×3], C5×D4 [×12], C22×C10 [×15], C2×C4⋊1D4, D4×C10 [×18], C23×C10, C5×C4⋊1D4 [×4], D4×C2×C10 [×3], C10×C4⋊1D4
Generators and relations
G = < a,b,c,d | a10=b4=c4=d2=1, ab=ba, ac=ca, ad=da, bc=cb, dbd=b-1, dcd=c-1 >
►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 16 46 154)(2 17 47 155)(3 18 48 156)(4 19 49 157)(5 20 50 158)(6 11 41 159)(7 12 42 160)(8 13 43 151)(9 14 44 152)(10 15 45 153)(21 68 40 57)(22 69 31 58)(23 70 32 59)(24 61 33 60)(25 62 34 51)(26 63 35 52)(27 64 36 53)(28 65 37 54)(29 66 38 55)(30 67 39 56)(71 150 110 111)(72 141 101 112)(73 142 102 113)(74 143 103 114)(75 144 104 115)(76 145 105 116)(77 146 106 117)(78 147 107 118)(79 148 108 119)(80 149 109 120)(81 134 94 121)(82 135 95 122)(83 136 96 123)(84 137 97 124)(85 138 98 125)(86 139 99 126)(87 140 100 127)(88 131 91 128)(89 132 92 129)(90 133 93 130)
(1 108 52 96)(2 109 53 97)(3 110 54 98)(4 101 55 99)(5 102 56 100)(6 103 57 91)(7 104 58 92)(8 105 59 93)(9 106 60 94)(10 107 51 95)(11 114 21 128)(12 115 22 129)(13 116 23 130)(14 117 24 121)(15 118 25 122)(16 119 26 123)(17 120 27 124)(18 111 28 125)(19 112 29 126)(20 113 30 127)(31 132 160 144)(32 133 151 145)(33 134 152 146)(34 135 153 147)(35 136 154 148)(36 137 155 149)(37 138 156 150)(38 139 157 141)(39 140 158 142)(40 131 159 143)(41 74 68 88)(42 75 69 89)(43 76 70 90)(44 77 61 81)(45 78 62 82)(46 79 63 83)(47 80 64 84)(48 71 65 85)(49 72 66 86)(50 73 67 87)
(1 114)(2 115)(3 116)(4 117)(5 118)(6 119)(7 120)(8 111)(9 112)(10 113)(11 108)(12 109)(13 110)(14 101)(15 102)(16 103)(17 104)(18 105)(19 106)(20 107)(21 96)(22 97)(23 98)(24 99)(25 100)(26 91)(27 92)(28 93)(29 94)(30 95)(31 84)(32 85)(33 86)(34 87)(35 88)(36 89)(37 90)(38 81)(39 82)(40 83)(41 148)(42 149)(43 150)(44 141)(45 142)(46 143)(47 144)(48 145)(49 146)(50 147)(51 127)(52 128)(53 129)(54 130)(55 121)(56 122)(57 123)(58 124)(59 125)(60 126)(61 139)(62 140)(63 131)(64 132)(65 133)(66 134)(67 135)(68 136)(69 137)(70 138)(71 151)(72 152)(73 153)(74 154)(75 155)(76 156)(77 157)(78 158)(79 159)(80 160)
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), (1,16,46,154)(2,17,47,155)(3,18,48,156)(4,19,49,157)(5,20,50,158)(6,11,41,159)(7,12,42,160)(8,13,43,151)(9,14,44,152)(10,15,45,153)(21,68,40,57)(22,69,31,58)(23,70,32,59)(24,61,33,60)(25,62,34,51)(26,63,35,52)(27,64,36,53)(28,65,37,54)(29,66,38,55)(30,67,39,56)(71,150,110,111)(72,141,101,112)(73,142,102,113)(74,143,103,114)(75,144,104,115)(76,145,105,116)(77,146,106,117)(78,147,107,118)(79,148,108,119)(80,149,109,120)(81,134,94,121)(82,135,95,122)(83,136,96,123)(84,137,97,124)(85,138,98,125)(86,139,99,126)(87,140,100,127)(88,131,91,128)(89,132,92,129)(90,133,93,130), (1,108,52,96)(2,109,53,97)(3,110,54,98)(4,101,55,99)(5,102,56,100)(6,103,57,91)(7,104,58,92)(8,105,59,93)(9,106,60,94)(10,107,51,95)(11,114,21,128)(12,115,22,129)(13,116,23,130)(14,117,24,121)(15,118,25,122)(16,119,26,123)(17,120,27,124)(18,111,28,125)(19,112,29,126)(20,113,30,127)(31,132,160,144)(32,133,151,145)(33,134,152,146)(34,135,153,147)(35,136,154,148)(36,137,155,149)(37,138,156,150)(38,139,157,141)(39,140,158,142)(40,131,159,143)(41,74,68,88)(42,75,69,89)(43,76,70,90)(44,77,61,81)(45,78,62,82)(46,79,63,83)(47,80,64,84)(48,71,65,85)(49,72,66,86)(50,73,67,87), (1,114)(2,115)(3,116)(4,117)(5,118)(6,119)(7,120)(8,111)(9,112)(10,113)(11,108)(12,109)(13,110)(14,101)(15,102)(16,103)(17,104)(18,105)(19,106)(20,107)(21,96)(22,97)(23,98)(24,99)(25,100)(26,91)(27,92)(28,93)(29,94)(30,95)(31,84)(32,85)(33,86)(34,87)(35,88)(36,89)(37,90)(38,81)(39,82)(40,83)(41,148)(42,149)(43,150)(44,141)(45,142)(46,143)(47,144)(48,145)(49,146)(50,147)(51,127)(52,128)(53,129)(54,130)(55,121)(56,122)(57,123)(58,124)(59,125)(60,126)(61,139)(62,140)(63,131)(64,132)(65,133)(66,134)(67,135)(68,136)(69,137)(70,138)(71,151)(72,152)(73,153)(74,154)(75,155)(76,156)(77,157)(78,158)(79,159)(80,160)>;
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), (1,16,46,154)(2,17,47,155)(3,18,48,156)(4,19,49,157)(5,20,50,158)(6,11,41,159)(7,12,42,160)(8,13,43,151)(9,14,44,152)(10,15,45,153)(21,68,40,57)(22,69,31,58)(23,70,32,59)(24,61,33,60)(25,62,34,51)(26,63,35,52)(27,64,36,53)(28,65,37,54)(29,66,38,55)(30,67,39,56)(71,150,110,111)(72,141,101,112)(73,142,102,113)(74,143,103,114)(75,144,104,115)(76,145,105,116)(77,146,106,117)(78,147,107,118)(79,148,108,119)(80,149,109,120)(81,134,94,121)(82,135,95,122)(83,136,96,123)(84,137,97,124)(85,138,98,125)(86,139,99,126)(87,140,100,127)(88,131,91,128)(89,132,92,129)(90,133,93,130), (1,108,52,96)(2,109,53,97)(3,110,54,98)(4,101,55,99)(5,102,56,100)(6,103,57,91)(7,104,58,92)(8,105,59,93)(9,106,60,94)(10,107,51,95)(11,114,21,128)(12,115,22,129)(13,116,23,130)(14,117,24,121)(15,118,25,122)(16,119,26,123)(17,120,27,124)(18,111,28,125)(19,112,29,126)(20,113,30,127)(31,132,160,144)(32,133,151,145)(33,134,152,146)(34,135,153,147)(35,136,154,148)(36,137,155,149)(37,138,156,150)(38,139,157,141)(39,140,158,142)(40,131,159,143)(41,74,68,88)(42,75,69,89)(43,76,70,90)(44,77,61,81)(45,78,62,82)(46,79,63,83)(47,80,64,84)(48,71,65,85)(49,72,66,86)(50,73,67,87), (1,114)(2,115)(3,116)(4,117)(5,118)(6,119)(7,120)(8,111)(9,112)(10,113)(11,108)(12,109)(13,110)(14,101)(15,102)(16,103)(17,104)(18,105)(19,106)(20,107)(21,96)(22,97)(23,98)(24,99)(25,100)(26,91)(27,92)(28,93)(29,94)(30,95)(31,84)(32,85)(33,86)(34,87)(35,88)(36,89)(37,90)(38,81)(39,82)(40,83)(41,148)(42,149)(43,150)(44,141)(45,142)(46,143)(47,144)(48,145)(49,146)(50,147)(51,127)(52,128)(53,129)(54,130)(55,121)(56,122)(57,123)(58,124)(59,125)(60,126)(61,139)(62,140)(63,131)(64,132)(65,133)(66,134)(67,135)(68,136)(69,137)(70,138)(71,151)(72,152)(73,153)(74,154)(75,155)(76,156)(77,157)(78,158)(79,159)(80,160) );
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)], [(1,16,46,154),(2,17,47,155),(3,18,48,156),(4,19,49,157),(5,20,50,158),(6,11,41,159),(7,12,42,160),(8,13,43,151),(9,14,44,152),(10,15,45,153),(21,68,40,57),(22,69,31,58),(23,70,32,59),(24,61,33,60),(25,62,34,51),(26,63,35,52),(27,64,36,53),(28,65,37,54),(29,66,38,55),(30,67,39,56),(71,150,110,111),(72,141,101,112),(73,142,102,113),(74,143,103,114),(75,144,104,115),(76,145,105,116),(77,146,106,117),(78,147,107,118),(79,148,108,119),(80,149,109,120),(81,134,94,121),(82,135,95,122),(83,136,96,123),(84,137,97,124),(85,138,98,125),(86,139,99,126),(87,140,100,127),(88,131,91,128),(89,132,92,129),(90,133,93,130)], [(1,108,52,96),(2,109,53,97),(3,110,54,98),(4,101,55,99),(5,102,56,100),(6,103,57,91),(7,104,58,92),(8,105,59,93),(9,106,60,94),(10,107,51,95),(11,114,21,128),(12,115,22,129),(13,116,23,130),(14,117,24,121),(15,118,25,122),(16,119,26,123),(17,120,27,124),(18,111,28,125),(19,112,29,126),(20,113,30,127),(31,132,160,144),(32,133,151,145),(33,134,152,146),(34,135,153,147),(35,136,154,148),(36,137,155,149),(37,138,156,150),(38,139,157,141),(39,140,158,142),(40,131,159,143),(41,74,68,88),(42,75,69,89),(43,76,70,90),(44,77,61,81),(45,78,62,82),(46,79,63,83),(47,80,64,84),(48,71,65,85),(49,72,66,86),(50,73,67,87)], [(1,114),(2,115),(3,116),(4,117),(5,118),(6,119),(7,120),(8,111),(9,112),(10,113),(11,108),(12,109),(13,110),(14,101),(15,102),(16,103),(17,104),(18,105),(19,106),(20,107),(21,96),(22,97),(23,98),(24,99),(25,100),(26,91),(27,92),(28,93),(29,94),(30,95),(31,84),(32,85),(33,86),(34,87),(35,88),(36,89),(37,90),(38,81),(39,82),(40,83),(41,148),(42,149),(43,150),(44,141),(45,142),(46,143),(47,144),(48,145),(49,146),(50,147),(51,127),(52,128),(53,129),(54,130),(55,121),(56,122),(57,123),(58,124),(59,125),(60,126),(61,139),(62,140),(63,131),(64,132),(65,133),(66,134),(67,135),(68,136),(69,137),(70,138),(71,151),(72,152),(73,153),(74,154),(75,155),(76,156),(77,157),(78,158),(79,159),(80,160)])
Matrix representation ►G ⊆ GL5(𝔽41)
| 40 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 37 | 0 |
| 0 | 0 | 0 | 0 | 37 |
| 40 | 0 | 0 | 0 | 0 |
| 0 | 40 | 2 | 0 | 0 |
| 0 | 40 | 1 | 0 | 0 |
| 0 | 0 | 0 | 34 | 2 |
| 0 | 0 | 0 | 16 | 7 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 34 | 2 |
| 0 | 0 | 0 | 16 | 7 |
| 40 | 0 | 0 | 0 | 0 |
| 0 | 1 | 39 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 34 | 1 |
G:=sub<GL(5,GF(41))| [40,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,37,0,0,0,0,0,37],[40,0,0,0,0,0,40,40,0,0,0,2,1,0,0,0,0,0,34,16,0,0,0,2,7],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,34,16,0,0,0,2,7],[40,0,0,0,0,0,1,0,0,0,0,39,40,0,0,0,0,0,40,34,0,0,0,0,1] >;
140 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | ··· | 2O | 4A | ··· | 4L | 5A | 5B | 5C | 5D | 10A | ··· | 10AB | 10AC | ··· | 10BH | 20A | ··· | 20AV |
| order | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 5 | 5 | 5 | 5 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | ··· | 20 |
| size | 1 | 1 | ··· | 1 | 4 | ··· | 4 | 2 | ··· | 2 | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 4 | ··· | 4 | 2 | ··· | 2 |
140 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | + | + | |||||
| image | C1 | C2 | C2 | C2 | C5 | C10 | C10 | C10 | D4 | C5×D4 |
| kernel | C10×C4⋊1D4 | C2×C4×C20 | C5×C4⋊1D4 | D4×C2×C10 | C2×C4⋊1D4 | C2×C42 | C4⋊1D4 | C22×D4 | C2×C20 | C2×C4 |
| # reps | 1 | 1 | 8 | 6 | 4 | 4 | 32 | 24 | 12 | 48 |
In GAP, Magma, Sage, TeX
C_{10}\times C_4\rtimes_1D_4 % in TeX
G:=Group("C10xC4:1D4"); // GroupNames label
G:=SmallGroup(320,1532);
// by ID
G=gap.SmallGroup(320,1532);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-5,-2,-2,1149,568,3446,856]);
// Polycyclic
G:=Group<a,b,c,d|a^10=b^4=c^4=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀