metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C4⋊C4.8D10, (C8×Dic5)⋊20C2, D4⋊C4.1D5, (C2×D4).22D10, C20.6(C4○D4), C10.Q16⋊3C2, (C2×C8).200D10, C4.Dic10⋊4C2, C4.23(C4○D20), C10.38(C4○D8), C2.8(D8⋊3D5), D4⋊Dic5.6C2, C22.170(D4×D5), C4.49(D4⋊2D5), C20.44D4⋊17C2, (C2×C40).181C22, (C2×C20).208C23, (C2×Dic5).132D4, C20.17D4.6C2, (D4×C10).29C22, C4⋊Dic5.66C22, C2.9(SD16⋊3D5), C10.25(C4.4D4), C5⋊2(C42.78C22), (C2×Dic10).58C22, (C4×Dic5).254C22, C2.15(Dic5.5D4), (C2×C10).221(C2×D4), (C5×C4⋊C4).13C22, (C5×D4⋊C4).10C2, (C2×C4).315(C22×D5), (C2×C5⋊2C8).218C22, SmallGroup(320,395)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for (C8×Dic5)⋊C2
G = < a,b,c,d | a8=b10=d2=1, c2=b5, ab=ba, ac=ca, dad=a-1b5, cbc-1=b-1, bd=db, dcd=a4b5c >
Subgroups: 350 in 96 conjugacy classes, 37 normal (all characteristic)
C1, C2 [×3], C2, C4 [×2], C4 [×5], C22, C22 [×3], C5, C8 [×2], C2×C4, C2×C4 [×5], D4 [×2], Q8 [×2], C23, C10 [×3], C10, C42, C22⋊C4 [×2], C4⋊C4, C4⋊C4 [×3], C2×C8, C2×C8, C2×D4, C2×Q8, Dic5 [×4], C20 [×2], C20, C2×C10, C2×C10 [×3], C4×C8, D4⋊C4, D4⋊C4, Q8⋊C4 [×2], C4.4D4, C42.C2, C5⋊2C8, C40, Dic10 [×2], C2×Dic5 [×2], C2×Dic5 [×2], C2×C20, C2×C20, C5×D4 [×2], C22×C10, C42.78C22, C2×C5⋊2C8, C4×Dic5, C10.D4, C4⋊Dic5, C4⋊Dic5, C23.D5 [×2], C5×C4⋊C4, C2×C40, C2×Dic10, D4×C10, C10.Q16, C8×Dic5, C20.44D4, D4⋊Dic5, C5×D4⋊C4, C4.Dic10, C20.17D4, (C8×Dic5)⋊C2
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], C23, D5, C2×D4, C4○D4 [×2], D10 [×3], C4.4D4, C4○D8 [×2], C22×D5, C42.78C22, C4○D20, D4×D5, D4⋊2D5, Dic5.5D4, D8⋊3D5, SD16⋊3D5, (C8×Dic5)⋊C2
►On 160 points
Generators in S160
(1 60 35 85 27 67 42 75)(2 51 36 86 28 68 43 76)(3 52 37 87 29 69 44 77)(4 53 38 88 30 70 45 78)(5 54 39 89 21 61 46 79)(6 55 40 90 22 62 47 80)(7 56 31 81 23 63 48 71)(8 57 32 82 24 64 49 72)(9 58 33 83 25 65 50 73)(10 59 34 84 26 66 41 74)(11 103 143 128 156 91 131 116)(12 104 144 129 157 92 132 117)(13 105 145 130 158 93 133 118)(14 106 146 121 159 94 134 119)(15 107 147 122 160 95 135 120)(16 108 148 123 151 96 136 111)(17 109 149 124 152 97 137 112)(18 110 150 125 153 98 138 113)(19 101 141 126 154 99 139 114)(20 102 142 127 155 100 140 115)
(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 91 6 96)(2 100 7 95)(3 99 8 94)(4 98 9 93)(5 97 10 92)(11 90 16 85)(12 89 17 84)(13 88 18 83)(14 87 19 82)(15 86 20 81)(21 109 26 104)(22 108 27 103)(23 107 28 102)(24 106 29 101)(25 105 30 110)(31 120 36 115)(32 119 37 114)(33 118 38 113)(34 117 39 112)(35 116 40 111)(41 129 46 124)(42 128 47 123)(43 127 48 122)(44 126 49 121)(45 125 50 130)(51 140 56 135)(52 139 57 134)(53 138 58 133)(54 137 59 132)(55 136 60 131)(61 149 66 144)(62 148 67 143)(63 147 68 142)(64 146 69 141)(65 145 70 150)(71 160 76 155)(72 159 77 154)(73 158 78 153)(74 157 79 152)(75 156 80 151)
(11 131)(12 132)(13 133)(14 134)(15 135)(16 136)(17 137)(18 138)(19 139)(20 140)(31 48)(32 49)(33 50)(34 41)(35 42)(36 43)(37 44)(38 45)(39 46)(40 47)(51 71)(52 72)(53 73)(54 74)(55 75)(56 76)(57 77)(58 78)(59 79)(60 80)(61 84)(62 85)(63 86)(64 87)(65 88)(66 89)(67 90)(68 81)(69 82)(70 83)(91 108)(92 109)(93 110)(94 101)(95 102)(96 103)(97 104)(98 105)(99 106)(100 107)(111 116)(112 117)(113 118)(114 119)(115 120)(121 126)(122 127)(123 128)(124 129)(125 130)(141 154)(142 155)(143 156)(144 157)(145 158)(146 159)(147 160)(148 151)(149 152)(150 153)
G:=sub<Sym(160)| (1,60,35,85,27,67,42,75)(2,51,36,86,28,68,43,76)(3,52,37,87,29,69,44,77)(4,53,38,88,30,70,45,78)(5,54,39,89,21,61,46,79)(6,55,40,90,22,62,47,80)(7,56,31,81,23,63,48,71)(8,57,32,82,24,64,49,72)(9,58,33,83,25,65,50,73)(10,59,34,84,26,66,41,74)(11,103,143,128,156,91,131,116)(12,104,144,129,157,92,132,117)(13,105,145,130,158,93,133,118)(14,106,146,121,159,94,134,119)(15,107,147,122,160,95,135,120)(16,108,148,123,151,96,136,111)(17,109,149,124,152,97,137,112)(18,110,150,125,153,98,138,113)(19,101,141,126,154,99,139,114)(20,102,142,127,155,100,140,115), (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), (1,91,6,96)(2,100,7,95)(3,99,8,94)(4,98,9,93)(5,97,10,92)(11,90,16,85)(12,89,17,84)(13,88,18,83)(14,87,19,82)(15,86,20,81)(21,109,26,104)(22,108,27,103)(23,107,28,102)(24,106,29,101)(25,105,30,110)(31,120,36,115)(32,119,37,114)(33,118,38,113)(34,117,39,112)(35,116,40,111)(41,129,46,124)(42,128,47,123)(43,127,48,122)(44,126,49,121)(45,125,50,130)(51,140,56,135)(52,139,57,134)(53,138,58,133)(54,137,59,132)(55,136,60,131)(61,149,66,144)(62,148,67,143)(63,147,68,142)(64,146,69,141)(65,145,70,150)(71,160,76,155)(72,159,77,154)(73,158,78,153)(74,157,79,152)(75,156,80,151), (11,131)(12,132)(13,133)(14,134)(15,135)(16,136)(17,137)(18,138)(19,139)(20,140)(31,48)(32,49)(33,50)(34,41)(35,42)(36,43)(37,44)(38,45)(39,46)(40,47)(51,71)(52,72)(53,73)(54,74)(55,75)(56,76)(57,77)(58,78)(59,79)(60,80)(61,84)(62,85)(63,86)(64,87)(65,88)(66,89)(67,90)(68,81)(69,82)(70,83)(91,108)(92,109)(93,110)(94,101)(95,102)(96,103)(97,104)(98,105)(99,106)(100,107)(111,116)(112,117)(113,118)(114,119)(115,120)(121,126)(122,127)(123,128)(124,129)(125,130)(141,154)(142,155)(143,156)(144,157)(145,158)(146,159)(147,160)(148,151)(149,152)(150,153)>;
G:=Group( (1,60,35,85,27,67,42,75)(2,51,36,86,28,68,43,76)(3,52,37,87,29,69,44,77)(4,53,38,88,30,70,45,78)(5,54,39,89,21,61,46,79)(6,55,40,90,22,62,47,80)(7,56,31,81,23,63,48,71)(8,57,32,82,24,64,49,72)(9,58,33,83,25,65,50,73)(10,59,34,84,26,66,41,74)(11,103,143,128,156,91,131,116)(12,104,144,129,157,92,132,117)(13,105,145,130,158,93,133,118)(14,106,146,121,159,94,134,119)(15,107,147,122,160,95,135,120)(16,108,148,123,151,96,136,111)(17,109,149,124,152,97,137,112)(18,110,150,125,153,98,138,113)(19,101,141,126,154,99,139,114)(20,102,142,127,155,100,140,115), (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), (1,91,6,96)(2,100,7,95)(3,99,8,94)(4,98,9,93)(5,97,10,92)(11,90,16,85)(12,89,17,84)(13,88,18,83)(14,87,19,82)(15,86,20,81)(21,109,26,104)(22,108,27,103)(23,107,28,102)(24,106,29,101)(25,105,30,110)(31,120,36,115)(32,119,37,114)(33,118,38,113)(34,117,39,112)(35,116,40,111)(41,129,46,124)(42,128,47,123)(43,127,48,122)(44,126,49,121)(45,125,50,130)(51,140,56,135)(52,139,57,134)(53,138,58,133)(54,137,59,132)(55,136,60,131)(61,149,66,144)(62,148,67,143)(63,147,68,142)(64,146,69,141)(65,145,70,150)(71,160,76,155)(72,159,77,154)(73,158,78,153)(74,157,79,152)(75,156,80,151), (11,131)(12,132)(13,133)(14,134)(15,135)(16,136)(17,137)(18,138)(19,139)(20,140)(31,48)(32,49)(33,50)(34,41)(35,42)(36,43)(37,44)(38,45)(39,46)(40,47)(51,71)(52,72)(53,73)(54,74)(55,75)(56,76)(57,77)(58,78)(59,79)(60,80)(61,84)(62,85)(63,86)(64,87)(65,88)(66,89)(67,90)(68,81)(69,82)(70,83)(91,108)(92,109)(93,110)(94,101)(95,102)(96,103)(97,104)(98,105)(99,106)(100,107)(111,116)(112,117)(113,118)(114,119)(115,120)(121,126)(122,127)(123,128)(124,129)(125,130)(141,154)(142,155)(143,156)(144,157)(145,158)(146,159)(147,160)(148,151)(149,152)(150,153) );
G=PermutationGroup([(1,60,35,85,27,67,42,75),(2,51,36,86,28,68,43,76),(3,52,37,87,29,69,44,77),(4,53,38,88,30,70,45,78),(5,54,39,89,21,61,46,79),(6,55,40,90,22,62,47,80),(7,56,31,81,23,63,48,71),(8,57,32,82,24,64,49,72),(9,58,33,83,25,65,50,73),(10,59,34,84,26,66,41,74),(11,103,143,128,156,91,131,116),(12,104,144,129,157,92,132,117),(13,105,145,130,158,93,133,118),(14,106,146,121,159,94,134,119),(15,107,147,122,160,95,135,120),(16,108,148,123,151,96,136,111),(17,109,149,124,152,97,137,112),(18,110,150,125,153,98,138,113),(19,101,141,126,154,99,139,114),(20,102,142,127,155,100,140,115)], [(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)], [(1,91,6,96),(2,100,7,95),(3,99,8,94),(4,98,9,93),(5,97,10,92),(11,90,16,85),(12,89,17,84),(13,88,18,83),(14,87,19,82),(15,86,20,81),(21,109,26,104),(22,108,27,103),(23,107,28,102),(24,106,29,101),(25,105,30,110),(31,120,36,115),(32,119,37,114),(33,118,38,113),(34,117,39,112),(35,116,40,111),(41,129,46,124),(42,128,47,123),(43,127,48,122),(44,126,49,121),(45,125,50,130),(51,140,56,135),(52,139,57,134),(53,138,58,133),(54,137,59,132),(55,136,60,131),(61,149,66,144),(62,148,67,143),(63,147,68,142),(64,146,69,141),(65,145,70,150),(71,160,76,155),(72,159,77,154),(73,158,78,153),(74,157,79,152),(75,156,80,151)], [(11,131),(12,132),(13,133),(14,134),(15,135),(16,136),(17,137),(18,138),(19,139),(20,140),(31,48),(32,49),(33,50),(34,41),(35,42),(36,43),(37,44),(38,45),(39,46),(40,47),(51,71),(52,72),(53,73),(54,74),(55,75),(56,76),(57,77),(58,78),(59,79),(60,80),(61,84),(62,85),(63,86),(64,87),(65,88),(66,89),(67,90),(68,81),(69,82),(70,83),(91,108),(92,109),(93,110),(94,101),(95,102),(96,103),(97,104),(98,105),(99,106),(100,107),(111,116),(112,117),(113,118),(114,119),(115,120),(121,126),(122,127),(123,128),(124,129),(125,130),(141,154),(142,155),(143,156),(144,157),(145,158),(146,159),(147,160),(148,151),(149,152),(150,153)])
50 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 5A | 5B | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 20A | 20B | 20C | 20D | 20E | 20F | 20G | 20H | 40A | ··· | 40H |
| order | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 8 | 2 | 2 | 8 | 10 | 10 | 10 | 10 | 40 | 40 | 2 | 2 | 2 | 2 | 2 | 2 | 10 | 10 | 10 | 10 | 2 | ··· | 2 | 8 | 8 | 8 | 8 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 4 | ··· | 4 |
50 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | - | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D5 | C4○D4 | D10 | D10 | D10 | C4○D8 | C4○D20 | D4⋊2D5 | D4×D5 | D8⋊3D5 | SD16⋊3D5 |
| kernel | (C8×Dic5)⋊C2 | C10.Q16 | C8×Dic5 | C20.44D4 | D4⋊Dic5 | C5×D4⋊C4 | C4.Dic10 | C20.17D4 | C2×Dic5 | D4⋊C4 | C20 | C4⋊C4 | C2×C8 | C2×D4 | C10 | C4 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 2 | 2 | 2 | 8 | 8 | 2 | 2 | 4 | 4 |
Matrix representation of (C8×Dic5)⋊C2 ►in GL6(𝔽41)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 32 | 0 | 0 | 0 |
| 0 | 0 | 0 | 32 | 0 | 0 |
| 0 | 0 | 0 | 0 | 30 | 30 |
| 0 | 0 | 0 | 0 | 26 | 0 |
| 40 | 40 | 0 | 0 | 0 | 0 |
| 8 | 7 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 38 | 33 | 0 | 0 | 0 | 0 |
| 1 | 3 | 0 | 0 | 0 | 0 |
| 0 | 0 | 7 | 18 | 0 | 0 |
| 0 | 0 | 20 | 34 | 0 | 0 |
| 0 | 0 | 0 | 0 | 32 | 0 |
| 0 | 0 | 0 | 0 | 0 | 32 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 22 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 40 | 40 |
G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,32,0,0,0,0,0,0,32,0,0,0,0,0,0,30,26,0,0,0,0,30,0],[40,8,0,0,0,0,40,7,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[38,1,0,0,0,0,33,3,0,0,0,0,0,0,7,20,0,0,0,0,18,34,0,0,0,0,0,0,32,0,0,0,0,0,0,32],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,22,0,0,0,0,0,40,0,0,0,0,0,0,1,40,0,0,0,0,0,40] >;
(C8×Dic5)⋊C2 in GAP, Magma, Sage, TeX
(C_8\times {\rm Dic}_5)\rtimes C_2 % in TeX
G:=Group("(C8xDic5):C2"); // GroupNames label
G:=SmallGroup(320,395);
// by ID
G=gap.SmallGroup(320,395);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,701,1094,135,100,570,297,136,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^8=b^10=d^2=1,c^2=b^5,a*b=b*a,a*c=c*a,d*a*d=a^-1*b^5,c*b*c^-1=b^-1,b*d=d*b,d*c*d=a^4*b^5*c>;
// generators/relations