metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D4⋊1Dic10, Dic5.14D8, C20⋊Q8⋊1C2, (C5×D4)⋊1Q8, C2.6(D5×D8), C4⋊C4.1D10, C40⋊5C4⋊6C2, (C2×C8).6D10, C20.1(C2×Q8), C5⋊1(D4⋊Q8), C10.19(C2×D8), D4⋊C4.2D5, C10.D8⋊3C2, (C2×C40).6C22, C20.8Q8⋊4C2, (D4×Dic5).3C2, C4.1(C2×Dic10), (C2×D4).126D10, D4⋊Dic5.1C2, C22.161(D4×D5), C10.7(C22⋊Q8), C20.145(C4○D4), C4.74(D4⋊2D5), (C2×C20).199C23, (C2×Dic5).190D4, C2.8(SD16⋊D5), (D4×C10).20C22, C4⋊Dic5.59C22, C10.25(C8.C22), (C4×Dic5).10C22, C2.12(Dic5.14D4), (C5×C4⋊C4).4C22, (C5×D4⋊C4).2C2, (C2×C10).212(C2×D4), (C2×C5⋊2C8).6C22, (C2×C4).306(C22×D5), SmallGroup(320,386)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic5.14D8
G = < a,b,c,d | a10=c8=d2=1, b2=a5, bab-1=a-1, ac=ca, ad=da, cbc-1=a5b, bd=db, dcd=a5c-1 >
Subgroups: 422 in 108 conjugacy classes, 43 normal (37 characteristic)
C1, C2 [×3], C2 [×2], C4 [×2], C4 [×6], C22, C22 [×4], C5, C8 [×2], C2×C4, C2×C4 [×7], D4 [×2], D4, Q8 [×2], C23, C10 [×3], C10 [×2], C42, C22⋊C4, C4⋊C4, C4⋊C4 [×3], C2×C8, C2×C8, C22×C4, C2×D4, C2×Q8, Dic5 [×2], Dic5 [×3], C20 [×2], C20, C2×C10, C2×C10 [×4], D4⋊C4, D4⋊C4, C4⋊C8, C2.D8 [×2], C4×D4, C4⋊Q8, C5⋊2C8, C40, Dic10 [×2], C2×Dic5 [×2], C2×Dic5 [×4], C2×C20, C2×C20, C5×D4 [×2], C5×D4, C22×C10, D4⋊Q8, C2×C5⋊2C8, C4×Dic5, C10.D4, C4⋊Dic5 [×2], C23.D5, C5×C4⋊C4, C2×C40, C2×Dic10, C22×Dic5, D4×C10, C10.D8, C20.8Q8, C40⋊5C4, D4⋊Dic5, C5×D4⋊C4, C20⋊Q8, D4×Dic5, Dic5.14D8
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], Q8 [×2], C23, D5, D8 [×2], C2×D4, C2×Q8, C4○D4, D10 [×3], C22⋊Q8, C2×D8, C8.C22, Dic10 [×2], C22×D5, D4⋊Q8, C2×Dic10, D4×D5, D4⋊2D5, Dic5.14D4, D5×D8, SD16⋊D5, Dic5.14D8
►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)
(1 116 6 111)(2 115 7 120)(3 114 8 119)(4 113 9 118)(5 112 10 117)(11 129 16 124)(12 128 17 123)(13 127 18 122)(14 126 19 121)(15 125 20 130)(21 69 26 64)(22 68 27 63)(23 67 28 62)(24 66 29 61)(25 65 30 70)(31 100 36 95)(32 99 37 94)(33 98 38 93)(34 97 39 92)(35 96 40 91)(41 108 46 103)(42 107 47 102)(43 106 48 101)(44 105 49 110)(45 104 50 109)(51 153 56 158)(52 152 57 157)(53 151 58 156)(54 160 59 155)(55 159 60 154)(71 140 76 135)(72 139 77 134)(73 138 78 133)(74 137 79 132)(75 136 80 131)(81 145 86 150)(82 144 87 149)(83 143 88 148)(84 142 89 147)(85 141 90 146)
(1 53 13 85 39 74 43 64)(2 54 14 86 40 75 44 65)(3 55 15 87 31 76 45 66)(4 56 16 88 32 77 46 67)(5 57 17 89 33 78 47 68)(6 58 18 90 34 79 48 69)(7 59 19 81 35 80 49 70)(8 60 20 82 36 71 50 61)(9 51 11 83 37 72 41 62)(10 52 12 84 38 73 42 63)(21 111 151 122 141 97 137 101)(22 112 152 123 142 98 138 102)(23 113 153 124 143 99 139 103)(24 114 154 125 144 100 140 104)(25 115 155 126 145 91 131 105)(26 116 156 127 146 92 132 106)(27 117 157 128 147 93 133 107)(28 118 158 129 148 94 134 108)(29 119 159 130 149 95 135 109)(30 120 160 121 150 96 136 110)
(11 41)(12 42)(13 43)(14 44)(15 45)(16 46)(17 47)(18 48)(19 49)(20 50)(21 156)(22 157)(23 158)(24 159)(25 160)(26 151)(27 152)(28 153)(29 154)(30 155)(51 67)(52 68)(53 69)(54 70)(55 61)(56 62)(57 63)(58 64)(59 65)(60 66)(71 87)(72 88)(73 89)(74 90)(75 81)(76 82)(77 83)(78 84)(79 85)(80 86)(101 122)(102 123)(103 124)(104 125)(105 126)(106 127)(107 128)(108 129)(109 130)(110 121)(131 150)(132 141)(133 142)(134 143)(135 144)(136 145)(137 146)(138 147)(139 148)(140 149)
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), (1,116,6,111)(2,115,7,120)(3,114,8,119)(4,113,9,118)(5,112,10,117)(11,129,16,124)(12,128,17,123)(13,127,18,122)(14,126,19,121)(15,125,20,130)(21,69,26,64)(22,68,27,63)(23,67,28,62)(24,66,29,61)(25,65,30,70)(31,100,36,95)(32,99,37,94)(33,98,38,93)(34,97,39,92)(35,96,40,91)(41,108,46,103)(42,107,47,102)(43,106,48,101)(44,105,49,110)(45,104,50,109)(51,153,56,158)(52,152,57,157)(53,151,58,156)(54,160,59,155)(55,159,60,154)(71,140,76,135)(72,139,77,134)(73,138,78,133)(74,137,79,132)(75,136,80,131)(81,145,86,150)(82,144,87,149)(83,143,88,148)(84,142,89,147)(85,141,90,146), (1,53,13,85,39,74,43,64)(2,54,14,86,40,75,44,65)(3,55,15,87,31,76,45,66)(4,56,16,88,32,77,46,67)(5,57,17,89,33,78,47,68)(6,58,18,90,34,79,48,69)(7,59,19,81,35,80,49,70)(8,60,20,82,36,71,50,61)(9,51,11,83,37,72,41,62)(10,52,12,84,38,73,42,63)(21,111,151,122,141,97,137,101)(22,112,152,123,142,98,138,102)(23,113,153,124,143,99,139,103)(24,114,154,125,144,100,140,104)(25,115,155,126,145,91,131,105)(26,116,156,127,146,92,132,106)(27,117,157,128,147,93,133,107)(28,118,158,129,148,94,134,108)(29,119,159,130,149,95,135,109)(30,120,160,121,150,96,136,110), (11,41)(12,42)(13,43)(14,44)(15,45)(16,46)(17,47)(18,48)(19,49)(20,50)(21,156)(22,157)(23,158)(24,159)(25,160)(26,151)(27,152)(28,153)(29,154)(30,155)(51,67)(52,68)(53,69)(54,70)(55,61)(56,62)(57,63)(58,64)(59,65)(60,66)(71,87)(72,88)(73,89)(74,90)(75,81)(76,82)(77,83)(78,84)(79,85)(80,86)(101,122)(102,123)(103,124)(104,125)(105,126)(106,127)(107,128)(108,129)(109,130)(110,121)(131,150)(132,141)(133,142)(134,143)(135,144)(136,145)(137,146)(138,147)(139,148)(140,149)>;
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), (1,116,6,111)(2,115,7,120)(3,114,8,119)(4,113,9,118)(5,112,10,117)(11,129,16,124)(12,128,17,123)(13,127,18,122)(14,126,19,121)(15,125,20,130)(21,69,26,64)(22,68,27,63)(23,67,28,62)(24,66,29,61)(25,65,30,70)(31,100,36,95)(32,99,37,94)(33,98,38,93)(34,97,39,92)(35,96,40,91)(41,108,46,103)(42,107,47,102)(43,106,48,101)(44,105,49,110)(45,104,50,109)(51,153,56,158)(52,152,57,157)(53,151,58,156)(54,160,59,155)(55,159,60,154)(71,140,76,135)(72,139,77,134)(73,138,78,133)(74,137,79,132)(75,136,80,131)(81,145,86,150)(82,144,87,149)(83,143,88,148)(84,142,89,147)(85,141,90,146), (1,53,13,85,39,74,43,64)(2,54,14,86,40,75,44,65)(3,55,15,87,31,76,45,66)(4,56,16,88,32,77,46,67)(5,57,17,89,33,78,47,68)(6,58,18,90,34,79,48,69)(7,59,19,81,35,80,49,70)(8,60,20,82,36,71,50,61)(9,51,11,83,37,72,41,62)(10,52,12,84,38,73,42,63)(21,111,151,122,141,97,137,101)(22,112,152,123,142,98,138,102)(23,113,153,124,143,99,139,103)(24,114,154,125,144,100,140,104)(25,115,155,126,145,91,131,105)(26,116,156,127,146,92,132,106)(27,117,157,128,147,93,133,107)(28,118,158,129,148,94,134,108)(29,119,159,130,149,95,135,109)(30,120,160,121,150,96,136,110), (11,41)(12,42)(13,43)(14,44)(15,45)(16,46)(17,47)(18,48)(19,49)(20,50)(21,156)(22,157)(23,158)(24,159)(25,160)(26,151)(27,152)(28,153)(29,154)(30,155)(51,67)(52,68)(53,69)(54,70)(55,61)(56,62)(57,63)(58,64)(59,65)(60,66)(71,87)(72,88)(73,89)(74,90)(75,81)(76,82)(77,83)(78,84)(79,85)(80,86)(101,122)(102,123)(103,124)(104,125)(105,126)(106,127)(107,128)(108,129)(109,130)(110,121)(131,150)(132,141)(133,142)(134,143)(135,144)(136,145)(137,146)(138,147)(139,148)(140,149) );
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)], [(1,116,6,111),(2,115,7,120),(3,114,8,119),(4,113,9,118),(5,112,10,117),(11,129,16,124),(12,128,17,123),(13,127,18,122),(14,126,19,121),(15,125,20,130),(21,69,26,64),(22,68,27,63),(23,67,28,62),(24,66,29,61),(25,65,30,70),(31,100,36,95),(32,99,37,94),(33,98,38,93),(34,97,39,92),(35,96,40,91),(41,108,46,103),(42,107,47,102),(43,106,48,101),(44,105,49,110),(45,104,50,109),(51,153,56,158),(52,152,57,157),(53,151,58,156),(54,160,59,155),(55,159,60,154),(71,140,76,135),(72,139,77,134),(73,138,78,133),(74,137,79,132),(75,136,80,131),(81,145,86,150),(82,144,87,149),(83,143,88,148),(84,142,89,147),(85,141,90,146)], [(1,53,13,85,39,74,43,64),(2,54,14,86,40,75,44,65),(3,55,15,87,31,76,45,66),(4,56,16,88,32,77,46,67),(5,57,17,89,33,78,47,68),(6,58,18,90,34,79,48,69),(7,59,19,81,35,80,49,70),(8,60,20,82,36,71,50,61),(9,51,11,83,37,72,41,62),(10,52,12,84,38,73,42,63),(21,111,151,122,141,97,137,101),(22,112,152,123,142,98,138,102),(23,113,153,124,143,99,139,103),(24,114,154,125,144,100,140,104),(25,115,155,126,145,91,131,105),(26,116,156,127,146,92,132,106),(27,117,157,128,147,93,133,107),(28,118,158,129,148,94,134,108),(29,119,159,130,149,95,135,109),(30,120,160,121,150,96,136,110)], [(11,41),(12,42),(13,43),(14,44),(15,45),(16,46),(17,47),(18,48),(19,49),(20,50),(21,156),(22,157),(23,158),(24,159),(25,160),(26,151),(27,152),(28,153),(29,154),(30,155),(51,67),(52,68),(53,69),(54,70),(55,61),(56,62),(57,63),(58,64),(59,65),(60,66),(71,87),(72,88),(73,89),(74,90),(75,81),(76,82),(77,83),(78,84),(79,85),(80,86),(101,122),(102,123),(103,124),(104,125),(105,126),(106,127),(107,128),(108,129),(109,130),(110,121),(131,150),(132,141),(133,142),(134,143),(135,144),(136,145),(137,146),(138,147),(139,148),(140,149)])
47 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 5A | 5B | 8A | 8B | 8C | 8D | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 20A | 20B | 20C | 20D | 20E | 20F | 20G | 20H | 40A | ··· | 40H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 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 | 4 | 4 | 2 | 2 | 8 | 10 | 10 | 20 | 20 | 20 | 40 | 2 | 2 | 4 | 4 | 20 | 20 | 2 | ··· | 2 | 8 | 8 | 8 | 8 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 4 | ··· | 4 |
47 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | - | + | + | + | + | + | - | - | - | + | + | - | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | Q8 | D5 | D8 | C4○D4 | D10 | D10 | D10 | Dic10 | C8.C22 | D4⋊2D5 | D4×D5 | D5×D8 | SD16⋊D5 |
| kernel | Dic5.14D8 | C10.D8 | C20.8Q8 | C40⋊5C4 | D4⋊Dic5 | C5×D4⋊C4 | C20⋊Q8 | D4×Dic5 | C2×Dic5 | C5×D4 | D4⋊C4 | Dic5 | C20 | C4⋊C4 | C2×C8 | C2×D4 | D4 | C10 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 2 | 2 | 2 | 2 | 8 | 1 | 2 | 2 | 4 | 4 |
Matrix representation of Dic5.14D8 ►in GL4(𝔽41) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 40 |
| 0 | 0 | 36 | 6 |
| 40 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 23 | 17 |
| 0 | 0 | 5 | 18 |
| 24 | 24 | 0 | 0 |
| 29 | 0 | 0 | 0 |
| 0 | 0 | 39 | 32 |
| 0 | 0 | 37 | 2 |
| 1 | 0 | 0 | 0 |
| 40 | 40 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(41))| [1,0,0,0,0,1,0,0,0,0,1,36,0,0,40,6],[40,0,0,0,0,40,0,0,0,0,23,5,0,0,17,18],[24,29,0,0,24,0,0,0,0,0,39,37,0,0,32,2],[1,40,0,0,0,40,0,0,0,0,1,0,0,0,0,1] >;
Dic5.14D8 in GAP, Magma, Sage, TeX
{\rm Dic}_5._{14}D_8 % in TeX
G:=Group("Dic5.14D8"); // GroupNames label
G:=SmallGroup(320,386);
// by ID
G=gap.SmallGroup(320,386);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,254,219,58,851,438,102,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^10=c^8=d^2=1,b^2=a^5,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,c*b*c^-1=a^5*b,b*d=d*b,d*c*d=a^5*c^-1>;
// generators/relations