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, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, D4, Q8, C23, C10, C10, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, C22×C4, C2×D4, C2×Q8, Dic5, Dic5, C20, C20, C2×C10, C2×C10, D4⋊C4, D4⋊C4, C4⋊C8, C2.D8, C4×D4, C4⋊Q8, C5⋊2C8, C40, Dic10, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C5×D4, C5×D4, C22×C10, D4⋊Q8, C2×C5⋊2C8, C4×Dic5, C10.D4, C4⋊Dic5, 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, C22, D4, Q8, C23, D5, D8, C2×D4, C2×Q8, C4○D4, D10, C22⋊Q8, C2×D8, C8.C22, Dic10, 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)(11 130 16 125)(12 129 17 124)(13 128 18 123)(14 127 19 122)(15 126 20 121)(21 63 26 68)(22 62 27 67)(23 61 28 66)(24 70 29 65)(25 69 30 64)(31 98 36 93)(32 97 37 92)(33 96 38 91)(34 95 39 100)(35 94 40 99)(41 104 46 109)(42 103 47 108)(43 102 48 107)(44 101 49 106)(45 110 50 105)(51 157 56 152)(52 156 57 151)(53 155 58 160)(54 154 59 159)(55 153 60 158)(71 138 76 133)(72 137 77 132)(73 136 78 131)(74 135 79 140)(75 134 80 139)(81 141 86 146)(82 150 87 145)(83 149 88 144)(84 148 89 143)(85 147 90 142)
(1 57 14 85 39 74 43 68)(2 58 15 86 40 75 44 69)(3 59 16 87 31 76 45 70)(4 60 17 88 32 77 46 61)(5 51 18 89 33 78 47 62)(6 52 19 90 34 79 48 63)(7 53 20 81 35 80 49 64)(8 54 11 82 36 71 50 65)(9 55 12 83 37 72 41 66)(10 56 13 84 38 73 42 67)(21 111 151 122 147 95 135 107)(22 112 152 123 148 96 136 108)(23 113 153 124 149 97 137 109)(24 114 154 125 150 98 138 110)(25 115 155 126 141 99 139 101)(26 116 156 127 142 100 140 102)(27 117 157 128 143 91 131 103)(28 118 158 129 144 92 132 104)(29 119 159 130 145 93 133 105)(30 120 160 121 146 94 134 106)
(11 50)(12 41)(13 42)(14 43)(15 44)(16 45)(17 46)(18 47)(19 48)(20 49)(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)(101 126)(102 127)(103 128)(104 129)(105 130)(106 121)(107 122)(108 123)(109 124)(110 125)(131 148)(132 149)(133 150)(134 141)(135 142)(136 143)(137 144)(138 145)(139 146)(140 147)
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)(11,130,16,125)(12,129,17,124)(13,128,18,123)(14,127,19,122)(15,126,20,121)(21,63,26,68)(22,62,27,67)(23,61,28,66)(24,70,29,65)(25,69,30,64)(31,98,36,93)(32,97,37,92)(33,96,38,91)(34,95,39,100)(35,94,40,99)(41,104,46,109)(42,103,47,108)(43,102,48,107)(44,101,49,106)(45,110,50,105)(51,157,56,152)(52,156,57,151)(53,155,58,160)(54,154,59,159)(55,153,60,158)(71,138,76,133)(72,137,77,132)(73,136,78,131)(74,135,79,140)(75,134,80,139)(81,141,86,146)(82,150,87,145)(83,149,88,144)(84,148,89,143)(85,147,90,142), (1,57,14,85,39,74,43,68)(2,58,15,86,40,75,44,69)(3,59,16,87,31,76,45,70)(4,60,17,88,32,77,46,61)(5,51,18,89,33,78,47,62)(6,52,19,90,34,79,48,63)(7,53,20,81,35,80,49,64)(8,54,11,82,36,71,50,65)(9,55,12,83,37,72,41,66)(10,56,13,84,38,73,42,67)(21,111,151,122,147,95,135,107)(22,112,152,123,148,96,136,108)(23,113,153,124,149,97,137,109)(24,114,154,125,150,98,138,110)(25,115,155,126,141,99,139,101)(26,116,156,127,142,100,140,102)(27,117,157,128,143,91,131,103)(28,118,158,129,144,92,132,104)(29,119,159,130,145,93,133,105)(30,120,160,121,146,94,134,106), (11,50)(12,41)(13,42)(14,43)(15,44)(16,45)(17,46)(18,47)(19,48)(20,49)(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)(101,126)(102,127)(103,128)(104,129)(105,130)(106,121)(107,122)(108,123)(109,124)(110,125)(131,148)(132,149)(133,150)(134,141)(135,142)(136,143)(137,144)(138,145)(139,146)(140,147)>;
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)(11,130,16,125)(12,129,17,124)(13,128,18,123)(14,127,19,122)(15,126,20,121)(21,63,26,68)(22,62,27,67)(23,61,28,66)(24,70,29,65)(25,69,30,64)(31,98,36,93)(32,97,37,92)(33,96,38,91)(34,95,39,100)(35,94,40,99)(41,104,46,109)(42,103,47,108)(43,102,48,107)(44,101,49,106)(45,110,50,105)(51,157,56,152)(52,156,57,151)(53,155,58,160)(54,154,59,159)(55,153,60,158)(71,138,76,133)(72,137,77,132)(73,136,78,131)(74,135,79,140)(75,134,80,139)(81,141,86,146)(82,150,87,145)(83,149,88,144)(84,148,89,143)(85,147,90,142), (1,57,14,85,39,74,43,68)(2,58,15,86,40,75,44,69)(3,59,16,87,31,76,45,70)(4,60,17,88,32,77,46,61)(5,51,18,89,33,78,47,62)(6,52,19,90,34,79,48,63)(7,53,20,81,35,80,49,64)(8,54,11,82,36,71,50,65)(9,55,12,83,37,72,41,66)(10,56,13,84,38,73,42,67)(21,111,151,122,147,95,135,107)(22,112,152,123,148,96,136,108)(23,113,153,124,149,97,137,109)(24,114,154,125,150,98,138,110)(25,115,155,126,141,99,139,101)(26,116,156,127,142,100,140,102)(27,117,157,128,143,91,131,103)(28,118,158,129,144,92,132,104)(29,119,159,130,145,93,133,105)(30,120,160,121,146,94,134,106), (11,50)(12,41)(13,42)(14,43)(15,44)(16,45)(17,46)(18,47)(19,48)(20,49)(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)(101,126)(102,127)(103,128)(104,129)(105,130)(106,121)(107,122)(108,123)(109,124)(110,125)(131,148)(132,149)(133,150)(134,141)(135,142)(136,143)(137,144)(138,145)(139,146)(140,147) );
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),(11,130,16,125),(12,129,17,124),(13,128,18,123),(14,127,19,122),(15,126,20,121),(21,63,26,68),(22,62,27,67),(23,61,28,66),(24,70,29,65),(25,69,30,64),(31,98,36,93),(32,97,37,92),(33,96,38,91),(34,95,39,100),(35,94,40,99),(41,104,46,109),(42,103,47,108),(43,102,48,107),(44,101,49,106),(45,110,50,105),(51,157,56,152),(52,156,57,151),(53,155,58,160),(54,154,59,159),(55,153,60,158),(71,138,76,133),(72,137,77,132),(73,136,78,131),(74,135,79,140),(75,134,80,139),(81,141,86,146),(82,150,87,145),(83,149,88,144),(84,148,89,143),(85,147,90,142)], [(1,57,14,85,39,74,43,68),(2,58,15,86,40,75,44,69),(3,59,16,87,31,76,45,70),(4,60,17,88,32,77,46,61),(5,51,18,89,33,78,47,62),(6,52,19,90,34,79,48,63),(7,53,20,81,35,80,49,64),(8,54,11,82,36,71,50,65),(9,55,12,83,37,72,41,66),(10,56,13,84,38,73,42,67),(21,111,151,122,147,95,135,107),(22,112,152,123,148,96,136,108),(23,113,153,124,149,97,137,109),(24,114,154,125,150,98,138,110),(25,115,155,126,141,99,139,101),(26,116,156,127,142,100,140,102),(27,117,157,128,143,91,131,103),(28,118,158,129,144,92,132,104),(29,119,159,130,145,93,133,105),(30,120,160,121,146,94,134,106)], [(11,50),(12,41),(13,42),(14,43),(15,44),(16,45),(17,46),(18,47),(19,48),(20,49),(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),(101,126),(102,127),(103,128),(104,129),(105,130),(106,121),(107,122),(108,123),(109,124),(110,125),(131,148),(132,149),(133,150),(134,141),(135,142),(136,143),(137,144),(138,145),(139,146),(140,147)]])
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