direct product, metabelian, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: D7×Dic5, D14.D5, C14.1D10, C10.1D14, Dic35⋊2C2, C70.1C22, C5⋊4(C4×D7), C35⋊4(C2×C4), (C5×D7)⋊2C4, (C10×D7).C2, C2.1(D5×D7), C7⋊1(C2×Dic5), (C7×Dic5)⋊1C2, SmallGroup(280,7)
Series: Derived ►Chief ►Lower central ►Upper central
| C35 — D7×Dic5 |
Generators and relations for D7×Dic5
G = < a,b,c,d | a7=b2=c10=1, d2=c5, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >
Smallest permutation representation of D7×Dic5
►On 140 points
Generators in S140
(1 76 128 96 112 104 88)(2 77 129 97 113 105 89)(3 78 130 98 114 106 90)(4 79 121 99 115 107 81)(5 80 122 100 116 108 82)(6 71 123 91 117 109 83)(7 72 124 92 118 110 84)(8 73 125 93 119 101 85)(9 74 126 94 120 102 86)(10 75 127 95 111 103 87)(11 43 27 132 51 37 63)(12 44 28 133 52 38 64)(13 45 29 134 53 39 65)(14 46 30 135 54 40 66)(15 47 21 136 55 31 67)(16 48 22 137 56 32 68)(17 49 23 138 57 33 69)(18 50 24 139 58 34 70)(19 41 25 140 59 35 61)(20 42 26 131 60 36 62)
(1 88)(2 89)(3 90)(4 81)(5 82)(6 83)(7 84)(8 85)(9 86)(10 87)(11 27)(12 28)(13 29)(14 30)(15 21)(16 22)(17 23)(18 24)(19 25)(20 26)(31 55)(32 56)(33 57)(34 58)(35 59)(36 60)(37 51)(38 52)(39 53)(40 54)(61 140)(62 131)(63 132)(64 133)(65 134)(66 135)(67 136)(68 137)(69 138)(70 139)(71 109)(72 110)(73 101)(74 102)(75 103)(76 104)(77 105)(78 106)(79 107)(80 108)(111 127)(112 128)(113 129)(114 130)(115 121)(116 122)(117 123)(118 124)(119 125)(120 126)
(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)
(1 34 6 39)(2 33 7 38)(3 32 8 37)(4 31 9 36)(5 40 10 35)(11 130 16 125)(12 129 17 124)(13 128 18 123)(14 127 19 122)(15 126 20 121)(21 120 26 115)(22 119 27 114)(23 118 28 113)(24 117 29 112)(25 116 30 111)(41 100 46 95)(42 99 47 94)(43 98 48 93)(44 97 49 92)(45 96 50 91)(51 90 56 85)(52 89 57 84)(53 88 58 83)(54 87 59 82)(55 86 60 81)(61 80 66 75)(62 79 67 74)(63 78 68 73)(64 77 69 72)(65 76 70 71)(101 132 106 137)(102 131 107 136)(103 140 108 135)(104 139 109 134)(105 138 110 133)
G:=sub<Sym(140)| (1,76,128,96,112,104,88)(2,77,129,97,113,105,89)(3,78,130,98,114,106,90)(4,79,121,99,115,107,81)(5,80,122,100,116,108,82)(6,71,123,91,117,109,83)(7,72,124,92,118,110,84)(8,73,125,93,119,101,85)(9,74,126,94,120,102,86)(10,75,127,95,111,103,87)(11,43,27,132,51,37,63)(12,44,28,133,52,38,64)(13,45,29,134,53,39,65)(14,46,30,135,54,40,66)(15,47,21,136,55,31,67)(16,48,22,137,56,32,68)(17,49,23,138,57,33,69)(18,50,24,139,58,34,70)(19,41,25,140,59,35,61)(20,42,26,131,60,36,62), (1,88)(2,89)(3,90)(4,81)(5,82)(6,83)(7,84)(8,85)(9,86)(10,87)(11,27)(12,28)(13,29)(14,30)(15,21)(16,22)(17,23)(18,24)(19,25)(20,26)(31,55)(32,56)(33,57)(34,58)(35,59)(36,60)(37,51)(38,52)(39,53)(40,54)(61,140)(62,131)(63,132)(64,133)(65,134)(66,135)(67,136)(68,137)(69,138)(70,139)(71,109)(72,110)(73,101)(74,102)(75,103)(76,104)(77,105)(78,106)(79,107)(80,108)(111,127)(112,128)(113,129)(114,130)(115,121)(116,122)(117,123)(118,124)(119,125)(120,126), (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), (1,34,6,39)(2,33,7,38)(3,32,8,37)(4,31,9,36)(5,40,10,35)(11,130,16,125)(12,129,17,124)(13,128,18,123)(14,127,19,122)(15,126,20,121)(21,120,26,115)(22,119,27,114)(23,118,28,113)(24,117,29,112)(25,116,30,111)(41,100,46,95)(42,99,47,94)(43,98,48,93)(44,97,49,92)(45,96,50,91)(51,90,56,85)(52,89,57,84)(53,88,58,83)(54,87,59,82)(55,86,60,81)(61,80,66,75)(62,79,67,74)(63,78,68,73)(64,77,69,72)(65,76,70,71)(101,132,106,137)(102,131,107,136)(103,140,108,135)(104,139,109,134)(105,138,110,133)>;
G:=Group( (1,76,128,96,112,104,88)(2,77,129,97,113,105,89)(3,78,130,98,114,106,90)(4,79,121,99,115,107,81)(5,80,122,100,116,108,82)(6,71,123,91,117,109,83)(7,72,124,92,118,110,84)(8,73,125,93,119,101,85)(9,74,126,94,120,102,86)(10,75,127,95,111,103,87)(11,43,27,132,51,37,63)(12,44,28,133,52,38,64)(13,45,29,134,53,39,65)(14,46,30,135,54,40,66)(15,47,21,136,55,31,67)(16,48,22,137,56,32,68)(17,49,23,138,57,33,69)(18,50,24,139,58,34,70)(19,41,25,140,59,35,61)(20,42,26,131,60,36,62), (1,88)(2,89)(3,90)(4,81)(5,82)(6,83)(7,84)(8,85)(9,86)(10,87)(11,27)(12,28)(13,29)(14,30)(15,21)(16,22)(17,23)(18,24)(19,25)(20,26)(31,55)(32,56)(33,57)(34,58)(35,59)(36,60)(37,51)(38,52)(39,53)(40,54)(61,140)(62,131)(63,132)(64,133)(65,134)(66,135)(67,136)(68,137)(69,138)(70,139)(71,109)(72,110)(73,101)(74,102)(75,103)(76,104)(77,105)(78,106)(79,107)(80,108)(111,127)(112,128)(113,129)(114,130)(115,121)(116,122)(117,123)(118,124)(119,125)(120,126), (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), (1,34,6,39)(2,33,7,38)(3,32,8,37)(4,31,9,36)(5,40,10,35)(11,130,16,125)(12,129,17,124)(13,128,18,123)(14,127,19,122)(15,126,20,121)(21,120,26,115)(22,119,27,114)(23,118,28,113)(24,117,29,112)(25,116,30,111)(41,100,46,95)(42,99,47,94)(43,98,48,93)(44,97,49,92)(45,96,50,91)(51,90,56,85)(52,89,57,84)(53,88,58,83)(54,87,59,82)(55,86,60,81)(61,80,66,75)(62,79,67,74)(63,78,68,73)(64,77,69,72)(65,76,70,71)(101,132,106,137)(102,131,107,136)(103,140,108,135)(104,139,109,134)(105,138,110,133) );
G=PermutationGroup([[(1,76,128,96,112,104,88),(2,77,129,97,113,105,89),(3,78,130,98,114,106,90),(4,79,121,99,115,107,81),(5,80,122,100,116,108,82),(6,71,123,91,117,109,83),(7,72,124,92,118,110,84),(8,73,125,93,119,101,85),(9,74,126,94,120,102,86),(10,75,127,95,111,103,87),(11,43,27,132,51,37,63),(12,44,28,133,52,38,64),(13,45,29,134,53,39,65),(14,46,30,135,54,40,66),(15,47,21,136,55,31,67),(16,48,22,137,56,32,68),(17,49,23,138,57,33,69),(18,50,24,139,58,34,70),(19,41,25,140,59,35,61),(20,42,26,131,60,36,62)], [(1,88),(2,89),(3,90),(4,81),(5,82),(6,83),(7,84),(8,85),(9,86),(10,87),(11,27),(12,28),(13,29),(14,30),(15,21),(16,22),(17,23),(18,24),(19,25),(20,26),(31,55),(32,56),(33,57),(34,58),(35,59),(36,60),(37,51),(38,52),(39,53),(40,54),(61,140),(62,131),(63,132),(64,133),(65,134),(66,135),(67,136),(68,137),(69,138),(70,139),(71,109),(72,110),(73,101),(74,102),(75,103),(76,104),(77,105),(78,106),(79,107),(80,108),(111,127),(112,128),(113,129),(114,130),(115,121),(116,122),(117,123),(118,124),(119,125),(120,126)], [(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)], [(1,34,6,39),(2,33,7,38),(3,32,8,37),(4,31,9,36),(5,40,10,35),(11,130,16,125),(12,129,17,124),(13,128,18,123),(14,127,19,122),(15,126,20,121),(21,120,26,115),(22,119,27,114),(23,118,28,113),(24,117,29,112),(25,116,30,111),(41,100,46,95),(42,99,47,94),(43,98,48,93),(44,97,49,92),(45,96,50,91),(51,90,56,85),(52,89,57,84),(53,88,58,83),(54,87,59,82),(55,86,60,81),(61,80,66,75),(62,79,67,74),(63,78,68,73),(64,77,69,72),(65,76,70,71),(101,132,106,137),(102,131,107,136),(103,140,108,135),(104,139,109,134),(105,138,110,133)]])
40 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 5A | 5B | 7A | 7B | 7C | 10A | 10B | 10C | 10D | 10E | 10F | 14A | 14B | 14C | 28A | ··· | 28F | 35A | ··· | 35F | 70A | ··· | 70F |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 5 | 5 | 7 | 7 | 7 | 10 | 10 | 10 | 10 | 10 | 10 | 14 | 14 | 14 | 28 | ··· | 28 | 35 | ··· | 35 | 70 | ··· | 70 |
| size | 1 | 1 | 7 | 7 | 5 | 5 | 35 | 35 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 14 | 14 | 14 | 14 | 2 | 2 | 2 | 10 | ··· | 10 | 4 | ··· | 4 | 4 | ··· | 4 |
40 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | - | + | + | + | - | ||
| image | C1 | C2 | C2 | C2 | C4 | D5 | D7 | Dic5 | D10 | D14 | C4×D7 | D5×D7 | D7×Dic5 |
| kernel | D7×Dic5 | C7×Dic5 | Dic35 | C10×D7 | C5×D7 | D14 | Dic5 | D7 | C14 | C10 | C5 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 4 | 2 | 3 | 4 | 2 | 3 | 6 | 6 | 6 |
Matrix representation of D7×Dic5 ►in GL4(𝔽281) generated by
| 0 | 1 | 0 | 0 |
| 280 | 47 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 280 | 0 | 0 | 0 |
| 0 | 280 | 0 | 0 |
| 0 | 0 | 280 | 45 |
| 0 | 0 | 124 | 39 |
| 53 | 0 | 0 | 0 |
| 0 | 53 | 0 | 0 |
| 0 | 0 | 88 | 152 |
| 0 | 0 | 84 | 193 |
G:=sub<GL(4,GF(281))| [0,280,0,0,1,47,0,0,0,0,1,0,0,0,0,1],[0,1,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[280,0,0,0,0,280,0,0,0,0,280,124,0,0,45,39],[53,0,0,0,0,53,0,0,0,0,88,84,0,0,152,193] >;
D7×Dic5 in GAP, Magma, Sage, TeX
D_7\times {\rm Dic}_5 % in TeX
G:=Group("D7xDic5"); // GroupNames label
G:=SmallGroup(280,7);
// by ID
G=gap.SmallGroup(280,7);
# by ID
G:=PCGroup([5,-2,-2,-2,-5,-7,26,328,6004]);
// Polycyclic
G:=Group<a,b,c,d|a^7=b^2=c^10=1,d^2=c^5,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations
Export