direct product, metabelian, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: D5×Dic7, D10.2D7, C14.2D10, C10.2D14, Dic35⋊3C2, C70.2C22, C7⋊3(C4×D5), C35⋊5(C2×C4), (C7×D5)⋊1C4, C2.2(D5×D7), C5⋊2(C2×Dic7), (C5×Dic7)⋊1C2, (D5×C14).1C2, SmallGroup(280,8)
Series: Derived ►Chief ►Lower central ►Upper central
| C35 — D5×Dic7 |
Generators and relations for D5×Dic7
G = < a,b,c,d | a5=b2=c14=1, d2=c7, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >
Smallest permutation representation of D5×Dic7
►On 140 points
Generators in S140
(1 62 36 135 98)(2 63 37 136 85)(3 64 38 137 86)(4 65 39 138 87)(5 66 40 139 88)(6 67 41 140 89)(7 68 42 127 90)(8 69 29 128 91)(9 70 30 129 92)(10 57 31 130 93)(11 58 32 131 94)(12 59 33 132 95)(13 60 34 133 96)(14 61 35 134 97)(15 104 121 51 71)(16 105 122 52 72)(17 106 123 53 73)(18 107 124 54 74)(19 108 125 55 75)(20 109 126 56 76)(21 110 113 43 77)(22 111 114 44 78)(23 112 115 45 79)(24 99 116 46 80)(25 100 117 47 81)(26 101 118 48 82)(27 102 119 49 83)(28 103 120 50 84)
(1 91)(2 92)(3 93)(4 94)(5 95)(6 96)(7 97)(8 98)(9 85)(10 86)(11 87)(12 88)(13 89)(14 90)(15 111)(16 112)(17 99)(18 100)(19 101)(20 102)(21 103)(22 104)(23 105)(24 106)(25 107)(26 108)(27 109)(28 110)(29 36)(30 37)(31 38)(32 39)(33 40)(34 41)(35 42)(43 50)(44 51)(45 52)(46 53)(47 54)(48 55)(49 56)(57 137)(58 138)(59 139)(60 140)(61 127)(62 128)(63 129)(64 130)(65 131)(66 132)(67 133)(68 134)(69 135)(70 136)(71 114)(72 115)(73 116)(74 117)(75 118)(76 119)(77 120)(78 121)(79 122)(80 123)(81 124)(82 125)(83 126)(84 113)
(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)
(1 105 8 112)(2 104 9 111)(3 103 10 110)(4 102 11 109)(5 101 12 108)(6 100 13 107)(7 99 14 106)(15 92 22 85)(16 91 23 98)(17 90 24 97)(18 89 25 96)(19 88 26 95)(20 87 27 94)(21 86 28 93)(29 45 36 52)(30 44 37 51)(31 43 38 50)(32 56 39 49)(33 55 40 48)(34 54 41 47)(35 53 42 46)(57 113 64 120)(58 126 65 119)(59 125 66 118)(60 124 67 117)(61 123 68 116)(62 122 69 115)(63 121 70 114)(71 129 78 136)(72 128 79 135)(73 127 80 134)(74 140 81 133)(75 139 82 132)(76 138 83 131)(77 137 84 130)
G:=sub<Sym(140)| (1,62,36,135,98)(2,63,37,136,85)(3,64,38,137,86)(4,65,39,138,87)(5,66,40,139,88)(6,67,41,140,89)(7,68,42,127,90)(8,69,29,128,91)(9,70,30,129,92)(10,57,31,130,93)(11,58,32,131,94)(12,59,33,132,95)(13,60,34,133,96)(14,61,35,134,97)(15,104,121,51,71)(16,105,122,52,72)(17,106,123,53,73)(18,107,124,54,74)(19,108,125,55,75)(20,109,126,56,76)(21,110,113,43,77)(22,111,114,44,78)(23,112,115,45,79)(24,99,116,46,80)(25,100,117,47,81)(26,101,118,48,82)(27,102,119,49,83)(28,103,120,50,84), (1,91)(2,92)(3,93)(4,94)(5,95)(6,96)(7,97)(8,98)(9,85)(10,86)(11,87)(12,88)(13,89)(14,90)(15,111)(16,112)(17,99)(18,100)(19,101)(20,102)(21,103)(22,104)(23,105)(24,106)(25,107)(26,108)(27,109)(28,110)(29,36)(30,37)(31,38)(32,39)(33,40)(34,41)(35,42)(43,50)(44,51)(45,52)(46,53)(47,54)(48,55)(49,56)(57,137)(58,138)(59,139)(60,140)(61,127)(62,128)(63,129)(64,130)(65,131)(66,132)(67,133)(68,134)(69,135)(70,136)(71,114)(72,115)(73,116)(74,117)(75,118)(76,119)(77,120)(78,121)(79,122)(80,123)(81,124)(82,125)(83,126)(84,113), (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), (1,105,8,112)(2,104,9,111)(3,103,10,110)(4,102,11,109)(5,101,12,108)(6,100,13,107)(7,99,14,106)(15,92,22,85)(16,91,23,98)(17,90,24,97)(18,89,25,96)(19,88,26,95)(20,87,27,94)(21,86,28,93)(29,45,36,52)(30,44,37,51)(31,43,38,50)(32,56,39,49)(33,55,40,48)(34,54,41,47)(35,53,42,46)(57,113,64,120)(58,126,65,119)(59,125,66,118)(60,124,67,117)(61,123,68,116)(62,122,69,115)(63,121,70,114)(71,129,78,136)(72,128,79,135)(73,127,80,134)(74,140,81,133)(75,139,82,132)(76,138,83,131)(77,137,84,130)>;
G:=Group( (1,62,36,135,98)(2,63,37,136,85)(3,64,38,137,86)(4,65,39,138,87)(5,66,40,139,88)(6,67,41,140,89)(7,68,42,127,90)(8,69,29,128,91)(9,70,30,129,92)(10,57,31,130,93)(11,58,32,131,94)(12,59,33,132,95)(13,60,34,133,96)(14,61,35,134,97)(15,104,121,51,71)(16,105,122,52,72)(17,106,123,53,73)(18,107,124,54,74)(19,108,125,55,75)(20,109,126,56,76)(21,110,113,43,77)(22,111,114,44,78)(23,112,115,45,79)(24,99,116,46,80)(25,100,117,47,81)(26,101,118,48,82)(27,102,119,49,83)(28,103,120,50,84), (1,91)(2,92)(3,93)(4,94)(5,95)(6,96)(7,97)(8,98)(9,85)(10,86)(11,87)(12,88)(13,89)(14,90)(15,111)(16,112)(17,99)(18,100)(19,101)(20,102)(21,103)(22,104)(23,105)(24,106)(25,107)(26,108)(27,109)(28,110)(29,36)(30,37)(31,38)(32,39)(33,40)(34,41)(35,42)(43,50)(44,51)(45,52)(46,53)(47,54)(48,55)(49,56)(57,137)(58,138)(59,139)(60,140)(61,127)(62,128)(63,129)(64,130)(65,131)(66,132)(67,133)(68,134)(69,135)(70,136)(71,114)(72,115)(73,116)(74,117)(75,118)(76,119)(77,120)(78,121)(79,122)(80,123)(81,124)(82,125)(83,126)(84,113), (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), (1,105,8,112)(2,104,9,111)(3,103,10,110)(4,102,11,109)(5,101,12,108)(6,100,13,107)(7,99,14,106)(15,92,22,85)(16,91,23,98)(17,90,24,97)(18,89,25,96)(19,88,26,95)(20,87,27,94)(21,86,28,93)(29,45,36,52)(30,44,37,51)(31,43,38,50)(32,56,39,49)(33,55,40,48)(34,54,41,47)(35,53,42,46)(57,113,64,120)(58,126,65,119)(59,125,66,118)(60,124,67,117)(61,123,68,116)(62,122,69,115)(63,121,70,114)(71,129,78,136)(72,128,79,135)(73,127,80,134)(74,140,81,133)(75,139,82,132)(76,138,83,131)(77,137,84,130) );
G=PermutationGroup([[(1,62,36,135,98),(2,63,37,136,85),(3,64,38,137,86),(4,65,39,138,87),(5,66,40,139,88),(6,67,41,140,89),(7,68,42,127,90),(8,69,29,128,91),(9,70,30,129,92),(10,57,31,130,93),(11,58,32,131,94),(12,59,33,132,95),(13,60,34,133,96),(14,61,35,134,97),(15,104,121,51,71),(16,105,122,52,72),(17,106,123,53,73),(18,107,124,54,74),(19,108,125,55,75),(20,109,126,56,76),(21,110,113,43,77),(22,111,114,44,78),(23,112,115,45,79),(24,99,116,46,80),(25,100,117,47,81),(26,101,118,48,82),(27,102,119,49,83),(28,103,120,50,84)], [(1,91),(2,92),(3,93),(4,94),(5,95),(6,96),(7,97),(8,98),(9,85),(10,86),(11,87),(12,88),(13,89),(14,90),(15,111),(16,112),(17,99),(18,100),(19,101),(20,102),(21,103),(22,104),(23,105),(24,106),(25,107),(26,108),(27,109),(28,110),(29,36),(30,37),(31,38),(32,39),(33,40),(34,41),(35,42),(43,50),(44,51),(45,52),(46,53),(47,54),(48,55),(49,56),(57,137),(58,138),(59,139),(60,140),(61,127),(62,128),(63,129),(64,130),(65,131),(66,132),(67,133),(68,134),(69,135),(70,136),(71,114),(72,115),(73,116),(74,117),(75,118),(76,119),(77,120),(78,121),(79,122),(80,123),(81,124),(82,125),(83,126),(84,113)], [(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)], [(1,105,8,112),(2,104,9,111),(3,103,10,110),(4,102,11,109),(5,101,12,108),(6,100,13,107),(7,99,14,106),(15,92,22,85),(16,91,23,98),(17,90,24,97),(18,89,25,96),(19,88,26,95),(20,87,27,94),(21,86,28,93),(29,45,36,52),(30,44,37,51),(31,43,38,50),(32,56,39,49),(33,55,40,48),(34,54,41,47),(35,53,42,46),(57,113,64,120),(58,126,65,119),(59,125,66,118),(60,124,67,117),(61,123,68,116),(62,122,69,115),(63,121,70,114),(71,129,78,136),(72,128,79,135),(73,127,80,134),(74,140,81,133),(75,139,82,132),(76,138,83,131),(77,137,84,130)]])
40 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 5A | 5B | 7A | 7B | 7C | 10A | 10B | 14A | 14B | 14C | 14D | ··· | 14I | 20A | 20B | 20C | 20D | 35A | ··· | 35F | 70A | ··· | 70F |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 5 | 5 | 7 | 7 | 7 | 10 | 10 | 14 | 14 | 14 | 14 | ··· | 14 | 20 | 20 | 20 | 20 | 35 | ··· | 35 | 70 | ··· | 70 |
| size | 1 | 1 | 5 | 5 | 7 | 7 | 35 | 35 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 10 | ··· | 10 | 14 | 14 | 14 | 14 | 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 | D10 | Dic7 | D14 | C4×D5 | D5×D7 | D5×Dic7 |
| kernel | D5×Dic7 | C5×Dic7 | Dic35 | D5×C14 | C7×D5 | Dic7 | D10 | C14 | D5 | C10 | C7 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 4 | 2 | 3 | 2 | 6 | 3 | 4 | 6 | 6 |
Matrix representation of D5×Dic7 ►in GL4(𝔽281) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 280 | 1 |
| 0 | 0 | 36 | 244 |
| 280 | 0 | 0 | 0 |
| 0 | 280 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 245 | 280 |
| 240 | 280 | 0 | 0 |
| 36 | 275 | 0 | 0 |
| 0 | 0 | 280 | 0 |
| 0 | 0 | 0 | 280 |
| 93 | 132 | 0 | 0 |
| 92 | 188 | 0 | 0 |
| 0 | 0 | 228 | 0 |
| 0 | 0 | 0 | 228 |
G:=sub<GL(4,GF(281))| [1,0,0,0,0,1,0,0,0,0,280,36,0,0,1,244],[280,0,0,0,0,280,0,0,0,0,1,245,0,0,0,280],[240,36,0,0,280,275,0,0,0,0,280,0,0,0,0,280],[93,92,0,0,132,188,0,0,0,0,228,0,0,0,0,228] >;
D5×Dic7 in GAP, Magma, Sage, TeX
D_5\times {\rm Dic}_7 % in TeX
G:=Group("D5xDic7"); // GroupNames label
G:=SmallGroup(280,8);
// by ID
G=gap.SmallGroup(280,8);
# by ID
G:=PCGroup([5,-2,-2,-2,-5,-7,20,328,6004]);
// Polycyclic
G:=Group<a,b,c,d|a^5=b^2=c^14=1,d^2=c^7,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