metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: Dic35, C7⋊Dic5, C35⋊3C4, C10.D7, C2.D35, C14.D5, C5⋊2Dic7, C70.1C2, SmallGroup(140,3)
Series: Derived ►Chief ►Lower central ►Upper central
| C35 — Dic35 |
Generators and relations for Dic35
G = < a,b | a70=1, b2=a35, bab-1=a-1 >
Smallest permutation representation of Dic35
►Regular action on 140 points
Generators in S140
(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 36 140)(2 104 37 139)(3 103 38 138)(4 102 39 137)(5 101 40 136)(6 100 41 135)(7 99 42 134)(8 98 43 133)(9 97 44 132)(10 96 45 131)(11 95 46 130)(12 94 47 129)(13 93 48 128)(14 92 49 127)(15 91 50 126)(16 90 51 125)(17 89 52 124)(18 88 53 123)(19 87 54 122)(20 86 55 121)(21 85 56 120)(22 84 57 119)(23 83 58 118)(24 82 59 117)(25 81 60 116)(26 80 61 115)(27 79 62 114)(28 78 63 113)(29 77 64 112)(30 76 65 111)(31 75 66 110)(32 74 67 109)(33 73 68 108)(34 72 69 107)(35 71 70 106)
G:=sub<Sym(140)| (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), (1,105,36,140)(2,104,37,139)(3,103,38,138)(4,102,39,137)(5,101,40,136)(6,100,41,135)(7,99,42,134)(8,98,43,133)(9,97,44,132)(10,96,45,131)(11,95,46,130)(12,94,47,129)(13,93,48,128)(14,92,49,127)(15,91,50,126)(16,90,51,125)(17,89,52,124)(18,88,53,123)(19,87,54,122)(20,86,55,121)(21,85,56,120)(22,84,57,119)(23,83,58,118)(24,82,59,117)(25,81,60,116)(26,80,61,115)(27,79,62,114)(28,78,63,113)(29,77,64,112)(30,76,65,111)(31,75,66,110)(32,74,67,109)(33,73,68,108)(34,72,69,107)(35,71,70,106)>;
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), (1,105,36,140)(2,104,37,139)(3,103,38,138)(4,102,39,137)(5,101,40,136)(6,100,41,135)(7,99,42,134)(8,98,43,133)(9,97,44,132)(10,96,45,131)(11,95,46,130)(12,94,47,129)(13,93,48,128)(14,92,49,127)(15,91,50,126)(16,90,51,125)(17,89,52,124)(18,88,53,123)(19,87,54,122)(20,86,55,121)(21,85,56,120)(22,84,57,119)(23,83,58,118)(24,82,59,117)(25,81,60,116)(26,80,61,115)(27,79,62,114)(28,78,63,113)(29,77,64,112)(30,76,65,111)(31,75,66,110)(32,74,67,109)(33,73,68,108)(34,72,69,107)(35,71,70,106) );
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)], [(1,105,36,140),(2,104,37,139),(3,103,38,138),(4,102,39,137),(5,101,40,136),(6,100,41,135),(7,99,42,134),(8,98,43,133),(9,97,44,132),(10,96,45,131),(11,95,46,130),(12,94,47,129),(13,93,48,128),(14,92,49,127),(15,91,50,126),(16,90,51,125),(17,89,52,124),(18,88,53,123),(19,87,54,122),(20,86,55,121),(21,85,56,120),(22,84,57,119),(23,83,58,118),(24,82,59,117),(25,81,60,116),(26,80,61,115),(27,79,62,114),(28,78,63,113),(29,77,64,112),(30,76,65,111),(31,75,66,110),(32,74,67,109),(33,73,68,108),(34,72,69,107),(35,71,70,106)]])
Dic35 is a maximal subgroup of
D7×Dic5 D5×Dic7 C35⋊D4 C35⋊Q8 Dic70 C4×D35 C35⋊7D4 C35⋊3C12 Dic105
Dic35 is a maximal quotient of
C35⋊3C8 Dic105
38 conjugacy classes
| class | 1 | 2 | 4A | 4B | 5A | 5B | 7A | 7B | 7C | 10A | 10B | 14A | 14B | 14C | 35A | ··· | 35L | 70A | ··· | 70L |
| order | 1 | 2 | 4 | 4 | 5 | 5 | 7 | 7 | 7 | 10 | 10 | 14 | 14 | 14 | 35 | ··· | 35 | 70 | ··· | 70 |
| size | 1 | 1 | 35 | 35 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
38 irreducible representations
| dim | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | - | + | - | |
| image | C1 | C2 | C4 | D5 | D7 | Dic5 | Dic7 | D35 | Dic35 |
| kernel | Dic35 | C70 | C35 | C14 | C10 | C7 | C5 | C2 | C1 |
| # reps | 1 | 1 | 2 | 2 | 3 | 2 | 3 | 12 | 12 |
Matrix representation of Dic35 ►in GL3(𝔽281) generated by
| 280 | 0 | 0 |
| 0 | 25 | 148 |
| 0 | 133 | 113 |
| 228 | 0 | 0 |
| 0 | 256 | 133 |
| 0 | 27 | 25 |
G:=sub<GL(3,GF(281))| [280,0,0,0,25,133,0,148,113],[228,0,0,0,256,27,0,133,25] >;
Dic35 in GAP, Magma, Sage, TeX
{\rm Dic}_{35} % in TeX
G:=Group("Dic35"); // GroupNames label
G:=SmallGroup(140,3);
// by ID
G=gap.SmallGroup(140,3);
# by ID
G:=PCGroup([4,-2,-2,-5,-7,8,194,1923]);
// Polycyclic
G:=Group<a,b|a^70=1,b^2=a^35,b*a*b^-1=a^-1>;
// generators/relations
Export