metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C35⋊1D4, D10⋊1D7, D14⋊1D5, C10.4D14, C14.4D10, Dic35⋊4C2, C70.4C22, C5⋊2(C7⋊D4), C7⋊2(C5⋊D4), C2.4(D5×D7), (D5×C14)⋊1C2, (C10×D7)⋊1C2, SmallGroup(280,10)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C35⋊D4
G = < a,b,c | a35=b4=c2=1, bab-1=a-1, cac=a6, cbc=b-1 >
Smallest permutation representation of C35⋊D4
►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 98 56 123)(2 97 57 122)(3 96 58 121)(4 95 59 120)(5 94 60 119)(6 93 61 118)(7 92 62 117)(8 91 63 116)(9 90 64 115)(10 89 65 114)(11 88 66 113)(12 87 67 112)(13 86 68 111)(14 85 69 110)(15 84 70 109)(16 83 36 108)(17 82 37 107)(18 81 38 106)(19 80 39 140)(20 79 40 139)(21 78 41 138)(22 77 42 137)(23 76 43 136)(24 75 44 135)(25 74 45 134)(26 73 46 133)(27 72 47 132)(28 71 48 131)(29 105 49 130)(30 104 50 129)(31 103 51 128)(32 102 52 127)(33 101 53 126)(34 100 54 125)(35 99 55 124)
(2 7)(3 13)(4 19)(5 25)(6 31)(9 14)(10 20)(11 26)(12 32)(16 21)(17 27)(18 33)(23 28)(24 34)(30 35)(36 41)(37 47)(38 53)(39 59)(40 65)(43 48)(44 54)(45 60)(46 66)(50 55)(51 61)(52 67)(57 62)(58 68)(64 69)(71 136)(72 107)(73 113)(74 119)(75 125)(76 131)(77 137)(78 108)(79 114)(80 120)(81 126)(82 132)(83 138)(84 109)(85 115)(86 121)(87 127)(88 133)(89 139)(90 110)(91 116)(92 122)(93 128)(94 134)(95 140)(96 111)(97 117)(98 123)(99 129)(100 135)(101 106)(102 112)(103 118)(104 124)(105 130)
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), (1,98,56,123)(2,97,57,122)(3,96,58,121)(4,95,59,120)(5,94,60,119)(6,93,61,118)(7,92,62,117)(8,91,63,116)(9,90,64,115)(10,89,65,114)(11,88,66,113)(12,87,67,112)(13,86,68,111)(14,85,69,110)(15,84,70,109)(16,83,36,108)(17,82,37,107)(18,81,38,106)(19,80,39,140)(20,79,40,139)(21,78,41,138)(22,77,42,137)(23,76,43,136)(24,75,44,135)(25,74,45,134)(26,73,46,133)(27,72,47,132)(28,71,48,131)(29,105,49,130)(30,104,50,129)(31,103,51,128)(32,102,52,127)(33,101,53,126)(34,100,54,125)(35,99,55,124), (2,7)(3,13)(4,19)(5,25)(6,31)(9,14)(10,20)(11,26)(12,32)(16,21)(17,27)(18,33)(23,28)(24,34)(30,35)(36,41)(37,47)(38,53)(39,59)(40,65)(43,48)(44,54)(45,60)(46,66)(50,55)(51,61)(52,67)(57,62)(58,68)(64,69)(71,136)(72,107)(73,113)(74,119)(75,125)(76,131)(77,137)(78,108)(79,114)(80,120)(81,126)(82,132)(83,138)(84,109)(85,115)(86,121)(87,127)(88,133)(89,139)(90,110)(91,116)(92,122)(93,128)(94,134)(95,140)(96,111)(97,117)(98,123)(99,129)(100,135)(101,106)(102,112)(103,118)(104,124)(105,130)>;
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), (1,98,56,123)(2,97,57,122)(3,96,58,121)(4,95,59,120)(5,94,60,119)(6,93,61,118)(7,92,62,117)(8,91,63,116)(9,90,64,115)(10,89,65,114)(11,88,66,113)(12,87,67,112)(13,86,68,111)(14,85,69,110)(15,84,70,109)(16,83,36,108)(17,82,37,107)(18,81,38,106)(19,80,39,140)(20,79,40,139)(21,78,41,138)(22,77,42,137)(23,76,43,136)(24,75,44,135)(25,74,45,134)(26,73,46,133)(27,72,47,132)(28,71,48,131)(29,105,49,130)(30,104,50,129)(31,103,51,128)(32,102,52,127)(33,101,53,126)(34,100,54,125)(35,99,55,124), (2,7)(3,13)(4,19)(5,25)(6,31)(9,14)(10,20)(11,26)(12,32)(16,21)(17,27)(18,33)(23,28)(24,34)(30,35)(36,41)(37,47)(38,53)(39,59)(40,65)(43,48)(44,54)(45,60)(46,66)(50,55)(51,61)(52,67)(57,62)(58,68)(64,69)(71,136)(72,107)(73,113)(74,119)(75,125)(76,131)(77,137)(78,108)(79,114)(80,120)(81,126)(82,132)(83,138)(84,109)(85,115)(86,121)(87,127)(88,133)(89,139)(90,110)(91,116)(92,122)(93,128)(94,134)(95,140)(96,111)(97,117)(98,123)(99,129)(100,135)(101,106)(102,112)(103,118)(104,124)(105,130) );
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)], [(1,98,56,123),(2,97,57,122),(3,96,58,121),(4,95,59,120),(5,94,60,119),(6,93,61,118),(7,92,62,117),(8,91,63,116),(9,90,64,115),(10,89,65,114),(11,88,66,113),(12,87,67,112),(13,86,68,111),(14,85,69,110),(15,84,70,109),(16,83,36,108),(17,82,37,107),(18,81,38,106),(19,80,39,140),(20,79,40,139),(21,78,41,138),(22,77,42,137),(23,76,43,136),(24,75,44,135),(25,74,45,134),(26,73,46,133),(27,72,47,132),(28,71,48,131),(29,105,49,130),(30,104,50,129),(31,103,51,128),(32,102,52,127),(33,101,53,126),(34,100,54,125),(35,99,55,124)], [(2,7),(3,13),(4,19),(5,25),(6,31),(9,14),(10,20),(11,26),(12,32),(16,21),(17,27),(18,33),(23,28),(24,34),(30,35),(36,41),(37,47),(38,53),(39,59),(40,65),(43,48),(44,54),(45,60),(46,66),(50,55),(51,61),(52,67),(57,62),(58,68),(64,69),(71,136),(72,107),(73,113),(74,119),(75,125),(76,131),(77,137),(78,108),(79,114),(80,120),(81,126),(82,132),(83,138),(84,109),(85,115),(86,121),(87,127),(88,133),(89,139),(90,110),(91,116),(92,122),(93,128),(94,134),(95,140),(96,111),(97,117),(98,123),(99,129),(100,135),(101,106),(102,112),(103,118),(104,124),(105,130)]])
37 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4 | 5A | 5B | 7A | 7B | 7C | 10A | 10B | 10C | 10D | 10E | 10F | 14A | 14B | 14C | 14D | ··· | 14I | 35A | ··· | 35F | 70A | ··· | 70F |
| order | 1 | 2 | 2 | 2 | 4 | 5 | 5 | 7 | 7 | 7 | 10 | 10 | 10 | 10 | 10 | 10 | 14 | 14 | 14 | 14 | ··· | 14 | 35 | ··· | 35 | 70 | ··· | 70 |
| size | 1 | 1 | 10 | 14 | 70 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 14 | 14 | 14 | 14 | 2 | 2 | 2 | 10 | ··· | 10 | 4 | ··· | 4 | 4 | ··· | 4 |
37 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | - | ||
| image | C1 | C2 | C2 | C2 | D4 | D5 | D7 | D10 | D14 | C5⋊D4 | C7⋊D4 | D5×D7 | C35⋊D4 |
| kernel | C35⋊D4 | Dic35 | C10×D7 | D5×C14 | C35 | D14 | D10 | C14 | C10 | C7 | C5 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 2 | 3 | 2 | 3 | 4 | 6 | 6 | 6 |
Matrix representation of C35⋊D4 ►in GL4(𝔽281) generated by
| 232 | 0 | 0 | 0 |
| 0 | 86 | 0 | 0 |
| 0 | 0 | 47 | 275 |
| 0 | 0 | 47 | 0 |
| 0 | 280 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 5 | 25 |
| 0 | 0 | 235 | 276 |
| 1 | 0 | 0 | 0 |
| 0 | 280 | 0 | 0 |
| 0 | 0 | 40 | 1 |
| 0 | 0 | 87 | 241 |
G:=sub<GL(4,GF(281))| [232,0,0,0,0,86,0,0,0,0,47,47,0,0,275,0],[0,1,0,0,280,0,0,0,0,0,5,235,0,0,25,276],[1,0,0,0,0,280,0,0,0,0,40,87,0,0,1,241] >;
C35⋊D4 in GAP, Magma, Sage, TeX
C_{35}\rtimes D_4 % in TeX
G:=Group("C35:D4"); // GroupNames label
G:=SmallGroup(280,10);
// by ID
G=gap.SmallGroup(280,10);
# by ID
G:=PCGroup([5,-2,-2,-2,-5,-7,61,328,6004]);
// Polycyclic
G:=Group<a,b,c|a^35=b^4=c^2=1,b*a*b^-1=a^-1,c*a*c=a^6,c*b*c=b^-1>;
// generators/relations
Export