metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C35⋊7D4, C22⋊D35, D70⋊2C2, C2.5D70, Dic35⋊1C2, C10.12D14, C14.12D10, C70.12C22, (C2×C70)⋊2C2, (C2×C14)⋊2D5, (C2×C10)⋊2D7, C5⋊3(C7⋊D4), C7⋊3(C5⋊D4), SmallGroup(280,28)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C35⋊7D4
G = < a,b,c | a35=b4=c2=1, bab-1=cac=a-1, cbc=b-1 >
Smallest permutation representation of C35⋊7D4
►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 88 40 113)(2 87 41 112)(3 86 42 111)(4 85 43 110)(5 84 44 109)(6 83 45 108)(7 82 46 107)(8 81 47 106)(9 80 48 140)(10 79 49 139)(11 78 50 138)(12 77 51 137)(13 76 52 136)(14 75 53 135)(15 74 54 134)(16 73 55 133)(17 72 56 132)(18 71 57 131)(19 105 58 130)(20 104 59 129)(21 103 60 128)(22 102 61 127)(23 101 62 126)(24 100 63 125)(25 99 64 124)(26 98 65 123)(27 97 66 122)(28 96 67 121)(29 95 68 120)(30 94 69 119)(31 93 70 118)(32 92 36 117)(33 91 37 116)(34 90 38 115)(35 89 39 114)
(2 35)(3 34)(4 33)(5 32)(6 31)(7 30)(8 29)(9 28)(10 27)(11 26)(12 25)(13 24)(14 23)(15 22)(16 21)(17 20)(18 19)(36 44)(37 43)(38 42)(39 41)(45 70)(46 69)(47 68)(48 67)(49 66)(50 65)(51 64)(52 63)(53 62)(54 61)(55 60)(56 59)(57 58)(71 130)(72 129)(73 128)(74 127)(75 126)(76 125)(77 124)(78 123)(79 122)(80 121)(81 120)(82 119)(83 118)(84 117)(85 116)(86 115)(87 114)(88 113)(89 112)(90 111)(91 110)(92 109)(93 108)(94 107)(95 106)(96 140)(97 139)(98 138)(99 137)(100 136)(101 135)(102 134)(103 133)(104 132)(105 131)
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), (1,88,40,113)(2,87,41,112)(3,86,42,111)(4,85,43,110)(5,84,44,109)(6,83,45,108)(7,82,46,107)(8,81,47,106)(9,80,48,140)(10,79,49,139)(11,78,50,138)(12,77,51,137)(13,76,52,136)(14,75,53,135)(15,74,54,134)(16,73,55,133)(17,72,56,132)(18,71,57,131)(19,105,58,130)(20,104,59,129)(21,103,60,128)(22,102,61,127)(23,101,62,126)(24,100,63,125)(25,99,64,124)(26,98,65,123)(27,97,66,122)(28,96,67,121)(29,95,68,120)(30,94,69,119)(31,93,70,118)(32,92,36,117)(33,91,37,116)(34,90,38,115)(35,89,39,114), (2,35)(3,34)(4,33)(5,32)(6,31)(7,30)(8,29)(9,28)(10,27)(11,26)(12,25)(13,24)(14,23)(15,22)(16,21)(17,20)(18,19)(36,44)(37,43)(38,42)(39,41)(45,70)(46,69)(47,68)(48,67)(49,66)(50,65)(51,64)(52,63)(53,62)(54,61)(55,60)(56,59)(57,58)(71,130)(72,129)(73,128)(74,127)(75,126)(76,125)(77,124)(78,123)(79,122)(80,121)(81,120)(82,119)(83,118)(84,117)(85,116)(86,115)(87,114)(88,113)(89,112)(90,111)(91,110)(92,109)(93,108)(94,107)(95,106)(96,140)(97,139)(98,138)(99,137)(100,136)(101,135)(102,134)(103,133)(104,132)(105,131)>;
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), (1,88,40,113)(2,87,41,112)(3,86,42,111)(4,85,43,110)(5,84,44,109)(6,83,45,108)(7,82,46,107)(8,81,47,106)(9,80,48,140)(10,79,49,139)(11,78,50,138)(12,77,51,137)(13,76,52,136)(14,75,53,135)(15,74,54,134)(16,73,55,133)(17,72,56,132)(18,71,57,131)(19,105,58,130)(20,104,59,129)(21,103,60,128)(22,102,61,127)(23,101,62,126)(24,100,63,125)(25,99,64,124)(26,98,65,123)(27,97,66,122)(28,96,67,121)(29,95,68,120)(30,94,69,119)(31,93,70,118)(32,92,36,117)(33,91,37,116)(34,90,38,115)(35,89,39,114), (2,35)(3,34)(4,33)(5,32)(6,31)(7,30)(8,29)(9,28)(10,27)(11,26)(12,25)(13,24)(14,23)(15,22)(16,21)(17,20)(18,19)(36,44)(37,43)(38,42)(39,41)(45,70)(46,69)(47,68)(48,67)(49,66)(50,65)(51,64)(52,63)(53,62)(54,61)(55,60)(56,59)(57,58)(71,130)(72,129)(73,128)(74,127)(75,126)(76,125)(77,124)(78,123)(79,122)(80,121)(81,120)(82,119)(83,118)(84,117)(85,116)(86,115)(87,114)(88,113)(89,112)(90,111)(91,110)(92,109)(93,108)(94,107)(95,106)(96,140)(97,139)(98,138)(99,137)(100,136)(101,135)(102,134)(103,133)(104,132)(105,131) );
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)], [(1,88,40,113),(2,87,41,112),(3,86,42,111),(4,85,43,110),(5,84,44,109),(6,83,45,108),(7,82,46,107),(8,81,47,106),(9,80,48,140),(10,79,49,139),(11,78,50,138),(12,77,51,137),(13,76,52,136),(14,75,53,135),(15,74,54,134),(16,73,55,133),(17,72,56,132),(18,71,57,131),(19,105,58,130),(20,104,59,129),(21,103,60,128),(22,102,61,127),(23,101,62,126),(24,100,63,125),(25,99,64,124),(26,98,65,123),(27,97,66,122),(28,96,67,121),(29,95,68,120),(30,94,69,119),(31,93,70,118),(32,92,36,117),(33,91,37,116),(34,90,38,115),(35,89,39,114)], [(2,35),(3,34),(4,33),(5,32),(6,31),(7,30),(8,29),(9,28),(10,27),(11,26),(12,25),(13,24),(14,23),(15,22),(16,21),(17,20),(18,19),(36,44),(37,43),(38,42),(39,41),(45,70),(46,69),(47,68),(48,67),(49,66),(50,65),(51,64),(52,63),(53,62),(54,61),(55,60),(56,59),(57,58),(71,130),(72,129),(73,128),(74,127),(75,126),(76,125),(77,124),(78,123),(79,122),(80,121),(81,120),(82,119),(83,118),(84,117),(85,116),(86,115),(87,114),(88,113),(89,112),(90,111),(91,110),(92,109),(93,108),(94,107),(95,106),(96,140),(97,139),(98,138),(99,137),(100,136),(101,135),(102,134),(103,133),(104,132),(105,131)]])
73 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4 | 5A | 5B | 7A | 7B | 7C | 10A | ··· | 10F | 14A | ··· | 14I | 35A | ··· | 35L | 70A | ··· | 70AJ |
| order | 1 | 2 | 2 | 2 | 4 | 5 | 5 | 7 | 7 | 7 | 10 | ··· | 10 | 14 | ··· | 14 | 35 | ··· | 35 | 70 | ··· | 70 |
| size | 1 | 1 | 2 | 70 | 70 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
73 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | D4 | D5 | D7 | D10 | D14 | C5⋊D4 | C7⋊D4 | D35 | D70 | C35⋊7D4 |
| kernel | C35⋊7D4 | Dic35 | D70 | C2×C70 | C35 | C2×C14 | C2×C10 | C14 | C10 | C7 | C5 | C22 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 2 | 3 | 2 | 3 | 4 | 6 | 12 | 12 | 24 |
Matrix representation of C35⋊7D4 ►in GL2(𝔽281) generated by
| 111 | 215 |
| 66 | 100 |
| 127 | 162 |
| 230 | 154 |
| 1 | 0 |
| 47 | 280 |
G:=sub<GL(2,GF(281))| [111,66,215,100],[127,230,162,154],[1,47,0,280] >;
C35⋊7D4 in GAP, Magma, Sage, TeX
C_{35}\rtimes_7D_4 % in TeX
G:=Group("C35:7D4"); // GroupNames label
G:=SmallGroup(280,28);
// by ID
G=gap.SmallGroup(280,28);
# by ID
G:=PCGroup([5,-2,-2,-2,-5,-7,61,643,6004]);
// Polycyclic
G:=Group<a,b,c|a^35=b^4=c^2=1,b*a*b^-1=c*a*c=a^-1,c*b*c=b^-1>;
// generators/relations
Export