direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C4×D35, C28⋊2D5, C20⋊2D7, C140⋊2C2, C2.1D70, D70.2C2, C10.9D14, C14.9D10, Dic35⋊5C2, C70.9C22, C5⋊3(C4×D7), C7⋊2(C4×D5), C35⋊7(C2×C4), SmallGroup(280,25)
Series: Derived ►Chief ►Lower central ►Upper central
| C35 — C4×D35 |
Generators and relations for C4×D35
G = < a,b,c | a4=b35=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of C4×D35
►On 140 points
Generators in S140
(1 119 48 102)(2 120 49 103)(3 121 50 104)(4 122 51 105)(5 123 52 71)(6 124 53 72)(7 125 54 73)(8 126 55 74)(9 127 56 75)(10 128 57 76)(11 129 58 77)(12 130 59 78)(13 131 60 79)(14 132 61 80)(15 133 62 81)(16 134 63 82)(17 135 64 83)(18 136 65 84)(19 137 66 85)(20 138 67 86)(21 139 68 87)(22 140 69 88)(23 106 70 89)(24 107 36 90)(25 108 37 91)(26 109 38 92)(27 110 39 93)(28 111 40 94)(29 112 41 95)(30 113 42 96)(31 114 43 97)(32 115 44 98)(33 116 45 99)(34 117 46 100)(35 118 47 101)
(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 47)(2 46)(3 45)(4 44)(5 43)(6 42)(7 41)(8 40)(9 39)(10 38)(11 37)(12 36)(13 70)(14 69)(15 68)(16 67)(17 66)(18 65)(19 64)(20 63)(21 62)(22 61)(23 60)(24 59)(25 58)(26 57)(27 56)(28 55)(29 54)(30 53)(31 52)(32 51)(33 50)(34 49)(35 48)(71 114)(72 113)(73 112)(74 111)(75 110)(76 109)(77 108)(78 107)(79 106)(80 140)(81 139)(82 138)(83 137)(84 136)(85 135)(86 134)(87 133)(88 132)(89 131)(90 130)(91 129)(92 128)(93 127)(94 126)(95 125)(96 124)(97 123)(98 122)(99 121)(100 120)(101 119)(102 118)(103 117)(104 116)(105 115)
G:=sub<Sym(140)| (1,119,48,102)(2,120,49,103)(3,121,50,104)(4,122,51,105)(5,123,52,71)(6,124,53,72)(7,125,54,73)(8,126,55,74)(9,127,56,75)(10,128,57,76)(11,129,58,77)(12,130,59,78)(13,131,60,79)(14,132,61,80)(15,133,62,81)(16,134,63,82)(17,135,64,83)(18,136,65,84)(19,137,66,85)(20,138,67,86)(21,139,68,87)(22,140,69,88)(23,106,70,89)(24,107,36,90)(25,108,37,91)(26,109,38,92)(27,110,39,93)(28,111,40,94)(29,112,41,95)(30,113,42,96)(31,114,43,97)(32,115,44,98)(33,116,45,99)(34,117,46,100)(35,118,47,101), (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), (1,47)(2,46)(3,45)(4,44)(5,43)(6,42)(7,41)(8,40)(9,39)(10,38)(11,37)(12,36)(13,70)(14,69)(15,68)(16,67)(17,66)(18,65)(19,64)(20,63)(21,62)(22,61)(23,60)(24,59)(25,58)(26,57)(27,56)(28,55)(29,54)(30,53)(31,52)(32,51)(33,50)(34,49)(35,48)(71,114)(72,113)(73,112)(74,111)(75,110)(76,109)(77,108)(78,107)(79,106)(80,140)(81,139)(82,138)(83,137)(84,136)(85,135)(86,134)(87,133)(88,132)(89,131)(90,130)(91,129)(92,128)(93,127)(94,126)(95,125)(96,124)(97,123)(98,122)(99,121)(100,120)(101,119)(102,118)(103,117)(104,116)(105,115)>;
G:=Group( (1,119,48,102)(2,120,49,103)(3,121,50,104)(4,122,51,105)(5,123,52,71)(6,124,53,72)(7,125,54,73)(8,126,55,74)(9,127,56,75)(10,128,57,76)(11,129,58,77)(12,130,59,78)(13,131,60,79)(14,132,61,80)(15,133,62,81)(16,134,63,82)(17,135,64,83)(18,136,65,84)(19,137,66,85)(20,138,67,86)(21,139,68,87)(22,140,69,88)(23,106,70,89)(24,107,36,90)(25,108,37,91)(26,109,38,92)(27,110,39,93)(28,111,40,94)(29,112,41,95)(30,113,42,96)(31,114,43,97)(32,115,44,98)(33,116,45,99)(34,117,46,100)(35,118,47,101), (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), (1,47)(2,46)(3,45)(4,44)(5,43)(6,42)(7,41)(8,40)(9,39)(10,38)(11,37)(12,36)(13,70)(14,69)(15,68)(16,67)(17,66)(18,65)(19,64)(20,63)(21,62)(22,61)(23,60)(24,59)(25,58)(26,57)(27,56)(28,55)(29,54)(30,53)(31,52)(32,51)(33,50)(34,49)(35,48)(71,114)(72,113)(73,112)(74,111)(75,110)(76,109)(77,108)(78,107)(79,106)(80,140)(81,139)(82,138)(83,137)(84,136)(85,135)(86,134)(87,133)(88,132)(89,131)(90,130)(91,129)(92,128)(93,127)(94,126)(95,125)(96,124)(97,123)(98,122)(99,121)(100,120)(101,119)(102,118)(103,117)(104,116)(105,115) );
G=PermutationGroup([[(1,119,48,102),(2,120,49,103),(3,121,50,104),(4,122,51,105),(5,123,52,71),(6,124,53,72),(7,125,54,73),(8,126,55,74),(9,127,56,75),(10,128,57,76),(11,129,58,77),(12,130,59,78),(13,131,60,79),(14,132,61,80),(15,133,62,81),(16,134,63,82),(17,135,64,83),(18,136,65,84),(19,137,66,85),(20,138,67,86),(21,139,68,87),(22,140,69,88),(23,106,70,89),(24,107,36,90),(25,108,37,91),(26,109,38,92),(27,110,39,93),(28,111,40,94),(29,112,41,95),(30,113,42,96),(31,114,43,97),(32,115,44,98),(33,116,45,99),(34,117,46,100),(35,118,47,101)], [(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)], [(1,47),(2,46),(3,45),(4,44),(5,43),(6,42),(7,41),(8,40),(9,39),(10,38),(11,37),(12,36),(13,70),(14,69),(15,68),(16,67),(17,66),(18,65),(19,64),(20,63),(21,62),(22,61),(23,60),(24,59),(25,58),(26,57),(27,56),(28,55),(29,54),(30,53),(31,52),(32,51),(33,50),(34,49),(35,48),(71,114),(72,113),(73,112),(74,111),(75,110),(76,109),(77,108),(78,107),(79,106),(80,140),(81,139),(82,138),(83,137),(84,136),(85,135),(86,134),(87,133),(88,132),(89,131),(90,130),(91,129),(92,128),(93,127),(94,126),(95,125),(96,124),(97,123),(98,122),(99,121),(100,120),(101,119),(102,118),(103,117),(104,116),(105,115)]])
76 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 5A | 5B | 7A | 7B | 7C | 10A | 10B | 14A | 14B | 14C | 20A | 20B | 20C | 20D | 28A | ··· | 28F | 35A | ··· | 35L | 70A | ··· | 70L | 140A | ··· | 140X |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 5 | 5 | 7 | 7 | 7 | 10 | 10 | 14 | 14 | 14 | 20 | 20 | 20 | 20 | 28 | ··· | 28 | 35 | ··· | 35 | 70 | ··· | 70 | 140 | ··· | 140 |
| size | 1 | 1 | 35 | 35 | 1 | 1 | 35 | 35 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
76 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | ||||
| image | C1 | C2 | C2 | C2 | C4 | D5 | D7 | D10 | D14 | C4×D5 | C4×D7 | D35 | D70 | C4×D35 |
| kernel | C4×D35 | Dic35 | C140 | D70 | D35 | C28 | C20 | C14 | C10 | C7 | C5 | C4 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 4 | 2 | 3 | 2 | 3 | 4 | 6 | 12 | 12 | 24 |
Matrix representation of C4×D35 ►in GL3(𝔽281) generated by
| 228 | 0 | 0 |
| 0 | 280 | 0 |
| 0 | 0 | 280 |
| 1 | 0 | 0 |
| 0 | 243 | 232 |
| 0 | 49 | 4 |
| 1 | 0 | 0 |
| 0 | 243 | 232 |
| 0 | 104 | 38 |
G:=sub<GL(3,GF(281))| [228,0,0,0,280,0,0,0,280],[1,0,0,0,243,49,0,232,4],[1,0,0,0,243,104,0,232,38] >;
C4×D35 in GAP, Magma, Sage, TeX
C_4\times D_{35} % in TeX
G:=Group("C4xD35"); // GroupNames label
G:=SmallGroup(280,25);
// by ID
G=gap.SmallGroup(280,25);
# by ID
G:=PCGroup([5,-2,-2,-2,-5,-7,26,643,6004]);
// Polycyclic
G:=Group<a,b,c|a^4=b^35=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export