direct product, metacyclic, nilpotent (class 2), monomial, 2-elementary
Aliases: D4×C35, C4⋊C70, C22⋊C70, C20⋊3C14, C28⋊3C10, C140⋊7C2, C70.23C22, (C2×C70)⋊1C2, (C2×C14)⋊1C10, (C2×C10)⋊1C14, C2.1(C2×C70), C14.6(C2×C10), C10.6(C2×C14), SmallGroup(280,30)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D4×C35
G = < a,b,c | a35=b4=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of D4×C35
►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 52 92 115)(2 53 93 116)(3 54 94 117)(4 55 95 118)(5 56 96 119)(6 57 97 120)(7 58 98 121)(8 59 99 122)(9 60 100 123)(10 61 101 124)(11 62 102 125)(12 63 103 126)(13 64 104 127)(14 65 105 128)(15 66 71 129)(16 67 72 130)(17 68 73 131)(18 69 74 132)(19 70 75 133)(20 36 76 134)(21 37 77 135)(22 38 78 136)(23 39 79 137)(24 40 80 138)(25 41 81 139)(26 42 82 140)(27 43 83 106)(28 44 84 107)(29 45 85 108)(30 46 86 109)(31 47 87 110)(32 48 88 111)(33 49 89 112)(34 50 90 113)(35 51 91 114)
(1 115)(2 116)(3 117)(4 118)(5 119)(6 120)(7 121)(8 122)(9 123)(10 124)(11 125)(12 126)(13 127)(14 128)(15 129)(16 130)(17 131)(18 132)(19 133)(20 134)(21 135)(22 136)(23 137)(24 138)(25 139)(26 140)(27 106)(28 107)(29 108)(30 109)(31 110)(32 111)(33 112)(34 113)(35 114)(36 76)(37 77)(38 78)(39 79)(40 80)(41 81)(42 82)(43 83)(44 84)(45 85)(46 86)(47 87)(48 88)(49 89)(50 90)(51 91)(52 92)(53 93)(54 94)(55 95)(56 96)(57 97)(58 98)(59 99)(60 100)(61 101)(62 102)(63 103)(64 104)(65 105)(66 71)(67 72)(68 73)(69 74)(70 75)
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), (1,52,92,115)(2,53,93,116)(3,54,94,117)(4,55,95,118)(5,56,96,119)(6,57,97,120)(7,58,98,121)(8,59,99,122)(9,60,100,123)(10,61,101,124)(11,62,102,125)(12,63,103,126)(13,64,104,127)(14,65,105,128)(15,66,71,129)(16,67,72,130)(17,68,73,131)(18,69,74,132)(19,70,75,133)(20,36,76,134)(21,37,77,135)(22,38,78,136)(23,39,79,137)(24,40,80,138)(25,41,81,139)(26,42,82,140)(27,43,83,106)(28,44,84,107)(29,45,85,108)(30,46,86,109)(31,47,87,110)(32,48,88,111)(33,49,89,112)(34,50,90,113)(35,51,91,114), (1,115)(2,116)(3,117)(4,118)(5,119)(6,120)(7,121)(8,122)(9,123)(10,124)(11,125)(12,126)(13,127)(14,128)(15,129)(16,130)(17,131)(18,132)(19,133)(20,134)(21,135)(22,136)(23,137)(24,138)(25,139)(26,140)(27,106)(28,107)(29,108)(30,109)(31,110)(32,111)(33,112)(34,113)(35,114)(36,76)(37,77)(38,78)(39,79)(40,80)(41,81)(42,82)(43,83)(44,84)(45,85)(46,86)(47,87)(48,88)(49,89)(50,90)(51,91)(52,92)(53,93)(54,94)(55,95)(56,96)(57,97)(58,98)(59,99)(60,100)(61,101)(62,102)(63,103)(64,104)(65,105)(66,71)(67,72)(68,73)(69,74)(70,75)>;
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), (1,52,92,115)(2,53,93,116)(3,54,94,117)(4,55,95,118)(5,56,96,119)(6,57,97,120)(7,58,98,121)(8,59,99,122)(9,60,100,123)(10,61,101,124)(11,62,102,125)(12,63,103,126)(13,64,104,127)(14,65,105,128)(15,66,71,129)(16,67,72,130)(17,68,73,131)(18,69,74,132)(19,70,75,133)(20,36,76,134)(21,37,77,135)(22,38,78,136)(23,39,79,137)(24,40,80,138)(25,41,81,139)(26,42,82,140)(27,43,83,106)(28,44,84,107)(29,45,85,108)(30,46,86,109)(31,47,87,110)(32,48,88,111)(33,49,89,112)(34,50,90,113)(35,51,91,114), (1,115)(2,116)(3,117)(4,118)(5,119)(6,120)(7,121)(8,122)(9,123)(10,124)(11,125)(12,126)(13,127)(14,128)(15,129)(16,130)(17,131)(18,132)(19,133)(20,134)(21,135)(22,136)(23,137)(24,138)(25,139)(26,140)(27,106)(28,107)(29,108)(30,109)(31,110)(32,111)(33,112)(34,113)(35,114)(36,76)(37,77)(38,78)(39,79)(40,80)(41,81)(42,82)(43,83)(44,84)(45,85)(46,86)(47,87)(48,88)(49,89)(50,90)(51,91)(52,92)(53,93)(54,94)(55,95)(56,96)(57,97)(58,98)(59,99)(60,100)(61,101)(62,102)(63,103)(64,104)(65,105)(66,71)(67,72)(68,73)(69,74)(70,75) );
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)], [(1,52,92,115),(2,53,93,116),(3,54,94,117),(4,55,95,118),(5,56,96,119),(6,57,97,120),(7,58,98,121),(8,59,99,122),(9,60,100,123),(10,61,101,124),(11,62,102,125),(12,63,103,126),(13,64,104,127),(14,65,105,128),(15,66,71,129),(16,67,72,130),(17,68,73,131),(18,69,74,132),(19,70,75,133),(20,36,76,134),(21,37,77,135),(22,38,78,136),(23,39,79,137),(24,40,80,138),(25,41,81,139),(26,42,82,140),(27,43,83,106),(28,44,84,107),(29,45,85,108),(30,46,86,109),(31,47,87,110),(32,48,88,111),(33,49,89,112),(34,50,90,113),(35,51,91,114)], [(1,115),(2,116),(3,117),(4,118),(5,119),(6,120),(7,121),(8,122),(9,123),(10,124),(11,125),(12,126),(13,127),(14,128),(15,129),(16,130),(17,131),(18,132),(19,133),(20,134),(21,135),(22,136),(23,137),(24,138),(25,139),(26,140),(27,106),(28,107),(29,108),(30,109),(31,110),(32,111),(33,112),(34,113),(35,114),(36,76),(37,77),(38,78),(39,79),(40,80),(41,81),(42,82),(43,83),(44,84),(45,85),(46,86),(47,87),(48,88),(49,89),(50,90),(51,91),(52,92),(53,93),(54,94),(55,95),(56,96),(57,97),(58,98),(59,99),(60,100),(61,101),(62,102),(63,103),(64,104),(65,105),(66,71),(67,72),(68,73),(69,74),(70,75)]])
175 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4 | 5A | 5B | 5C | 5D | 7A | ··· | 7F | 10A | 10B | 10C | 10D | 10E | ··· | 10L | 14A | ··· | 14F | 14G | ··· | 14R | 20A | 20B | 20C | 20D | 28A | ··· | 28F | 35A | ··· | 35X | 70A | ··· | 70X | 70Y | ··· | 70BT | 140A | ··· | 140X |
| order | 1 | 2 | 2 | 2 | 4 | 5 | 5 | 5 | 5 | 7 | ··· | 7 | 10 | 10 | 10 | 10 | 10 | ··· | 10 | 14 | ··· | 14 | 14 | ··· | 14 | 20 | 20 | 20 | 20 | 28 | ··· | 28 | 35 | ··· | 35 | 70 | ··· | 70 | 70 | ··· | 70 | 140 | ··· | 140 |
| size | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 |
175 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | ||||||||||||
| image | C1 | C2 | C2 | C5 | C7 | C10 | C10 | C14 | C14 | C35 | C70 | C70 | D4 | C5×D4 | C7×D4 | D4×C35 |
| kernel | D4×C35 | C140 | C2×C70 | C7×D4 | C5×D4 | C28 | C2×C14 | C20 | C2×C10 | D4 | C4 | C22 | C35 | C7 | C5 | C1 |
| # reps | 1 | 1 | 2 | 4 | 6 | 4 | 8 | 6 | 12 | 24 | 24 | 48 | 1 | 4 | 6 | 24 |
Matrix representation of D4×C35 ►in GL4(𝔽281) generated by
| 232 | 0 | 0 | 0 |
| 0 | 79 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 38 | 279 |
| 0 | 0 | 20 | 243 |
| 280 | 0 | 0 | 0 |
| 0 | 280 | 0 | 0 |
| 0 | 0 | 38 | 279 |
| 0 | 0 | 19 | 243 |
G:=sub<GL(4,GF(281))| [232,0,0,0,0,79,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,38,20,0,0,279,243],[280,0,0,0,0,280,0,0,0,0,38,19,0,0,279,243] >;
D4×C35 in GAP, Magma, Sage, TeX
D_4\times C_{35} % in TeX
G:=Group("D4xC35"); // GroupNames label
G:=SmallGroup(280,30);
// by ID
G=gap.SmallGroup(280,30);
# by ID
G:=PCGroup([5,-2,-2,-5,-7,-2,1421]);
// Polycyclic
G:=Group<a,b,c|a^35=b^4=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
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