direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C5×C7⋊D4, C35⋊8D4, Dic7⋊C10, D14⋊2C10, C10.17D14, C70.17C22, C7⋊2(C5×D4), C22⋊(C5×D7), (C2×C70)⋊4C2, (C2×C10)⋊1D7, (C2×C14)⋊2C10, (C10×D7)⋊5C2, C2.5(C10×D7), C14.5(C2×C10), (C5×Dic7)⋊4C2, SmallGroup(280,18)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5×C7⋊D4
G = < a,b,c,d | a5=b7=c4=d2=1, ab=ba, ac=ca, ad=da, cbc-1=dbd=b-1, dcd=c-1 >
Smallest permutation representation of C5×C7⋊D4
►On 140 points
Generators in S140
(1 57 43 29 15)(2 58 44 30 16)(3 59 45 31 17)(4 60 46 32 18)(5 61 47 33 19)(6 62 48 34 20)(7 63 49 35 21)(8 64 50 36 22)(9 65 51 37 23)(10 66 52 38 24)(11 67 53 39 25)(12 68 54 40 26)(13 69 55 41 27)(14 70 56 42 28)(71 127 113 99 85)(72 128 114 100 86)(73 129 115 101 87)(74 130 116 102 88)(75 131 117 103 89)(76 132 118 104 90)(77 133 119 105 91)(78 134 120 106 92)(79 135 121 107 93)(80 136 122 108 94)(81 137 123 109 95)(82 138 124 110 96)(83 139 125 111 97)(84 140 126 112 98)
(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 71 8 78)(2 77 9 84)(3 76 10 83)(4 75 11 82)(5 74 12 81)(6 73 13 80)(7 72 14 79)(15 85 22 92)(16 91 23 98)(17 90 24 97)(18 89 25 96)(19 88 26 95)(20 87 27 94)(21 86 28 93)(29 99 36 106)(30 105 37 112)(31 104 38 111)(32 103 39 110)(33 102 40 109)(34 101 41 108)(35 100 42 107)(43 113 50 120)(44 119 51 126)(45 118 52 125)(46 117 53 124)(47 116 54 123)(48 115 55 122)(49 114 56 121)(57 127 64 134)(58 133 65 140)(59 132 66 139)(60 131 67 138)(61 130 68 137)(62 129 69 136)(63 128 70 135)
(2 7)(3 6)(4 5)(9 14)(10 13)(11 12)(16 21)(17 20)(18 19)(23 28)(24 27)(25 26)(30 35)(31 34)(32 33)(37 42)(38 41)(39 40)(44 49)(45 48)(46 47)(51 56)(52 55)(53 54)(58 63)(59 62)(60 61)(65 70)(66 69)(67 68)(71 78)(72 84)(73 83)(74 82)(75 81)(76 80)(77 79)(85 92)(86 98)(87 97)(88 96)(89 95)(90 94)(91 93)(99 106)(100 112)(101 111)(102 110)(103 109)(104 108)(105 107)(113 120)(114 126)(115 125)(116 124)(117 123)(118 122)(119 121)(127 134)(128 140)(129 139)(130 138)(131 137)(132 136)(133 135)
G:=sub<Sym(140)| (1,57,43,29,15)(2,58,44,30,16)(3,59,45,31,17)(4,60,46,32,18)(5,61,47,33,19)(6,62,48,34,20)(7,63,49,35,21)(8,64,50,36,22)(9,65,51,37,23)(10,66,52,38,24)(11,67,53,39,25)(12,68,54,40,26)(13,69,55,41,27)(14,70,56,42,28)(71,127,113,99,85)(72,128,114,100,86)(73,129,115,101,87)(74,130,116,102,88)(75,131,117,103,89)(76,132,118,104,90)(77,133,119,105,91)(78,134,120,106,92)(79,135,121,107,93)(80,136,122,108,94)(81,137,123,109,95)(82,138,124,110,96)(83,139,125,111,97)(84,140,126,112,98), (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), (1,71,8,78)(2,77,9,84)(3,76,10,83)(4,75,11,82)(5,74,12,81)(6,73,13,80)(7,72,14,79)(15,85,22,92)(16,91,23,98)(17,90,24,97)(18,89,25,96)(19,88,26,95)(20,87,27,94)(21,86,28,93)(29,99,36,106)(30,105,37,112)(31,104,38,111)(32,103,39,110)(33,102,40,109)(34,101,41,108)(35,100,42,107)(43,113,50,120)(44,119,51,126)(45,118,52,125)(46,117,53,124)(47,116,54,123)(48,115,55,122)(49,114,56,121)(57,127,64,134)(58,133,65,140)(59,132,66,139)(60,131,67,138)(61,130,68,137)(62,129,69,136)(63,128,70,135), (2,7)(3,6)(4,5)(9,14)(10,13)(11,12)(16,21)(17,20)(18,19)(23,28)(24,27)(25,26)(30,35)(31,34)(32,33)(37,42)(38,41)(39,40)(44,49)(45,48)(46,47)(51,56)(52,55)(53,54)(58,63)(59,62)(60,61)(65,70)(66,69)(67,68)(71,78)(72,84)(73,83)(74,82)(75,81)(76,80)(77,79)(85,92)(86,98)(87,97)(88,96)(89,95)(90,94)(91,93)(99,106)(100,112)(101,111)(102,110)(103,109)(104,108)(105,107)(113,120)(114,126)(115,125)(116,124)(117,123)(118,122)(119,121)(127,134)(128,140)(129,139)(130,138)(131,137)(132,136)(133,135)>;
G:=Group( (1,57,43,29,15)(2,58,44,30,16)(3,59,45,31,17)(4,60,46,32,18)(5,61,47,33,19)(6,62,48,34,20)(7,63,49,35,21)(8,64,50,36,22)(9,65,51,37,23)(10,66,52,38,24)(11,67,53,39,25)(12,68,54,40,26)(13,69,55,41,27)(14,70,56,42,28)(71,127,113,99,85)(72,128,114,100,86)(73,129,115,101,87)(74,130,116,102,88)(75,131,117,103,89)(76,132,118,104,90)(77,133,119,105,91)(78,134,120,106,92)(79,135,121,107,93)(80,136,122,108,94)(81,137,123,109,95)(82,138,124,110,96)(83,139,125,111,97)(84,140,126,112,98), (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), (1,71,8,78)(2,77,9,84)(3,76,10,83)(4,75,11,82)(5,74,12,81)(6,73,13,80)(7,72,14,79)(15,85,22,92)(16,91,23,98)(17,90,24,97)(18,89,25,96)(19,88,26,95)(20,87,27,94)(21,86,28,93)(29,99,36,106)(30,105,37,112)(31,104,38,111)(32,103,39,110)(33,102,40,109)(34,101,41,108)(35,100,42,107)(43,113,50,120)(44,119,51,126)(45,118,52,125)(46,117,53,124)(47,116,54,123)(48,115,55,122)(49,114,56,121)(57,127,64,134)(58,133,65,140)(59,132,66,139)(60,131,67,138)(61,130,68,137)(62,129,69,136)(63,128,70,135), (2,7)(3,6)(4,5)(9,14)(10,13)(11,12)(16,21)(17,20)(18,19)(23,28)(24,27)(25,26)(30,35)(31,34)(32,33)(37,42)(38,41)(39,40)(44,49)(45,48)(46,47)(51,56)(52,55)(53,54)(58,63)(59,62)(60,61)(65,70)(66,69)(67,68)(71,78)(72,84)(73,83)(74,82)(75,81)(76,80)(77,79)(85,92)(86,98)(87,97)(88,96)(89,95)(90,94)(91,93)(99,106)(100,112)(101,111)(102,110)(103,109)(104,108)(105,107)(113,120)(114,126)(115,125)(116,124)(117,123)(118,122)(119,121)(127,134)(128,140)(129,139)(130,138)(131,137)(132,136)(133,135) );
G=PermutationGroup([[(1,57,43,29,15),(2,58,44,30,16),(3,59,45,31,17),(4,60,46,32,18),(5,61,47,33,19),(6,62,48,34,20),(7,63,49,35,21),(8,64,50,36,22),(9,65,51,37,23),(10,66,52,38,24),(11,67,53,39,25),(12,68,54,40,26),(13,69,55,41,27),(14,70,56,42,28),(71,127,113,99,85),(72,128,114,100,86),(73,129,115,101,87),(74,130,116,102,88),(75,131,117,103,89),(76,132,118,104,90),(77,133,119,105,91),(78,134,120,106,92),(79,135,121,107,93),(80,136,122,108,94),(81,137,123,109,95),(82,138,124,110,96),(83,139,125,111,97),(84,140,126,112,98)], [(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)], [(1,71,8,78),(2,77,9,84),(3,76,10,83),(4,75,11,82),(5,74,12,81),(6,73,13,80),(7,72,14,79),(15,85,22,92),(16,91,23,98),(17,90,24,97),(18,89,25,96),(19,88,26,95),(20,87,27,94),(21,86,28,93),(29,99,36,106),(30,105,37,112),(31,104,38,111),(32,103,39,110),(33,102,40,109),(34,101,41,108),(35,100,42,107),(43,113,50,120),(44,119,51,126),(45,118,52,125),(46,117,53,124),(47,116,54,123),(48,115,55,122),(49,114,56,121),(57,127,64,134),(58,133,65,140),(59,132,66,139),(60,131,67,138),(61,130,68,137),(62,129,69,136),(63,128,70,135)], [(2,7),(3,6),(4,5),(9,14),(10,13),(11,12),(16,21),(17,20),(18,19),(23,28),(24,27),(25,26),(30,35),(31,34),(32,33),(37,42),(38,41),(39,40),(44,49),(45,48),(46,47),(51,56),(52,55),(53,54),(58,63),(59,62),(60,61),(65,70),(66,69),(67,68),(71,78),(72,84),(73,83),(74,82),(75,81),(76,80),(77,79),(85,92),(86,98),(87,97),(88,96),(89,95),(90,94),(91,93),(99,106),(100,112),(101,111),(102,110),(103,109),(104,108),(105,107),(113,120),(114,126),(115,125),(116,124),(117,123),(118,122),(119,121),(127,134),(128,140),(129,139),(130,138),(131,137),(132,136),(133,135)]])
85 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4 | 5A | 5B | 5C | 5D | 7A | 7B | 7C | 10A | 10B | 10C | 10D | 10E | 10F | 10G | 10H | 10I | 10J | 10K | 10L | 14A | ··· | 14I | 20A | 20B | 20C | 20D | 35A | ··· | 35L | 70A | ··· | 70AJ |
| order | 1 | 2 | 2 | 2 | 4 | 5 | 5 | 5 | 5 | 7 | 7 | 7 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 14 | ··· | 14 | 20 | 20 | 20 | 20 | 35 | ··· | 35 | 70 | ··· | 70 |
| size | 1 | 1 | 2 | 14 | 14 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 14 | 14 | 14 | 14 | 2 | ··· | 2 | 14 | 14 | 14 | 14 | 2 | ··· | 2 | 2 | ··· | 2 |
85 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | |||||||||
| image | C1 | C2 | C2 | C2 | C5 | C10 | C10 | C10 | D4 | D7 | D14 | C5×D4 | C7⋊D4 | C5×D7 | C10×D7 | C5×C7⋊D4 |
| kernel | C5×C7⋊D4 | C5×Dic7 | C10×D7 | C2×C70 | C7⋊D4 | Dic7 | D14 | C2×C14 | C35 | C2×C10 | C10 | C7 | C5 | C22 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 4 | 1 | 3 | 3 | 4 | 6 | 12 | 12 | 24 |
Matrix representation of C5×C7⋊D4 ►in GL2(𝔽281) generated by
| 153 | 0 |
| 0 | 153 |
| 240 | 280 |
| 1 | 0 |
| 25 | 152 |
| 251 | 256 |
| 1 | 0 |
| 240 | 280 |
G:=sub<GL(2,GF(281))| [153,0,0,153],[240,1,280,0],[25,251,152,256],[1,240,0,280] >;
C5×C7⋊D4 in GAP, Magma, Sage, TeX
C_5\times C_7\rtimes D_4
% in TeX
G:=Group("C5xC7:D4"); // GroupNames label
G:=SmallGroup(280,18);
// by ID
G=gap.SmallGroup(280,18);
# by ID
G:=PCGroup([5,-2,-2,-5,-2,-7,221,6004]);
// Polycyclic
G:=Group<a,b,c,d|a^5=b^7=c^4=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations
Export