direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C7×C5⋊D4, C35⋊9D4, Dic5⋊C14, D10⋊2C14, C14.17D10, C70.22C22, C5⋊2(C7×D4), C22⋊(C7×D5), (C2×C70)⋊6C2, (C2×C14)⋊1D5, (C2×C10)⋊2C14, (D5×C14)⋊5C2, C2.5(D5×C14), C10.5(C2×C14), (C7×Dic5)⋊4C2, SmallGroup(280,23)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C7×C5⋊D4
G = < a,b,c,d | a7=b5=c4=d2=1, ab=ba, ac=ca, ad=da, cbc-1=dbd=b-1, dcd=c-1 >
Smallest permutation representation of C7×C5⋊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 97 89 24 76)(2 98 90 25 77)(3 92 91 26 71)(4 93 85 27 72)(5 94 86 28 73)(6 95 87 22 74)(7 96 88 23 75)(8 133 19 51 122)(9 127 20 52 123)(10 128 21 53 124)(11 129 15 54 125)(12 130 16 55 126)(13 131 17 56 120)(14 132 18 50 121)(29 61 40 111 136)(30 62 41 112 137)(31 63 42 106 138)(32 57 36 107 139)(33 58 37 108 140)(34 59 38 109 134)(35 60 39 110 135)(43 66 118 100 79)(44 67 119 101 80)(45 68 113 102 81)(46 69 114 103 82)(47 70 115 104 83)(48 64 116 105 84)(49 65 117 99 78)
(1 132 78 39)(2 133 79 40)(3 127 80 41)(4 128 81 42)(5 129 82 36)(6 130 83 37)(7 131 84 38)(8 43 61 98)(9 44 62 92)(10 45 63 93)(11 46 57 94)(12 47 58 95)(13 48 59 96)(14 49 60 97)(15 103 107 73)(16 104 108 74)(17 105 109 75)(18 99 110 76)(19 100 111 77)(20 101 112 71)(21 102 106 72)(22 55 115 140)(23 56 116 134)(24 50 117 135)(25 51 118 136)(26 52 119 137)(27 53 113 138)(28 54 114 139)(29 90 122 66)(30 91 123 67)(31 85 124 68)(32 86 125 69)(33 87 126 70)(34 88 120 64)(35 89 121 65)
(8 111)(9 112)(10 106)(11 107)(12 108)(13 109)(14 110)(15 57)(16 58)(17 59)(18 60)(19 61)(20 62)(21 63)(22 87)(23 88)(24 89)(25 90)(26 91)(27 85)(28 86)(29 51)(30 52)(31 53)(32 54)(33 55)(34 56)(35 50)(36 129)(37 130)(38 131)(39 132)(40 133)(41 127)(42 128)(43 100)(44 101)(45 102)(46 103)(47 104)(48 105)(49 99)(64 116)(65 117)(66 118)(67 119)(68 113)(69 114)(70 115)(71 92)(72 93)(73 94)(74 95)(75 96)(76 97)(77 98)(120 134)(121 135)(122 136)(123 137)(124 138)(125 139)(126 140)
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), (1,97,89,24,76)(2,98,90,25,77)(3,92,91,26,71)(4,93,85,27,72)(5,94,86,28,73)(6,95,87,22,74)(7,96,88,23,75)(8,133,19,51,122)(9,127,20,52,123)(10,128,21,53,124)(11,129,15,54,125)(12,130,16,55,126)(13,131,17,56,120)(14,132,18,50,121)(29,61,40,111,136)(30,62,41,112,137)(31,63,42,106,138)(32,57,36,107,139)(33,58,37,108,140)(34,59,38,109,134)(35,60,39,110,135)(43,66,118,100,79)(44,67,119,101,80)(45,68,113,102,81)(46,69,114,103,82)(47,70,115,104,83)(48,64,116,105,84)(49,65,117,99,78), (1,132,78,39)(2,133,79,40)(3,127,80,41)(4,128,81,42)(5,129,82,36)(6,130,83,37)(7,131,84,38)(8,43,61,98)(9,44,62,92)(10,45,63,93)(11,46,57,94)(12,47,58,95)(13,48,59,96)(14,49,60,97)(15,103,107,73)(16,104,108,74)(17,105,109,75)(18,99,110,76)(19,100,111,77)(20,101,112,71)(21,102,106,72)(22,55,115,140)(23,56,116,134)(24,50,117,135)(25,51,118,136)(26,52,119,137)(27,53,113,138)(28,54,114,139)(29,90,122,66)(30,91,123,67)(31,85,124,68)(32,86,125,69)(33,87,126,70)(34,88,120,64)(35,89,121,65), (8,111)(9,112)(10,106)(11,107)(12,108)(13,109)(14,110)(15,57)(16,58)(17,59)(18,60)(19,61)(20,62)(21,63)(22,87)(23,88)(24,89)(25,90)(26,91)(27,85)(28,86)(29,51)(30,52)(31,53)(32,54)(33,55)(34,56)(35,50)(36,129)(37,130)(38,131)(39,132)(40,133)(41,127)(42,128)(43,100)(44,101)(45,102)(46,103)(47,104)(48,105)(49,99)(64,116)(65,117)(66,118)(67,119)(68,113)(69,114)(70,115)(71,92)(72,93)(73,94)(74,95)(75,96)(76,97)(77,98)(120,134)(121,135)(122,136)(123,137)(124,138)(125,139)(126,140)>;
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), (1,97,89,24,76)(2,98,90,25,77)(3,92,91,26,71)(4,93,85,27,72)(5,94,86,28,73)(6,95,87,22,74)(7,96,88,23,75)(8,133,19,51,122)(9,127,20,52,123)(10,128,21,53,124)(11,129,15,54,125)(12,130,16,55,126)(13,131,17,56,120)(14,132,18,50,121)(29,61,40,111,136)(30,62,41,112,137)(31,63,42,106,138)(32,57,36,107,139)(33,58,37,108,140)(34,59,38,109,134)(35,60,39,110,135)(43,66,118,100,79)(44,67,119,101,80)(45,68,113,102,81)(46,69,114,103,82)(47,70,115,104,83)(48,64,116,105,84)(49,65,117,99,78), (1,132,78,39)(2,133,79,40)(3,127,80,41)(4,128,81,42)(5,129,82,36)(6,130,83,37)(7,131,84,38)(8,43,61,98)(9,44,62,92)(10,45,63,93)(11,46,57,94)(12,47,58,95)(13,48,59,96)(14,49,60,97)(15,103,107,73)(16,104,108,74)(17,105,109,75)(18,99,110,76)(19,100,111,77)(20,101,112,71)(21,102,106,72)(22,55,115,140)(23,56,116,134)(24,50,117,135)(25,51,118,136)(26,52,119,137)(27,53,113,138)(28,54,114,139)(29,90,122,66)(30,91,123,67)(31,85,124,68)(32,86,125,69)(33,87,126,70)(34,88,120,64)(35,89,121,65), (8,111)(9,112)(10,106)(11,107)(12,108)(13,109)(14,110)(15,57)(16,58)(17,59)(18,60)(19,61)(20,62)(21,63)(22,87)(23,88)(24,89)(25,90)(26,91)(27,85)(28,86)(29,51)(30,52)(31,53)(32,54)(33,55)(34,56)(35,50)(36,129)(37,130)(38,131)(39,132)(40,133)(41,127)(42,128)(43,100)(44,101)(45,102)(46,103)(47,104)(48,105)(49,99)(64,116)(65,117)(66,118)(67,119)(68,113)(69,114)(70,115)(71,92)(72,93)(73,94)(74,95)(75,96)(76,97)(77,98)(120,134)(121,135)(122,136)(123,137)(124,138)(125,139)(126,140) );
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)], [(1,97,89,24,76),(2,98,90,25,77),(3,92,91,26,71),(4,93,85,27,72),(5,94,86,28,73),(6,95,87,22,74),(7,96,88,23,75),(8,133,19,51,122),(9,127,20,52,123),(10,128,21,53,124),(11,129,15,54,125),(12,130,16,55,126),(13,131,17,56,120),(14,132,18,50,121),(29,61,40,111,136),(30,62,41,112,137),(31,63,42,106,138),(32,57,36,107,139),(33,58,37,108,140),(34,59,38,109,134),(35,60,39,110,135),(43,66,118,100,79),(44,67,119,101,80),(45,68,113,102,81),(46,69,114,103,82),(47,70,115,104,83),(48,64,116,105,84),(49,65,117,99,78)], [(1,132,78,39),(2,133,79,40),(3,127,80,41),(4,128,81,42),(5,129,82,36),(6,130,83,37),(7,131,84,38),(8,43,61,98),(9,44,62,92),(10,45,63,93),(11,46,57,94),(12,47,58,95),(13,48,59,96),(14,49,60,97),(15,103,107,73),(16,104,108,74),(17,105,109,75),(18,99,110,76),(19,100,111,77),(20,101,112,71),(21,102,106,72),(22,55,115,140),(23,56,116,134),(24,50,117,135),(25,51,118,136),(26,52,119,137),(27,53,113,138),(28,54,114,139),(29,90,122,66),(30,91,123,67),(31,85,124,68),(32,86,125,69),(33,87,126,70),(34,88,120,64),(35,89,121,65)], [(8,111),(9,112),(10,106),(11,107),(12,108),(13,109),(14,110),(15,57),(16,58),(17,59),(18,60),(19,61),(20,62),(21,63),(22,87),(23,88),(24,89),(25,90),(26,91),(27,85),(28,86),(29,51),(30,52),(31,53),(32,54),(33,55),(34,56),(35,50),(36,129),(37,130),(38,131),(39,132),(40,133),(41,127),(42,128),(43,100),(44,101),(45,102),(46,103),(47,104),(48,105),(49,99),(64,116),(65,117),(66,118),(67,119),(68,113),(69,114),(70,115),(71,92),(72,93),(73,94),(74,95),(75,96),(76,97),(77,98),(120,134),(121,135),(122,136),(123,137),(124,138),(125,139),(126,140)]])
91 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4 | 5A | 5B | 7A | ··· | 7F | 10A | ··· | 10F | 14A | ··· | 14F | 14G | ··· | 14L | 14M | ··· | 14R | 28A | ··· | 28F | 35A | ··· | 35L | 70A | ··· | 70AJ |
| order | 1 | 2 | 2 | 2 | 4 | 5 | 5 | 7 | ··· | 7 | 10 | ··· | 10 | 14 | ··· | 14 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 35 | ··· | 35 | 70 | ··· | 70 |
| size | 1 | 1 | 2 | 10 | 10 | 2 | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 10 | ··· | 10 | 10 | ··· | 10 | 2 | ··· | 2 | 2 | ··· | 2 |
91 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 | C7 | C14 | C14 | C14 | D4 | D5 | D10 | C5⋊D4 | C7×D4 | C7×D5 | D5×C14 | C7×C5⋊D4 |
| kernel | C7×C5⋊D4 | C7×Dic5 | D5×C14 | C2×C70 | C5⋊D4 | Dic5 | D10 | C2×C10 | C35 | C2×C14 | C14 | C7 | C5 | C22 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 6 | 6 | 6 | 6 | 1 | 2 | 2 | 4 | 6 | 12 | 12 | 24 |
Matrix representation of C7×C5⋊D4 ►in GL2(𝔽281) generated by
| 249 | 0 |
| 0 | 249 |
| 243 | 280 |
| 1 | 0 |
| 237 | 116 |
| 102 | 44 |
| 1 | 0 |
| 243 | 280 |
G:=sub<GL(2,GF(281))| [249,0,0,249],[243,1,280,0],[237,102,116,44],[1,243,0,280] >;
C7×C5⋊D4 in GAP, Magma, Sage, TeX
C_7\times C_5\rtimes D_4
% in TeX
G:=Group("C7xC5:D4"); // GroupNames label
G:=SmallGroup(280,23);
// by ID
G=gap.SmallGroup(280,23);
# by ID
G:=PCGroup([5,-2,-2,-7,-2,-5,301,5604]);
// Polycyclic
G:=Group<a,b,c,d|a^7=b^5=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