direct product, metabelian, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: D5×C2×C14, C35⋊4C23, C70⋊4C22, C10⋊(C2×C14), C5⋊(C22×C14), (C2×C70)⋊7C2, (C2×C10)⋊3C14, SmallGroup(280,38)
Series: Derived ►Chief ►Lower central ►Upper central
| C5 — D5×C2×C14 |
Generators and relations for D5×C2×C14
G = < a,b,c,d | a2=b14=c5=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >
Subgroups: 152 in 64 conjugacy classes, 42 normal (10 characteristic)
C1, C2, C2, C22, C22, C5, C7, C23, D5, C10, C14, C14, D10, C2×C10, C2×C14, C2×C14, C35, C22×D5, C22×C14, C7×D5, C70, D5×C14, C2×C70, D5×C2×C14
Quotients: C1, C2, C22, C7, C23, D5, C14, D10, C2×C14, C22×D5, C22×C14, C7×D5, D5×C14, D5×C2×C14
►On 140 points
Generators in S140
(1 133)(2 134)(3 135)(4 136)(5 137)(6 138)(7 139)(8 140)(9 127)(10 128)(11 129)(12 130)(13 131)(14 132)(15 68)(16 69)(17 70)(18 57)(19 58)(20 59)(21 60)(22 61)(23 62)(24 63)(25 64)(26 65)(27 66)(28 67)(29 88)(30 89)(31 90)(32 91)(33 92)(34 93)(35 94)(36 95)(37 96)(38 97)(39 98)(40 85)(41 86)(42 87)(43 74)(44 75)(45 76)(46 77)(47 78)(48 79)(49 80)(50 81)(51 82)(52 83)(53 84)(54 71)(55 72)(56 73)(99 125)(100 126)(101 113)(102 114)(103 115)(104 116)(105 117)(106 118)(107 119)(108 120)(109 121)(110 122)(111 123)(112 124)
(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 39 70 79 101)(2 40 57 80 102)(3 41 58 81 103)(4 42 59 82 104)(5 29 60 83 105)(6 30 61 84 106)(7 31 62 71 107)(8 32 63 72 108)(9 33 64 73 109)(10 34 65 74 110)(11 35 66 75 111)(12 36 67 76 112)(13 37 68 77 99)(14 38 69 78 100)(15 46 125 131 96)(16 47 126 132 97)(17 48 113 133 98)(18 49 114 134 85)(19 50 115 135 86)(20 51 116 136 87)(21 52 117 137 88)(22 53 118 138 89)(23 54 119 139 90)(24 55 120 140 91)(25 56 121 127 92)(26 43 122 128 93)(27 44 123 129 94)(28 45 124 130 95)
(1 113)(2 114)(3 115)(4 116)(5 117)(6 118)(7 119)(8 120)(9 121)(10 122)(11 123)(12 124)(13 125)(14 126)(15 68)(16 69)(17 70)(18 57)(19 58)(20 59)(21 60)(22 61)(23 62)(24 63)(25 64)(26 65)(27 66)(28 67)(29 52)(30 53)(31 54)(32 55)(33 56)(34 43)(35 44)(36 45)(37 46)(38 47)(39 48)(40 49)(41 50)(42 51)(71 90)(72 91)(73 92)(74 93)(75 94)(76 95)(77 96)(78 97)(79 98)(80 85)(81 86)(82 87)(83 88)(84 89)(99 131)(100 132)(101 133)(102 134)(103 135)(104 136)(105 137)(106 138)(107 139)(108 140)(109 127)(110 128)(111 129)(112 130)
G:=sub<Sym(140)| (1,133)(2,134)(3,135)(4,136)(5,137)(6,138)(7,139)(8,140)(9,127)(10,128)(11,129)(12,130)(13,131)(14,132)(15,68)(16,69)(17,70)(18,57)(19,58)(20,59)(21,60)(22,61)(23,62)(24,63)(25,64)(26,65)(27,66)(28,67)(29,88)(30,89)(31,90)(32,91)(33,92)(34,93)(35,94)(36,95)(37,96)(38,97)(39,98)(40,85)(41,86)(42,87)(43,74)(44,75)(45,76)(46,77)(47,78)(48,79)(49,80)(50,81)(51,82)(52,83)(53,84)(54,71)(55,72)(56,73)(99,125)(100,126)(101,113)(102,114)(103,115)(104,116)(105,117)(106,118)(107,119)(108,120)(109,121)(110,122)(111,123)(112,124), (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), (1,39,70,79,101)(2,40,57,80,102)(3,41,58,81,103)(4,42,59,82,104)(5,29,60,83,105)(6,30,61,84,106)(7,31,62,71,107)(8,32,63,72,108)(9,33,64,73,109)(10,34,65,74,110)(11,35,66,75,111)(12,36,67,76,112)(13,37,68,77,99)(14,38,69,78,100)(15,46,125,131,96)(16,47,126,132,97)(17,48,113,133,98)(18,49,114,134,85)(19,50,115,135,86)(20,51,116,136,87)(21,52,117,137,88)(22,53,118,138,89)(23,54,119,139,90)(24,55,120,140,91)(25,56,121,127,92)(26,43,122,128,93)(27,44,123,129,94)(28,45,124,130,95), (1,113)(2,114)(3,115)(4,116)(5,117)(6,118)(7,119)(8,120)(9,121)(10,122)(11,123)(12,124)(13,125)(14,126)(15,68)(16,69)(17,70)(18,57)(19,58)(20,59)(21,60)(22,61)(23,62)(24,63)(25,64)(26,65)(27,66)(28,67)(29,52)(30,53)(31,54)(32,55)(33,56)(34,43)(35,44)(36,45)(37,46)(38,47)(39,48)(40,49)(41,50)(42,51)(71,90)(72,91)(73,92)(74,93)(75,94)(76,95)(77,96)(78,97)(79,98)(80,85)(81,86)(82,87)(83,88)(84,89)(99,131)(100,132)(101,133)(102,134)(103,135)(104,136)(105,137)(106,138)(107,139)(108,140)(109,127)(110,128)(111,129)(112,130)>;
G:=Group( (1,133)(2,134)(3,135)(4,136)(5,137)(6,138)(7,139)(8,140)(9,127)(10,128)(11,129)(12,130)(13,131)(14,132)(15,68)(16,69)(17,70)(18,57)(19,58)(20,59)(21,60)(22,61)(23,62)(24,63)(25,64)(26,65)(27,66)(28,67)(29,88)(30,89)(31,90)(32,91)(33,92)(34,93)(35,94)(36,95)(37,96)(38,97)(39,98)(40,85)(41,86)(42,87)(43,74)(44,75)(45,76)(46,77)(47,78)(48,79)(49,80)(50,81)(51,82)(52,83)(53,84)(54,71)(55,72)(56,73)(99,125)(100,126)(101,113)(102,114)(103,115)(104,116)(105,117)(106,118)(107,119)(108,120)(109,121)(110,122)(111,123)(112,124), (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), (1,39,70,79,101)(2,40,57,80,102)(3,41,58,81,103)(4,42,59,82,104)(5,29,60,83,105)(6,30,61,84,106)(7,31,62,71,107)(8,32,63,72,108)(9,33,64,73,109)(10,34,65,74,110)(11,35,66,75,111)(12,36,67,76,112)(13,37,68,77,99)(14,38,69,78,100)(15,46,125,131,96)(16,47,126,132,97)(17,48,113,133,98)(18,49,114,134,85)(19,50,115,135,86)(20,51,116,136,87)(21,52,117,137,88)(22,53,118,138,89)(23,54,119,139,90)(24,55,120,140,91)(25,56,121,127,92)(26,43,122,128,93)(27,44,123,129,94)(28,45,124,130,95), (1,113)(2,114)(3,115)(4,116)(5,117)(6,118)(7,119)(8,120)(9,121)(10,122)(11,123)(12,124)(13,125)(14,126)(15,68)(16,69)(17,70)(18,57)(19,58)(20,59)(21,60)(22,61)(23,62)(24,63)(25,64)(26,65)(27,66)(28,67)(29,52)(30,53)(31,54)(32,55)(33,56)(34,43)(35,44)(36,45)(37,46)(38,47)(39,48)(40,49)(41,50)(42,51)(71,90)(72,91)(73,92)(74,93)(75,94)(76,95)(77,96)(78,97)(79,98)(80,85)(81,86)(82,87)(83,88)(84,89)(99,131)(100,132)(101,133)(102,134)(103,135)(104,136)(105,137)(106,138)(107,139)(108,140)(109,127)(110,128)(111,129)(112,130) );
G=PermutationGroup([[(1,133),(2,134),(3,135),(4,136),(5,137),(6,138),(7,139),(8,140),(9,127),(10,128),(11,129),(12,130),(13,131),(14,132),(15,68),(16,69),(17,70),(18,57),(19,58),(20,59),(21,60),(22,61),(23,62),(24,63),(25,64),(26,65),(27,66),(28,67),(29,88),(30,89),(31,90),(32,91),(33,92),(34,93),(35,94),(36,95),(37,96),(38,97),(39,98),(40,85),(41,86),(42,87),(43,74),(44,75),(45,76),(46,77),(47,78),(48,79),(49,80),(50,81),(51,82),(52,83),(53,84),(54,71),(55,72),(56,73),(99,125),(100,126),(101,113),(102,114),(103,115),(104,116),(105,117),(106,118),(107,119),(108,120),(109,121),(110,122),(111,123),(112,124)], [(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)], [(1,39,70,79,101),(2,40,57,80,102),(3,41,58,81,103),(4,42,59,82,104),(5,29,60,83,105),(6,30,61,84,106),(7,31,62,71,107),(8,32,63,72,108),(9,33,64,73,109),(10,34,65,74,110),(11,35,66,75,111),(12,36,67,76,112),(13,37,68,77,99),(14,38,69,78,100),(15,46,125,131,96),(16,47,126,132,97),(17,48,113,133,98),(18,49,114,134,85),(19,50,115,135,86),(20,51,116,136,87),(21,52,117,137,88),(22,53,118,138,89),(23,54,119,139,90),(24,55,120,140,91),(25,56,121,127,92),(26,43,122,128,93),(27,44,123,129,94),(28,45,124,130,95)], [(1,113),(2,114),(3,115),(4,116),(5,117),(6,118),(7,119),(8,120),(9,121),(10,122),(11,123),(12,124),(13,125),(14,126),(15,68),(16,69),(17,70),(18,57),(19,58),(20,59),(21,60),(22,61),(23,62),(24,63),(25,64),(26,65),(27,66),(28,67),(29,52),(30,53),(31,54),(32,55),(33,56),(34,43),(35,44),(36,45),(37,46),(38,47),(39,48),(40,49),(41,50),(42,51),(71,90),(72,91),(73,92),(74,93),(75,94),(76,95),(77,96),(78,97),(79,98),(80,85),(81,86),(82,87),(83,88),(84,89),(99,131),(100,132),(101,133),(102,134),(103,135),(104,136),(105,137),(106,138),(107,139),(108,140),(109,127),(110,128),(111,129),(112,130)]])
112 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 5A | 5B | 7A | ··· | 7F | 10A | ··· | 10F | 14A | ··· | 14R | 14S | ··· | 14AP | 35A | ··· | 35L | 70A | ··· | 70AJ |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 5 | 5 | 7 | ··· | 7 | 10 | ··· | 10 | 14 | ··· | 14 | 14 | ··· | 14 | 35 | ··· | 35 | 70 | ··· | 70 |
| size | 1 | 1 | 1 | 1 | 5 | 5 | 5 | 5 | 2 | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 5 | ··· | 5 | 2 | ··· | 2 | 2 | ··· | 2 |
112 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | |||||
| image | C1 | C2 | C2 | C7 | C14 | C14 | D5 | D10 | C7×D5 | D5×C14 |
| kernel | D5×C2×C14 | D5×C14 | C2×C70 | C22×D5 | D10 | C2×C10 | C2×C14 | C14 | C22 | C2 |
| # reps | 1 | 6 | 1 | 6 | 36 | 6 | 2 | 6 | 12 | 36 |
Matrix representation of D5×C2×C14 ►in GL3(𝔽71) generated by
| 70 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 1 |
| 70 | 0 | 0 |
| 0 | 26 | 0 |
| 0 | 0 | 26 |
| 1 | 0 | 0 |
| 0 | 8 | 1 |
| 0 | 70 | 0 |
| 1 | 0 | 0 |
| 0 | 1 | 8 |
| 0 | 0 | 70 |
G:=sub<GL(3,GF(71))| [70,0,0,0,1,0,0,0,1],[70,0,0,0,26,0,0,0,26],[1,0,0,0,8,70,0,1,0],[1,0,0,0,1,0,0,8,70] >;
D5×C2×C14 in GAP, Magma, Sage, TeX
D_5\times C_2\times C_{14} % in TeX
G:=Group("D5xC2xC14"); // GroupNames label
G:=SmallGroup(280,38);
// by ID
G=gap.SmallGroup(280,38);
# by ID
G:=PCGroup([5,-2,-2,-2,-7,-5,5604]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^14=c^5=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations