direct product, metabelian, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: D7×C2×C10, C35⋊3C23, C70⋊3C22, C14⋊(C2×C10), C7⋊(C22×C10), (C2×C70)⋊5C2, (C2×C14)⋊3C10, SmallGroup(280,37)
Series: Derived ►Chief ►Lower central ►Upper central
| C7 — D7×C2×C10 |
Generators and relations for D7×C2×C10
G = < a,b,c,d | a2=b10=c7=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >
Subgroups: 196 in 64 conjugacy classes, 42 normal (10 characteristic)
C1, C2, C2, C22, C22, C5, C7, C23, C10, C10, D7, C14, C2×C10, C2×C10, D14, C2×C14, C35, C22×C10, C22×D7, C5×D7, C70, C10×D7, C2×C70, D7×C2×C10
Quotients: C1, C2, C22, C5, C23, C10, D7, C2×C10, D14, C22×C10, C22×D7, C5×D7, C10×D7, D7×C2×C10
►On 140 points
Generators in S140
(1 51)(2 52)(3 53)(4 54)(5 55)(6 56)(7 57)(8 58)(9 59)(10 60)(11 78)(12 79)(13 80)(14 71)(15 72)(16 73)(17 74)(18 75)(19 76)(20 77)(21 123)(22 124)(23 125)(24 126)(25 127)(26 128)(27 129)(28 130)(29 121)(30 122)(31 105)(32 106)(33 107)(34 108)(35 109)(36 110)(37 101)(38 102)(39 103)(40 104)(41 90)(42 81)(43 82)(44 83)(45 84)(46 85)(47 86)(48 87)(49 88)(50 89)(61 100)(62 91)(63 92)(64 93)(65 94)(66 95)(67 96)(68 97)(69 98)(70 99)(111 133)(112 134)(113 135)(114 136)(115 137)(116 138)(117 139)(118 140)(119 131)(120 132)
(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 87 109 121 99 115 75)(2 88 110 122 100 116 76)(3 89 101 123 91 117 77)(4 90 102 124 92 118 78)(5 81 103 125 93 119 79)(6 82 104 126 94 120 80)(7 83 105 127 95 111 71)(8 84 106 128 96 112 72)(9 85 107 129 97 113 73)(10 86 108 130 98 114 74)(11 54 41 38 22 63 140)(12 55 42 39 23 64 131)(13 56 43 40 24 65 132)(14 57 44 31 25 66 133)(15 58 45 32 26 67 134)(16 59 46 33 27 68 135)(17 60 47 34 28 69 136)(18 51 48 35 29 70 137)(19 52 49 36 30 61 138)(20 53 50 37 21 62 139)
(1 80)(2 71)(3 72)(4 73)(5 74)(6 75)(7 76)(8 77)(9 78)(10 79)(11 59)(12 60)(13 51)(14 52)(15 53)(16 54)(17 55)(18 56)(19 57)(20 58)(21 26)(22 27)(23 28)(24 29)(25 30)(31 61)(32 62)(33 63)(34 64)(35 65)(36 66)(37 67)(38 68)(39 69)(40 70)(41 135)(42 136)(43 137)(44 138)(45 139)(46 140)(47 131)(48 132)(49 133)(50 134)(81 114)(82 115)(83 116)(84 117)(85 118)(86 119)(87 120)(88 111)(89 112)(90 113)(91 106)(92 107)(93 108)(94 109)(95 110)(96 101)(97 102)(98 103)(99 104)(100 105)(121 126)(122 127)(123 128)(124 129)(125 130)
G:=sub<Sym(140)| (1,51)(2,52)(3,53)(4,54)(5,55)(6,56)(7,57)(8,58)(9,59)(10,60)(11,78)(12,79)(13,80)(14,71)(15,72)(16,73)(17,74)(18,75)(19,76)(20,77)(21,123)(22,124)(23,125)(24,126)(25,127)(26,128)(27,129)(28,130)(29,121)(30,122)(31,105)(32,106)(33,107)(34,108)(35,109)(36,110)(37,101)(38,102)(39,103)(40,104)(41,90)(42,81)(43,82)(44,83)(45,84)(46,85)(47,86)(48,87)(49,88)(50,89)(61,100)(62,91)(63,92)(64,93)(65,94)(66,95)(67,96)(68,97)(69,98)(70,99)(111,133)(112,134)(113,135)(114,136)(115,137)(116,138)(117,139)(118,140)(119,131)(120,132), (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), (1,87,109,121,99,115,75)(2,88,110,122,100,116,76)(3,89,101,123,91,117,77)(4,90,102,124,92,118,78)(5,81,103,125,93,119,79)(6,82,104,126,94,120,80)(7,83,105,127,95,111,71)(8,84,106,128,96,112,72)(9,85,107,129,97,113,73)(10,86,108,130,98,114,74)(11,54,41,38,22,63,140)(12,55,42,39,23,64,131)(13,56,43,40,24,65,132)(14,57,44,31,25,66,133)(15,58,45,32,26,67,134)(16,59,46,33,27,68,135)(17,60,47,34,28,69,136)(18,51,48,35,29,70,137)(19,52,49,36,30,61,138)(20,53,50,37,21,62,139), (1,80)(2,71)(3,72)(4,73)(5,74)(6,75)(7,76)(8,77)(9,78)(10,79)(11,59)(12,60)(13,51)(14,52)(15,53)(16,54)(17,55)(18,56)(19,57)(20,58)(21,26)(22,27)(23,28)(24,29)(25,30)(31,61)(32,62)(33,63)(34,64)(35,65)(36,66)(37,67)(38,68)(39,69)(40,70)(41,135)(42,136)(43,137)(44,138)(45,139)(46,140)(47,131)(48,132)(49,133)(50,134)(81,114)(82,115)(83,116)(84,117)(85,118)(86,119)(87,120)(88,111)(89,112)(90,113)(91,106)(92,107)(93,108)(94,109)(95,110)(96,101)(97,102)(98,103)(99,104)(100,105)(121,126)(122,127)(123,128)(124,129)(125,130)>;
G:=Group( (1,51)(2,52)(3,53)(4,54)(5,55)(6,56)(7,57)(8,58)(9,59)(10,60)(11,78)(12,79)(13,80)(14,71)(15,72)(16,73)(17,74)(18,75)(19,76)(20,77)(21,123)(22,124)(23,125)(24,126)(25,127)(26,128)(27,129)(28,130)(29,121)(30,122)(31,105)(32,106)(33,107)(34,108)(35,109)(36,110)(37,101)(38,102)(39,103)(40,104)(41,90)(42,81)(43,82)(44,83)(45,84)(46,85)(47,86)(48,87)(49,88)(50,89)(61,100)(62,91)(63,92)(64,93)(65,94)(66,95)(67,96)(68,97)(69,98)(70,99)(111,133)(112,134)(113,135)(114,136)(115,137)(116,138)(117,139)(118,140)(119,131)(120,132), (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), (1,87,109,121,99,115,75)(2,88,110,122,100,116,76)(3,89,101,123,91,117,77)(4,90,102,124,92,118,78)(5,81,103,125,93,119,79)(6,82,104,126,94,120,80)(7,83,105,127,95,111,71)(8,84,106,128,96,112,72)(9,85,107,129,97,113,73)(10,86,108,130,98,114,74)(11,54,41,38,22,63,140)(12,55,42,39,23,64,131)(13,56,43,40,24,65,132)(14,57,44,31,25,66,133)(15,58,45,32,26,67,134)(16,59,46,33,27,68,135)(17,60,47,34,28,69,136)(18,51,48,35,29,70,137)(19,52,49,36,30,61,138)(20,53,50,37,21,62,139), (1,80)(2,71)(3,72)(4,73)(5,74)(6,75)(7,76)(8,77)(9,78)(10,79)(11,59)(12,60)(13,51)(14,52)(15,53)(16,54)(17,55)(18,56)(19,57)(20,58)(21,26)(22,27)(23,28)(24,29)(25,30)(31,61)(32,62)(33,63)(34,64)(35,65)(36,66)(37,67)(38,68)(39,69)(40,70)(41,135)(42,136)(43,137)(44,138)(45,139)(46,140)(47,131)(48,132)(49,133)(50,134)(81,114)(82,115)(83,116)(84,117)(85,118)(86,119)(87,120)(88,111)(89,112)(90,113)(91,106)(92,107)(93,108)(94,109)(95,110)(96,101)(97,102)(98,103)(99,104)(100,105)(121,126)(122,127)(123,128)(124,129)(125,130) );
G=PermutationGroup([[(1,51),(2,52),(3,53),(4,54),(5,55),(6,56),(7,57),(8,58),(9,59),(10,60),(11,78),(12,79),(13,80),(14,71),(15,72),(16,73),(17,74),(18,75),(19,76),(20,77),(21,123),(22,124),(23,125),(24,126),(25,127),(26,128),(27,129),(28,130),(29,121),(30,122),(31,105),(32,106),(33,107),(34,108),(35,109),(36,110),(37,101),(38,102),(39,103),(40,104),(41,90),(42,81),(43,82),(44,83),(45,84),(46,85),(47,86),(48,87),(49,88),(50,89),(61,100),(62,91),(63,92),(64,93),(65,94),(66,95),(67,96),(68,97),(69,98),(70,99),(111,133),(112,134),(113,135),(114,136),(115,137),(116,138),(117,139),(118,140),(119,131),(120,132)], [(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)], [(1,87,109,121,99,115,75),(2,88,110,122,100,116,76),(3,89,101,123,91,117,77),(4,90,102,124,92,118,78),(5,81,103,125,93,119,79),(6,82,104,126,94,120,80),(7,83,105,127,95,111,71),(8,84,106,128,96,112,72),(9,85,107,129,97,113,73),(10,86,108,130,98,114,74),(11,54,41,38,22,63,140),(12,55,42,39,23,64,131),(13,56,43,40,24,65,132),(14,57,44,31,25,66,133),(15,58,45,32,26,67,134),(16,59,46,33,27,68,135),(17,60,47,34,28,69,136),(18,51,48,35,29,70,137),(19,52,49,36,30,61,138),(20,53,50,37,21,62,139)], [(1,80),(2,71),(3,72),(4,73),(5,74),(6,75),(7,76),(8,77),(9,78),(10,79),(11,59),(12,60),(13,51),(14,52),(15,53),(16,54),(17,55),(18,56),(19,57),(20,58),(21,26),(22,27),(23,28),(24,29),(25,30),(31,61),(32,62),(33,63),(34,64),(35,65),(36,66),(37,67),(38,68),(39,69),(40,70),(41,135),(42,136),(43,137),(44,138),(45,139),(46,140),(47,131),(48,132),(49,133),(50,134),(81,114),(82,115),(83,116),(84,117),(85,118),(86,119),(87,120),(88,111),(89,112),(90,113),(91,106),(92,107),(93,108),(94,109),(95,110),(96,101),(97,102),(98,103),(99,104),(100,105),(121,126),(122,127),(123,128),(124,129),(125,130)]])
100 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 5A | 5B | 5C | 5D | 7A | 7B | 7C | 10A | ··· | 10L | 10M | ··· | 10AB | 14A | ··· | 14I | 35A | ··· | 35L | 70A | ··· | 70AJ |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 5 | 5 | 5 | 5 | 7 | 7 | 7 | 10 | ··· | 10 | 10 | ··· | 10 | 14 | ··· | 14 | 35 | ··· | 35 | 70 | ··· | 70 |
| size | 1 | 1 | 1 | 1 | 7 | 7 | 7 | 7 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 1 | ··· | 1 | 7 | ··· | 7 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
100 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | |||||
| image | C1 | C2 | C2 | C5 | C10 | C10 | D7 | D14 | C5×D7 | C10×D7 |
| kernel | D7×C2×C10 | C10×D7 | C2×C70 | C22×D7 | D14 | C2×C14 | C2×C10 | C10 | C22 | C2 |
| # reps | 1 | 6 | 1 | 4 | 24 | 4 | 3 | 9 | 12 | 36 |
Matrix representation of D7×C2×C10 ►in GL3(𝔽71) generated by
| 70 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 1 |
| 70 | 0 | 0 |
| 0 | 66 | 0 |
| 0 | 0 | 66 |
| 1 | 0 | 0 |
| 0 | 67 | 2 |
| 0 | 70 | 18 |
| 1 | 0 | 0 |
| 0 | 14 | 19 |
| 0 | 57 | 57 |
G:=sub<GL(3,GF(71))| [70,0,0,0,1,0,0,0,1],[70,0,0,0,66,0,0,0,66],[1,0,0,0,67,70,0,2,18],[1,0,0,0,14,57,0,19,57] >;
D7×C2×C10 in GAP, Magma, Sage, TeX
D_7\times C_2\times C_{10} % in TeX
G:=Group("D7xC2xC10"); // GroupNames label
G:=SmallGroup(280,37);
// by ID
G=gap.SmallGroup(280,37);
# by ID
G:=PCGroup([5,-2,-2,-2,-5,-7,6004]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^10=c^7=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