direct product, metabelian, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C22×D35, C10⋊2D14, C14⋊2D10, C35⋊2C23, C70⋊2C22, (C2×C70)⋊3C2, (C2×C14)⋊3D5, (C2×C10)⋊3D7, C7⋊2(C22×D5), C5⋊2(C22×D7), SmallGroup(280,39)
Series: Derived ►Chief ►Lower central ►Upper central
| C35 — C22×D35 |
Generators and relations for C22×D35
G = < a,b,c,d | a2=b2=c35=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >
Subgroups: 548 in 64 conjugacy classes, 31 normal (9 characteristic)
C1, C2, C2, C22, C22, C5, C7, C23, D5, C10, D7, C14, D10, C2×C10, D14, C2×C14, C35, C22×D5, C22×D7, D35, C70, D70, C2×C70, C22×D35
Quotients: C1, C2, C22, C23, D5, D7, D10, D14, C22×D5, C22×D7, D35, D70, C22×D35
►On 140 points
Generators in S140
(1 98)(2 99)(3 100)(4 101)(5 102)(6 103)(7 104)(8 105)(9 71)(10 72)(11 73)(12 74)(13 75)(14 76)(15 77)(16 78)(17 79)(18 80)(19 81)(20 82)(21 83)(22 84)(23 85)(24 86)(25 87)(26 88)(27 89)(28 90)(29 91)(30 92)(31 93)(32 94)(33 95)(34 96)(35 97)(36 134)(37 135)(38 136)(39 137)(40 138)(41 139)(42 140)(43 106)(44 107)(45 108)(46 109)(47 110)(48 111)(49 112)(50 113)(51 114)(52 115)(53 116)(54 117)(55 118)(56 119)(57 120)(58 121)(59 122)(60 123)(61 124)(62 125)(63 126)(64 127)(65 128)(66 129)(67 130)(68 131)(69 132)(70 133)
(1 49)(2 50)(3 51)(4 52)(5 53)(6 54)(7 55)(8 56)(9 57)(10 58)(11 59)(12 60)(13 61)(14 62)(15 63)(16 64)(17 65)(18 66)(19 67)(20 68)(21 69)(22 70)(23 36)(24 37)(25 38)(26 39)(27 40)(28 41)(29 42)(30 43)(31 44)(32 45)(33 46)(34 47)(35 48)(71 120)(72 121)(73 122)(74 123)(75 124)(76 125)(77 126)(78 127)(79 128)(80 129)(81 130)(82 131)(83 132)(84 133)(85 134)(86 135)(87 136)(88 137)(89 138)(90 139)(91 140)(92 106)(93 107)(94 108)(95 109)(96 110)(97 111)(98 112)(99 113)(100 114)(101 115)(102 116)(103 117)(104 118)(105 119)
(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)
(1 111)(2 110)(3 109)(4 108)(5 107)(6 106)(7 140)(8 139)(9 138)(10 137)(11 136)(12 135)(13 134)(14 133)(15 132)(16 131)(17 130)(18 129)(19 128)(20 127)(21 126)(22 125)(23 124)(24 123)(25 122)(26 121)(27 120)(28 119)(29 118)(30 117)(31 116)(32 115)(33 114)(34 113)(35 112)(36 75)(37 74)(38 73)(39 72)(40 71)(41 105)(42 104)(43 103)(44 102)(45 101)(46 100)(47 99)(48 98)(49 97)(50 96)(51 95)(52 94)(53 93)(54 92)(55 91)(56 90)(57 89)(58 88)(59 87)(60 86)(61 85)(62 84)(63 83)(64 82)(65 81)(66 80)(67 79)(68 78)(69 77)(70 76)
G:=sub<Sym(140)| (1,98)(2,99)(3,100)(4,101)(5,102)(6,103)(7,104)(8,105)(9,71)(10,72)(11,73)(12,74)(13,75)(14,76)(15,77)(16,78)(17,79)(18,80)(19,81)(20,82)(21,83)(22,84)(23,85)(24,86)(25,87)(26,88)(27,89)(28,90)(29,91)(30,92)(31,93)(32,94)(33,95)(34,96)(35,97)(36,134)(37,135)(38,136)(39,137)(40,138)(41,139)(42,140)(43,106)(44,107)(45,108)(46,109)(47,110)(48,111)(49,112)(50,113)(51,114)(52,115)(53,116)(54,117)(55,118)(56,119)(57,120)(58,121)(59,122)(60,123)(61,124)(62,125)(63,126)(64,127)(65,128)(66,129)(67,130)(68,131)(69,132)(70,133), (1,49)(2,50)(3,51)(4,52)(5,53)(6,54)(7,55)(8,56)(9,57)(10,58)(11,59)(12,60)(13,61)(14,62)(15,63)(16,64)(17,65)(18,66)(19,67)(20,68)(21,69)(22,70)(23,36)(24,37)(25,38)(26,39)(27,40)(28,41)(29,42)(30,43)(31,44)(32,45)(33,46)(34,47)(35,48)(71,120)(72,121)(73,122)(74,123)(75,124)(76,125)(77,126)(78,127)(79,128)(80,129)(81,130)(82,131)(83,132)(84,133)(85,134)(86,135)(87,136)(88,137)(89,138)(90,139)(91,140)(92,106)(93,107)(94,108)(95,109)(96,110)(97,111)(98,112)(99,113)(100,114)(101,115)(102,116)(103,117)(104,118)(105,119), (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), (1,111)(2,110)(3,109)(4,108)(5,107)(6,106)(7,140)(8,139)(9,138)(10,137)(11,136)(12,135)(13,134)(14,133)(15,132)(16,131)(17,130)(18,129)(19,128)(20,127)(21,126)(22,125)(23,124)(24,123)(25,122)(26,121)(27,120)(28,119)(29,118)(30,117)(31,116)(32,115)(33,114)(34,113)(35,112)(36,75)(37,74)(38,73)(39,72)(40,71)(41,105)(42,104)(43,103)(44,102)(45,101)(46,100)(47,99)(48,98)(49,97)(50,96)(51,95)(52,94)(53,93)(54,92)(55,91)(56,90)(57,89)(58,88)(59,87)(60,86)(61,85)(62,84)(63,83)(64,82)(65,81)(66,80)(67,79)(68,78)(69,77)(70,76)>;
G:=Group( (1,98)(2,99)(3,100)(4,101)(5,102)(6,103)(7,104)(8,105)(9,71)(10,72)(11,73)(12,74)(13,75)(14,76)(15,77)(16,78)(17,79)(18,80)(19,81)(20,82)(21,83)(22,84)(23,85)(24,86)(25,87)(26,88)(27,89)(28,90)(29,91)(30,92)(31,93)(32,94)(33,95)(34,96)(35,97)(36,134)(37,135)(38,136)(39,137)(40,138)(41,139)(42,140)(43,106)(44,107)(45,108)(46,109)(47,110)(48,111)(49,112)(50,113)(51,114)(52,115)(53,116)(54,117)(55,118)(56,119)(57,120)(58,121)(59,122)(60,123)(61,124)(62,125)(63,126)(64,127)(65,128)(66,129)(67,130)(68,131)(69,132)(70,133), (1,49)(2,50)(3,51)(4,52)(5,53)(6,54)(7,55)(8,56)(9,57)(10,58)(11,59)(12,60)(13,61)(14,62)(15,63)(16,64)(17,65)(18,66)(19,67)(20,68)(21,69)(22,70)(23,36)(24,37)(25,38)(26,39)(27,40)(28,41)(29,42)(30,43)(31,44)(32,45)(33,46)(34,47)(35,48)(71,120)(72,121)(73,122)(74,123)(75,124)(76,125)(77,126)(78,127)(79,128)(80,129)(81,130)(82,131)(83,132)(84,133)(85,134)(86,135)(87,136)(88,137)(89,138)(90,139)(91,140)(92,106)(93,107)(94,108)(95,109)(96,110)(97,111)(98,112)(99,113)(100,114)(101,115)(102,116)(103,117)(104,118)(105,119), (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), (1,111)(2,110)(3,109)(4,108)(5,107)(6,106)(7,140)(8,139)(9,138)(10,137)(11,136)(12,135)(13,134)(14,133)(15,132)(16,131)(17,130)(18,129)(19,128)(20,127)(21,126)(22,125)(23,124)(24,123)(25,122)(26,121)(27,120)(28,119)(29,118)(30,117)(31,116)(32,115)(33,114)(34,113)(35,112)(36,75)(37,74)(38,73)(39,72)(40,71)(41,105)(42,104)(43,103)(44,102)(45,101)(46,100)(47,99)(48,98)(49,97)(50,96)(51,95)(52,94)(53,93)(54,92)(55,91)(56,90)(57,89)(58,88)(59,87)(60,86)(61,85)(62,84)(63,83)(64,82)(65,81)(66,80)(67,79)(68,78)(69,77)(70,76) );
G=PermutationGroup([[(1,98),(2,99),(3,100),(4,101),(5,102),(6,103),(7,104),(8,105),(9,71),(10,72),(11,73),(12,74),(13,75),(14,76),(15,77),(16,78),(17,79),(18,80),(19,81),(20,82),(21,83),(22,84),(23,85),(24,86),(25,87),(26,88),(27,89),(28,90),(29,91),(30,92),(31,93),(32,94),(33,95),(34,96),(35,97),(36,134),(37,135),(38,136),(39,137),(40,138),(41,139),(42,140),(43,106),(44,107),(45,108),(46,109),(47,110),(48,111),(49,112),(50,113),(51,114),(52,115),(53,116),(54,117),(55,118),(56,119),(57,120),(58,121),(59,122),(60,123),(61,124),(62,125),(63,126),(64,127),(65,128),(66,129),(67,130),(68,131),(69,132),(70,133)], [(1,49),(2,50),(3,51),(4,52),(5,53),(6,54),(7,55),(8,56),(9,57),(10,58),(11,59),(12,60),(13,61),(14,62),(15,63),(16,64),(17,65),(18,66),(19,67),(20,68),(21,69),(22,70),(23,36),(24,37),(25,38),(26,39),(27,40),(28,41),(29,42),(30,43),(31,44),(32,45),(33,46),(34,47),(35,48),(71,120),(72,121),(73,122),(74,123),(75,124),(76,125),(77,126),(78,127),(79,128),(80,129),(81,130),(82,131),(83,132),(84,133),(85,134),(86,135),(87,136),(88,137),(89,138),(90,139),(91,140),(92,106),(93,107),(94,108),(95,109),(96,110),(97,111),(98,112),(99,113),(100,114),(101,115),(102,116),(103,117),(104,118),(105,119)], [(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)], [(1,111),(2,110),(3,109),(4,108),(5,107),(6,106),(7,140),(8,139),(9,138),(10,137),(11,136),(12,135),(13,134),(14,133),(15,132),(16,131),(17,130),(18,129),(19,128),(20,127),(21,126),(22,125),(23,124),(24,123),(25,122),(26,121),(27,120),(28,119),(29,118),(30,117),(31,116),(32,115),(33,114),(34,113),(35,112),(36,75),(37,74),(38,73),(39,72),(40,71),(41,105),(42,104),(43,103),(44,102),(45,101),(46,100),(47,99),(48,98),(49,97),(50,96),(51,95),(52,94),(53,93),(54,92),(55,91),(56,90),(57,89),(58,88),(59,87),(60,86),(61,85),(62,84),(63,83),(64,82),(65,81),(66,80),(67,79),(68,78),(69,77),(70,76)]])
76 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 5A | 5B | 7A | 7B | 7C | 10A | ··· | 10F | 14A | ··· | 14I | 35A | ··· | 35L | 70A | ··· | 70AJ |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 5 | 5 | 7 | 7 | 7 | 10 | ··· | 10 | 14 | ··· | 14 | 35 | ··· | 35 | 70 | ··· | 70 |
| size | 1 | 1 | 1 | 1 | 35 | 35 | 35 | 35 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
76 irreducible representations
| dim | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + |
| image | C1 | C2 | C2 | D5 | D7 | D10 | D14 | D35 | D70 |
| kernel | C22×D35 | D70 | C2×C70 | C2×C14 | C2×C10 | C14 | C10 | C22 | C2 |
| # reps | 1 | 6 | 1 | 2 | 3 | 6 | 9 | 12 | 36 |
Matrix representation of C22×D35 ►in GL3(𝔽71) generated by
| 1 | 0 | 0 |
| 0 | 70 | 0 |
| 0 | 0 | 70 |
| 70 | 0 | 0 |
| 0 | 70 | 0 |
| 0 | 0 | 70 |
| 1 | 0 | 0 |
| 0 | 7 | 5 |
| 0 | 66 | 27 |
| 1 | 0 | 0 |
| 0 | 7 | 5 |
| 0 | 33 | 64 |
G:=sub<GL(3,GF(71))| [1,0,0,0,70,0,0,0,70],[70,0,0,0,70,0,0,0,70],[1,0,0,0,7,66,0,5,27],[1,0,0,0,7,33,0,5,64] >;
C22×D35 in GAP, Magma, Sage, TeX
C_2^2\times D_{35} % in TeX
G:=Group("C2^2xD35"); // GroupNames label
G:=SmallGroup(280,39);
// by ID
G=gap.SmallGroup(280,39);
# by ID
G:=PCGroup([5,-2,-2,-2,-5,-7,643,6004]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^2=c^35=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