direct product, abelian, monomial, 2-elementary
Aliases: C2×C70, SmallGroup(140,11)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C2×C70 |
| C1 — C2×C70 |
| C1 — C2×C70 |
Generators and relations for C2×C70
G = < a,b | a2=b70=1, ab=ba >
Smallest permutation representation of C2×C70
►Regular action on 140 points
Generators in S140
(1 132)(2 133)(3 134)(4 135)(5 136)(6 137)(7 138)(8 139)(9 140)(10 71)(11 72)(12 73)(13 74)(14 75)(15 76)(16 77)(17 78)(18 79)(19 80)(20 81)(21 82)(22 83)(23 84)(24 85)(25 86)(26 87)(27 88)(28 89)(29 90)(30 91)(31 92)(32 93)(33 94)(34 95)(35 96)(36 97)(37 98)(38 99)(39 100)(40 101)(41 102)(42 103)(43 104)(44 105)(45 106)(46 107)(47 108)(48 109)(49 110)(50 111)(51 112)(52 113)(53 114)(54 115)(55 116)(56 117)(57 118)(58 119)(59 120)(60 121)(61 122)(62 123)(63 124)(64 125)(65 126)(66 127)(67 128)(68 129)(69 130)(70 131)
(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)
G:=sub<Sym(140)| (1,132)(2,133)(3,134)(4,135)(5,136)(6,137)(7,138)(8,139)(9,140)(10,71)(11,72)(12,73)(13,74)(14,75)(15,76)(16,77)(17,78)(18,79)(19,80)(20,81)(21,82)(22,83)(23,84)(24,85)(25,86)(26,87)(27,88)(28,89)(29,90)(30,91)(31,92)(32,93)(33,94)(34,95)(35,96)(36,97)(37,98)(38,99)(39,100)(40,101)(41,102)(42,103)(43,104)(44,105)(45,106)(46,107)(47,108)(48,109)(49,110)(50,111)(51,112)(52,113)(53,114)(54,115)(55,116)(56,117)(57,118)(58,119)(59,120)(60,121)(61,122)(62,123)(63,124)(64,125)(65,126)(66,127)(67,128)(68,129)(69,130)(70,131), (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)>;
G:=Group( (1,132)(2,133)(3,134)(4,135)(5,136)(6,137)(7,138)(8,139)(9,140)(10,71)(11,72)(12,73)(13,74)(14,75)(15,76)(16,77)(17,78)(18,79)(19,80)(20,81)(21,82)(22,83)(23,84)(24,85)(25,86)(26,87)(27,88)(28,89)(29,90)(30,91)(31,92)(32,93)(33,94)(34,95)(35,96)(36,97)(37,98)(38,99)(39,100)(40,101)(41,102)(42,103)(43,104)(44,105)(45,106)(46,107)(47,108)(48,109)(49,110)(50,111)(51,112)(52,113)(53,114)(54,115)(55,116)(56,117)(57,118)(58,119)(59,120)(60,121)(61,122)(62,123)(63,124)(64,125)(65,126)(66,127)(67,128)(68,129)(69,130)(70,131), (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) );
G=PermutationGroup([[(1,132),(2,133),(3,134),(4,135),(5,136),(6,137),(7,138),(8,139),(9,140),(10,71),(11,72),(12,73),(13,74),(14,75),(15,76),(16,77),(17,78),(18,79),(19,80),(20,81),(21,82),(22,83),(23,84),(24,85),(25,86),(26,87),(27,88),(28,89),(29,90),(30,91),(31,92),(32,93),(33,94),(34,95),(35,96),(36,97),(37,98),(38,99),(39,100),(40,101),(41,102),(42,103),(43,104),(44,105),(45,106),(46,107),(47,108),(48,109),(49,110),(50,111),(51,112),(52,113),(53,114),(54,115),(55,116),(56,117),(57,118),(58,119),(59,120),(60,121),(61,122),(62,123),(63,124),(64,125),(65,126),(66,127),(67,128),(68,129),(69,130),(70,131)], [(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)]])
C2×C70 is a maximal subgroup of
C35⋊7D4
140 conjugacy classes
| class | 1 | 2A | 2B | 2C | 5A | 5B | 5C | 5D | 7A | ··· | 7F | 10A | ··· | 10L | 14A | ··· | 14R | 35A | ··· | 35X | 70A | ··· | 70BT |
| order | 1 | 2 | 2 | 2 | 5 | 5 | 5 | 5 | 7 | ··· | 7 | 10 | ··· | 10 | 14 | ··· | 14 | 35 | ··· | 35 | 70 | ··· | 70 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
140 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| type | + | + | ||||||
| image | C1 | C2 | C5 | C7 | C10 | C14 | C35 | C70 |
| kernel | C2×C70 | C70 | C2×C14 | C2×C10 | C14 | C10 | C22 | C2 |
| # reps | 1 | 3 | 4 | 6 | 12 | 18 | 24 | 72 |
Matrix representation of C2×C70 ►in GL2(𝔽71) generated by
| 70 | 0 |
| 0 | 1 |
| 59 | 0 |
| 0 | 34 |
G:=sub<GL(2,GF(71))| [70,0,0,1],[59,0,0,34] >;
C2×C70 in GAP, Magma, Sage, TeX
C_2\times C_{70} % in TeX
G:=Group("C2xC70"); // GroupNames label
G:=SmallGroup(140,11);
// by ID
G=gap.SmallGroup(140,11);
# by ID
G:=PCGroup([4,-2,-2,-5,-7]);
// Polycyclic
G:=Group<a,b|a^2=b^70=1,a*b=b*a>;
// generators/relations
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