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G = C2×C70order 140 = 22·5·7

Abelian group of type [2,70]

direct product, abelian, monomial, 2-elementary

Aliases: C2×C70, SmallGroup(140,11)

Series: Derived Chief Lower central Upper central

C1 — C2×C70
C1C7C35C70 — 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
(1 123)(2 124)(3 125)(4 126)(5 127)(6 128)(7 129)(8 130)(9 131)(10 132)(11 133)(12 134)(13 135)(14 136)(15 137)(16 138)(17 139)(18 140)(19 71)(20 72)(21 73)(22 74)(23 75)(24 76)(25 77)(26 78)(27 79)(28 80)(29 81)(30 82)(31 83)(32 84)(33 85)(34 86)(35 87)(36 88)(37 89)(38 90)(39 91)(40 92)(41 93)(42 94)(43 95)(44 96)(45 97)(46 98)(47 99)(48 100)(49 101)(50 102)(51 103)(52 104)(53 105)(54 106)(55 107)(56 108)(57 109)(58 110)(59 111)(60 112)(61 113)(62 114)(63 115)(64 116)(65 117)(66 118)(67 119)(68 120)(69 121)(70 122)
(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)| (1,123)(2,124)(3,125)(4,126)(5,127)(6,128)(7,129)(8,130)(9,131)(10,132)(11,133)(12,134)(13,135)(14,136)(15,137)(16,138)(17,139)(18,140)(19,71)(20,72)(21,73)(22,74)(23,75)(24,76)(25,77)(26,78)(27,79)(28,80)(29,81)(30,82)(31,83)(32,84)(33,85)(34,86)(35,87)(36,88)(37,89)(38,90)(39,91)(40,92)(41,93)(42,94)(43,95)(44,96)(45,97)(46,98)(47,99)(48,100)(49,101)(50,102)(51,103)(52,104)(53,105)(54,106)(55,107)(56,108)(57,109)(58,110)(59,111)(60,112)(61,113)(62,114)(63,115)(64,116)(65,117)(66,118)(67,119)(68,120)(69,121)(70,122), (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( (1,123)(2,124)(3,125)(4,126)(5,127)(6,128)(7,129)(8,130)(9,131)(10,132)(11,133)(12,134)(13,135)(14,136)(15,137)(16,138)(17,139)(18,140)(19,71)(20,72)(21,73)(22,74)(23,75)(24,76)(25,77)(26,78)(27,79)(28,80)(29,81)(30,82)(31,83)(32,84)(33,85)(34,86)(35,87)(36,88)(37,89)(38,90)(39,91)(40,92)(41,93)(42,94)(43,95)(44,96)(45,97)(46,98)(47,99)(48,100)(49,101)(50,102)(51,103)(52,104)(53,105)(54,106)(55,107)(56,108)(57,109)(58,110)(59,111)(60,112)(61,113)(62,114)(63,115)(64,116)(65,117)(66,118)(67,119)(68,120)(69,121)(70,122), (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([(1,123),(2,124),(3,125),(4,126),(5,127),(6,128),(7,129),(8,130),(9,131),(10,132),(11,133),(12,134),(13,135),(14,136),(15,137),(16,138),(17,139),(18,140),(19,71),(20,72),(21,73),(22,74),(23,75),(24,76),(25,77),(26,78),(27,79),(28,80),(29,81),(30,82),(31,83),(32,84),(33,85),(34,86),(35,87),(36,88),(37,89),(38,90),(39,91),(40,92),(41,93),(42,94),(43,95),(44,96),(45,97),(46,98),(47,99),(48,100),(49,101),(50,102),(51,103),(52,104),(53,105),(54,106),(55,107),(56,108),(57,109),(58,110),(59,111),(60,112),(61,113),(62,114),(63,115),(64,116),(65,117),(66,118),(67,119),(68,120),(69,121),(70,122)], [(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   C357D4

140 conjugacy classes

class 1 2A2B2C5A5B5C5D7A···7F10A···10L14A···14R35A···35X70A···70BT
order122255557···710···1014···1435···3570···70
size111111111···11···11···11···11···1

140 irreducible representations

dim11111111
type++
imageC1C2C5C7C10C14C35C70
kernelC2×C70C70C2×C14C2×C10C14C10C22C2
# reps134612182472

Matrix representation of C2×C70 in GL2(𝔽71) generated by

700
01
,
590
034
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

Export

Subgroup lattice of C2×C70 in TeX

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