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G = D7×C20order 280 = 23·5·7

Direct product of C20 and D7

direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary

Aliases: D7×C20, C282C10, C1405C2, D14.C10, Dic72C10, C10.14D14, C70.14C22, C71(C2×C20), C358(C2×C4), C2.1(C10×D7), C14.2(C2×C10), (C5×Dic7)⋊5C2, (C10×D7).2C2, SmallGroup(280,15)

Series: Derived Chief Lower central Upper central

C1C7 — D7×C20
C1C7C14C70C10×D7 — D7×C20
C7 — D7×C20
C1C20

Generators and relations for D7×C20
 G = < a,b,c | a20=b7=c2=1, ab=ba, ac=ca, cbc=b-1 >

7C2
7C2
7C4
7C22
7C10
7C10
7C2×C4
7C20
7C2×C10
7C2×C20

Smallest permutation representation of D7×C20
On 140 points
Generators in S140
(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 97 133 29 65 46 105)(2 98 134 30 66 47 106)(3 99 135 31 67 48 107)(4 100 136 32 68 49 108)(5 81 137 33 69 50 109)(6 82 138 34 70 51 110)(7 83 139 35 71 52 111)(8 84 140 36 72 53 112)(9 85 121 37 73 54 113)(10 86 122 38 74 55 114)(11 87 123 39 75 56 115)(12 88 124 40 76 57 116)(13 89 125 21 77 58 117)(14 90 126 22 78 59 118)(15 91 127 23 79 60 119)(16 92 128 24 80 41 120)(17 93 129 25 61 42 101)(18 94 130 26 62 43 102)(19 95 131 27 63 44 103)(20 96 132 28 64 45 104)
(1 105)(2 106)(3 107)(4 108)(5 109)(6 110)(7 111)(8 112)(9 113)(10 114)(11 115)(12 116)(13 117)(14 118)(15 119)(16 120)(17 101)(18 102)(19 103)(20 104)(41 92)(42 93)(43 94)(44 95)(45 96)(46 97)(47 98)(48 99)(49 100)(50 81)(51 82)(52 83)(53 84)(54 85)(55 86)(56 87)(57 88)(58 89)(59 90)(60 91)(61 129)(62 130)(63 131)(64 132)(65 133)(66 134)(67 135)(68 136)(69 137)(70 138)(71 139)(72 140)(73 121)(74 122)(75 123)(76 124)(77 125)(78 126)(79 127)(80 128)

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

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

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

100 conjugacy classes

class 1 2A2B2C4A4B4C4D5A5B5C5D7A7B7C10A10B10C10D10E···10L14A14B14C20A···20H20I···20P28A···28F35A···35L70A···70L140A···140X
order1222444455557771010101010···1014141420···2020···2028···2835···3570···70140···140
size11771177111122211117···72221···17···72···22···22···22···2

100 irreducible representations

dim1111111111222222
type++++++
imageC1C2C2C2C4C5C10C10C10C20D7D14C4×D7C5×D7C10×D7D7×C20
kernelD7×C20C5×Dic7C140C10×D7C5×D7C4×D7Dic7C28D14D7C20C10C5C4C2C1
# reps11114444416336121224

Matrix representation of D7×C20 in GL2(𝔽41) generated by

80
08
,
4020
3215
,
2634
3215
G:=sub<GL(2,GF(41))| [8,0,0,8],[40,32,20,15],[26,32,34,15] >;

D7×C20 in GAP, Magma, Sage, TeX

D_7\times C_{20}
% in TeX

G:=Group("D7xC20");
// GroupNames label

G:=SmallGroup(280,15);
// by ID

G=gap.SmallGroup(280,15);
# by ID

G:=PCGroup([5,-2,-2,-5,-2,-7,106,6004]);
// Polycyclic

G:=Group<a,b,c|a^20=b^7=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

Export

Subgroup lattice of D7×C20 in TeX

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