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G = C7×Dic5order 140 = 22·5·7

Direct product of C7 and Dic5

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

Aliases: C7×Dic5, C52C28, C355C4, C10.C14, C70.3C2, C14.2D5, C2.(C7×D5), SmallGroup(140,1)

Series: Derived Chief Lower central Upper central

C1C5 — C7×Dic5
C1C5C10C70 — C7×Dic5
C5 — C7×Dic5
C1C14

Generators and relations for C7×Dic5
 G = < a,b,c | a7=b10=1, c2=b5, ab=ba, ac=ca, cbc-1=b-1 >

5C4
5C28

Smallest permutation representation of C7×Dic5
Regular action on 140 points
Generators in S140
(1 69 59 49 39 29 19)(2 70 60 50 40 30 20)(3 61 51 41 31 21 11)(4 62 52 42 32 22 12)(5 63 53 43 33 23 13)(6 64 54 44 34 24 14)(7 65 55 45 35 25 15)(8 66 56 46 36 26 16)(9 67 57 47 37 27 17)(10 68 58 48 38 28 18)(71 131 121 111 101 91 81)(72 132 122 112 102 92 82)(73 133 123 113 103 93 83)(74 134 124 114 104 94 84)(75 135 125 115 105 95 85)(76 136 126 116 106 96 86)(77 137 127 117 107 97 87)(78 138 128 118 108 98 88)(79 139 129 119 109 99 89)(80 140 130 120 110 100 90)
(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 76 6 71)(2 75 7 80)(3 74 8 79)(4 73 9 78)(5 72 10 77)(11 84 16 89)(12 83 17 88)(13 82 18 87)(14 81 19 86)(15 90 20 85)(21 94 26 99)(22 93 27 98)(23 92 28 97)(24 91 29 96)(25 100 30 95)(31 104 36 109)(32 103 37 108)(33 102 38 107)(34 101 39 106)(35 110 40 105)(41 114 46 119)(42 113 47 118)(43 112 48 117)(44 111 49 116)(45 120 50 115)(51 124 56 129)(52 123 57 128)(53 122 58 127)(54 121 59 126)(55 130 60 125)(61 134 66 139)(62 133 67 138)(63 132 68 137)(64 131 69 136)(65 140 70 135)

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

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

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

C7×Dic5 is a maximal subgroup of   C35⋊C8  D70.C2  C5⋊D28  C35⋊Q8  D5×C28

56 conjugacy classes

class 1  2 4A4B5A5B7A···7F10A10B14A···14F28A···28L35A···35L70A···70L
order1244557···7101014···1428···2835···3570···70
size1155221···1221···15···52···22···2

56 irreducible representations

dim1111112222
type+++-
imageC1C2C4C7C14C28D5Dic5C7×D5C7×Dic5
kernelC7×Dic5C70C35Dic5C10C5C14C7C2C1
# reps1126612221212

Matrix representation of C7×Dic5 in GL2(𝔽29) generated by

230
023
,
124
1323
,
1227
017
G:=sub<GL(2,GF(29))| [23,0,0,23],[1,13,24,23],[12,0,27,17] >;

C7×Dic5 in GAP, Magma, Sage, TeX

C_7\times {\rm Dic}_5
% in TeX

G:=Group("C7xDic5");
// GroupNames label

G:=SmallGroup(140,1);
// by ID

G=gap.SmallGroup(140,1);
# by ID

G:=PCGroup([4,-2,-7,-2,-5,56,1795]);
// Polycyclic

G:=Group<a,b,c|a^7=b^10=1,c^2=b^5,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

Export

Subgroup lattice of C7×Dic5 in TeX

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