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G = C5×Dic7order 140 = 22·5·7

Direct product of C5 and Dic7

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

Aliases: C5×Dic7, C7⋊C20, C354C4, C14.C10, C70.2C2, C10.2D7, C2.(C5×D7), SmallGroup(140,2)

Series: Derived Chief Lower central Upper central

C1C7 — C5×Dic7
C1C7C14C70 — C5×Dic7
C7 — C5×Dic7
C1C10

Generators and relations for C5×Dic7
 G = < a,b,c | a5=b14=1, c2=b7, ab=ba, ac=ca, cbc-1=b-1 >

7C4
7C20

Smallest permutation representation of C5×Dic7
Regular action on 140 points
Generators in S140
(1 60 55 37 18)(2 61 56 38 19)(3 62 43 39 20)(4 63 44 40 21)(5 64 45 41 22)(6 65 46 42 23)(7 66 47 29 24)(8 67 48 30 25)(9 68 49 31 26)(10 69 50 32 27)(11 70 51 33 28)(12 57 52 34 15)(13 58 53 35 16)(14 59 54 36 17)(71 127 120 106 85)(72 128 121 107 86)(73 129 122 108 87)(74 130 123 109 88)(75 131 124 110 89)(76 132 125 111 90)(77 133 126 112 91)(78 134 113 99 92)(79 135 114 100 93)(80 136 115 101 94)(81 137 116 102 95)(82 138 117 103 96)(83 139 118 104 97)(84 140 119 105 98)
(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 71 8 78)(2 84 9 77)(3 83 10 76)(4 82 11 75)(5 81 12 74)(6 80 13 73)(7 79 14 72)(15 88 22 95)(16 87 23 94)(17 86 24 93)(18 85 25 92)(19 98 26 91)(20 97 27 90)(21 96 28 89)(29 100 36 107)(30 99 37 106)(31 112 38 105)(32 111 39 104)(33 110 40 103)(34 109 41 102)(35 108 42 101)(43 118 50 125)(44 117 51 124)(45 116 52 123)(46 115 53 122)(47 114 54 121)(48 113 55 120)(49 126 56 119)(57 130 64 137)(58 129 65 136)(59 128 66 135)(60 127 67 134)(61 140 68 133)(62 139 69 132)(63 138 70 131)

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

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

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

C5×Dic7 is a maximal subgroup of   D70.C2  C7⋊D20  C35⋊Q8  D7×C20

50 conjugacy classes

class 1  2 4A4B5A5B5C5D7A7B7C10A10B10C10D14A14B14C20A···20H35A···35L70A···70L
order124455557771010101014141420···2035···3570···70
size1177111122211112227···72···22···2

50 irreducible representations

dim1111112222
type+++-
imageC1C2C4C5C10C20D7Dic7C5×D7C5×Dic7
kernelC5×Dic7C70C35Dic7C14C7C10C5C2C1
# reps112448331212

Matrix representation of C5×Dic7 in GL2(𝔽41) generated by

100
010
,
117
1510
,
930
032
G:=sub<GL(2,GF(41))| [10,0,0,10],[1,15,17,10],[9,0,30,32] >;

C5×Dic7 in GAP, Magma, Sage, TeX

C_5\times {\rm Dic}_7
% in TeX

G:=Group("C5xDic7");
// GroupNames label

G:=SmallGroup(140,2);
// by ID

G=gap.SmallGroup(140,2);
# by ID

G:=PCGroup([4,-2,-5,-2,-7,40,1923]);
// Polycyclic

G:=Group<a,b,c|a^5=b^14=1,c^2=b^7,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

Export

Subgroup lattice of C5×Dic7 in TeX

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