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G = C7×D20order 280 = 23·5·7

Direct product of C7 and D20

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

Aliases: C7×D20, C356D4, C283D5, C201C14, C1404C2, D101C14, C14.15D10, C70.20C22, C4⋊(C7×D5), C51(C7×D4), (D5×C14)⋊4C2, C2.4(D5×C14), C10.3(C2×C14), SmallGroup(280,21)

Series: Derived Chief Lower central Upper central

C1C10 — C7×D20
C1C5C10C70D5×C14 — C7×D20
C5C10 — C7×D20
C1C14C28

Generators and relations for C7×D20
 G = < a,b,c | a7=b20=c2=1, ab=ba, ac=ca, cbc=b-1 >

10C2
10C2
5C22
5C22
2D5
2D5
10C14
10C14
5D4
5C2×C14
5C2×C14
2C7×D5
2C7×D5
5C7×D4

Smallest permutation representation of C7×D20
On 140 points
Generators in S140
(1 39 121 73 98 114 60)(2 40 122 74 99 115 41)(3 21 123 75 100 116 42)(4 22 124 76 81 117 43)(5 23 125 77 82 118 44)(6 24 126 78 83 119 45)(7 25 127 79 84 120 46)(8 26 128 80 85 101 47)(9 27 129 61 86 102 48)(10 28 130 62 87 103 49)(11 29 131 63 88 104 50)(12 30 132 64 89 105 51)(13 31 133 65 90 106 52)(14 32 134 66 91 107 53)(15 33 135 67 92 108 54)(16 34 136 68 93 109 55)(17 35 137 69 94 110 56)(18 36 138 70 95 111 57)(19 37 139 71 96 112 58)(20 38 140 72 97 113 59)
(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 20)(2 19)(3 18)(4 17)(5 16)(6 15)(7 14)(8 13)(9 12)(10 11)(21 36)(22 35)(23 34)(24 33)(25 32)(26 31)(27 30)(28 29)(37 40)(38 39)(41 58)(42 57)(43 56)(44 55)(45 54)(46 53)(47 52)(48 51)(49 50)(59 60)(61 64)(62 63)(65 80)(66 79)(67 78)(68 77)(69 76)(70 75)(71 74)(72 73)(81 94)(82 93)(83 92)(84 91)(85 90)(86 89)(87 88)(95 100)(96 99)(97 98)(101 106)(102 105)(103 104)(107 120)(108 119)(109 118)(110 117)(111 116)(112 115)(113 114)(121 140)(122 139)(123 138)(124 137)(125 136)(126 135)(127 134)(128 133)(129 132)(130 131)

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

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

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

91 conjugacy classes

class 1 2A2B2C 4 5A5B7A···7F10A10B14A···14F14G···14R20A20B20C20D28A···28F35A···35L70A···70L140A···140X
order12224557···7101014···1414···142020202028···2835···3570···70140···140
size1110102221···1221···110···1022222···22···22···22···2

91 irreducible representations

dim11111122222222
type+++++++
imageC1C2C2C7C14C14D4D5D10D20C7×D4C7×D5D5×C14C7×D20
kernelC7×D20C140D5×C14D20C20D10C35C28C14C7C5C4C2C1
# reps112661212246121224

Matrix representation of C7×D20 in GL2(𝔽281) generated by

1810
0181
,
13450
23117
,
13450
231147
G:=sub<GL(2,GF(281))| [181,0,0,181],[134,231,50,17],[134,231,50,147] >;

C7×D20 in GAP, Magma, Sage, TeX

C_7\times D_{20}
% in TeX

G:=Group("C7xD20");
// GroupNames label

G:=SmallGroup(280,21);
// by ID

G=gap.SmallGroup(280,21);
# by ID

G:=PCGroup([5,-2,-2,-7,-2,-5,301,146,5604]);
// Polycyclic

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

Export

Subgroup lattice of C7×D20 in TeX

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