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## G = D7×C2×C10order 280 = 23·5·7

### Direct product of C2×C10 and D7

Aliases: D7×C2×C10, C353C23, C703C22, C14⋊(C2×C10), C7⋊(C22×C10), (C2×C70)⋊5C2, (C2×C14)⋊3C10, SmallGroup(280,37)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C7 — D7×C2×C10
 Chief series C1 — C7 — C35 — C5×D7 — C10×D7 — D7×C2×C10
 Lower central C7 — D7×C2×C10
 Upper central C1 — C2×C10

Generators and relations for D7×C2×C10
G = < a,b,c,d | a2=b10=c7=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

Subgroups: 196 in 64 conjugacy classes, 42 normal (10 characteristic)
C1, C2, C2, C22, C22, C5, C7, C23, C10, C10, D7, C14, C2×C10, C2×C10, D14, C2×C14, C35, C22×C10, C22×D7, C5×D7, C70, C10×D7, C2×C70, D7×C2×C10
Quotients: C1, C2, C22, C5, C23, C10, D7, C2×C10, D14, C22×C10, C22×D7, C5×D7, C10×D7, D7×C2×C10

Smallest permutation representation of D7×C2×C10
On 140 points
Generators in S140
(1 51)(2 52)(3 53)(4 54)(5 55)(6 56)(7 57)(8 58)(9 59)(10 60)(11 78)(12 79)(13 80)(14 71)(15 72)(16 73)(17 74)(18 75)(19 76)(20 77)(21 123)(22 124)(23 125)(24 126)(25 127)(26 128)(27 129)(28 130)(29 121)(30 122)(31 105)(32 106)(33 107)(34 108)(35 109)(36 110)(37 101)(38 102)(39 103)(40 104)(41 90)(42 81)(43 82)(44 83)(45 84)(46 85)(47 86)(48 87)(49 88)(50 89)(61 100)(62 91)(63 92)(64 93)(65 94)(66 95)(67 96)(68 97)(69 98)(70 99)(111 133)(112 134)(113 135)(114 136)(115 137)(116 138)(117 139)(118 140)(119 131)(120 132)
(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 87 109 121 99 115 75)(2 88 110 122 100 116 76)(3 89 101 123 91 117 77)(4 90 102 124 92 118 78)(5 81 103 125 93 119 79)(6 82 104 126 94 120 80)(7 83 105 127 95 111 71)(8 84 106 128 96 112 72)(9 85 107 129 97 113 73)(10 86 108 130 98 114 74)(11 54 41 38 22 63 140)(12 55 42 39 23 64 131)(13 56 43 40 24 65 132)(14 57 44 31 25 66 133)(15 58 45 32 26 67 134)(16 59 46 33 27 68 135)(17 60 47 34 28 69 136)(18 51 48 35 29 70 137)(19 52 49 36 30 61 138)(20 53 50 37 21 62 139)
(1 80)(2 71)(3 72)(4 73)(5 74)(6 75)(7 76)(8 77)(9 78)(10 79)(11 59)(12 60)(13 51)(14 52)(15 53)(16 54)(17 55)(18 56)(19 57)(20 58)(21 26)(22 27)(23 28)(24 29)(25 30)(31 61)(32 62)(33 63)(34 64)(35 65)(36 66)(37 67)(38 68)(39 69)(40 70)(41 135)(42 136)(43 137)(44 138)(45 139)(46 140)(47 131)(48 132)(49 133)(50 134)(81 114)(82 115)(83 116)(84 117)(85 118)(86 119)(87 120)(88 111)(89 112)(90 113)(91 106)(92 107)(93 108)(94 109)(95 110)(96 101)(97 102)(98 103)(99 104)(100 105)(121 126)(122 127)(123 128)(124 129)(125 130)

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

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

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

100 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 5A 5B 5C 5D 7A 7B 7C 10A ··· 10L 10M ··· 10AB 14A ··· 14I 35A ··· 35L 70A ··· 70AJ order 1 2 2 2 2 2 2 2 5 5 5 5 7 7 7 10 ··· 10 10 ··· 10 14 ··· 14 35 ··· 35 70 ··· 70 size 1 1 1 1 7 7 7 7 1 1 1 1 2 2 2 1 ··· 1 7 ··· 7 2 ··· 2 2 ··· 2 2 ··· 2

100 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 type + + + + + image C1 C2 C2 C5 C10 C10 D7 D14 C5×D7 C10×D7 kernel D7×C2×C10 C10×D7 C2×C70 C22×D7 D14 C2×C14 C2×C10 C10 C22 C2 # reps 1 6 1 4 24 4 3 9 12 36

Matrix representation of D7×C2×C10 in GL3(𝔽71) generated by

 70 0 0 0 1 0 0 0 1
,
 70 0 0 0 66 0 0 0 66
,
 1 0 0 0 67 2 0 70 18
,
 1 0 0 0 14 19 0 57 57
G:=sub<GL(3,GF(71))| [70,0,0,0,1,0,0,0,1],[70,0,0,0,66,0,0,0,66],[1,0,0,0,67,70,0,2,18],[1,0,0,0,14,57,0,19,57] >;

D7×C2×C10 in GAP, Magma, Sage, TeX

D_7\times C_2\times C_{10}
% in TeX

G:=Group("D7xC2xC10");
// GroupNames label

G:=SmallGroup(280,37);
// by ID

G=gap.SmallGroup(280,37);
# by ID

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

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

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