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## G = D5×Dic7order 280 = 23·5·7

### Direct product of D5 and Dic7

Aliases: D5×Dic7, D10.2D7, C14.2D10, C10.2D14, Dic353C2, C70.2C22, C73(C4×D5), C355(C2×C4), (C7×D5)⋊1C4, C2.2(D5×D7), C52(C2×Dic7), (C5×Dic7)⋊1C2, (D5×C14).1C2, SmallGroup(280,8)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C35 — D5×Dic7
 Chief series C1 — C7 — C35 — C70 — C5×Dic7 — D5×Dic7
 Lower central C35 — D5×Dic7
 Upper central C1 — C2

Generators and relations for D5×Dic7
G = < a,b,c,d | a5=b2=c14=1, d2=c7, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >

Smallest permutation representation of D5×Dic7
On 140 points
Generators in S140
(1 62 36 135 98)(2 63 37 136 85)(3 64 38 137 86)(4 65 39 138 87)(5 66 40 139 88)(6 67 41 140 89)(7 68 42 127 90)(8 69 29 128 91)(9 70 30 129 92)(10 57 31 130 93)(11 58 32 131 94)(12 59 33 132 95)(13 60 34 133 96)(14 61 35 134 97)(15 104 121 51 71)(16 105 122 52 72)(17 106 123 53 73)(18 107 124 54 74)(19 108 125 55 75)(20 109 126 56 76)(21 110 113 43 77)(22 111 114 44 78)(23 112 115 45 79)(24 99 116 46 80)(25 100 117 47 81)(26 101 118 48 82)(27 102 119 49 83)(28 103 120 50 84)
(1 91)(2 92)(3 93)(4 94)(5 95)(6 96)(7 97)(8 98)(9 85)(10 86)(11 87)(12 88)(13 89)(14 90)(15 111)(16 112)(17 99)(18 100)(19 101)(20 102)(21 103)(22 104)(23 105)(24 106)(25 107)(26 108)(27 109)(28 110)(29 36)(30 37)(31 38)(32 39)(33 40)(34 41)(35 42)(43 50)(44 51)(45 52)(46 53)(47 54)(48 55)(49 56)(57 137)(58 138)(59 139)(60 140)(61 127)(62 128)(63 129)(64 130)(65 131)(66 132)(67 133)(68 134)(69 135)(70 136)(71 114)(72 115)(73 116)(74 117)(75 118)(76 119)(77 120)(78 121)(79 122)(80 123)(81 124)(82 125)(83 126)(84 113)
(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)
(1 105 8 112)(2 104 9 111)(3 103 10 110)(4 102 11 109)(5 101 12 108)(6 100 13 107)(7 99 14 106)(15 92 22 85)(16 91 23 98)(17 90 24 97)(18 89 25 96)(19 88 26 95)(20 87 27 94)(21 86 28 93)(29 45 36 52)(30 44 37 51)(31 43 38 50)(32 56 39 49)(33 55 40 48)(34 54 41 47)(35 53 42 46)(57 113 64 120)(58 126 65 119)(59 125 66 118)(60 124 67 117)(61 123 68 116)(62 122 69 115)(63 121 70 114)(71 129 78 136)(72 128 79 135)(73 127 80 134)(74 140 81 133)(75 139 82 132)(76 138 83 131)(77 137 84 130)

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

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

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

40 conjugacy classes

 class 1 2A 2B 2C 4A 4B 4C 4D 5A 5B 7A 7B 7C 10A 10B 14A 14B 14C 14D ··· 14I 20A 20B 20C 20D 35A ··· 35F 70A ··· 70F order 1 2 2 2 4 4 4 4 5 5 7 7 7 10 10 14 14 14 14 ··· 14 20 20 20 20 35 ··· 35 70 ··· 70 size 1 1 5 5 7 7 35 35 2 2 2 2 2 2 2 2 2 2 10 ··· 10 14 14 14 14 4 ··· 4 4 ··· 4

40 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 4 4 type + + + + + + + - + + - image C1 C2 C2 C2 C4 D5 D7 D10 Dic7 D14 C4×D5 D5×D7 D5×Dic7 kernel D5×Dic7 C5×Dic7 Dic35 D5×C14 C7×D5 Dic7 D10 C14 D5 C10 C7 C2 C1 # reps 1 1 1 1 4 2 3 2 6 3 4 6 6

Matrix representation of D5×Dic7 in GL4(𝔽281) generated by

 1 0 0 0 0 1 0 0 0 0 280 1 0 0 36 244
,
 280 0 0 0 0 280 0 0 0 0 1 0 0 0 245 280
,
 240 280 0 0 36 275 0 0 0 0 280 0 0 0 0 280
,
 93 132 0 0 92 188 0 0 0 0 228 0 0 0 0 228
G:=sub<GL(4,GF(281))| [1,0,0,0,0,1,0,0,0,0,280,36,0,0,1,244],[280,0,0,0,0,280,0,0,0,0,1,245,0,0,0,280],[240,36,0,0,280,275,0,0,0,0,280,0,0,0,0,280],[93,92,0,0,132,188,0,0,0,0,228,0,0,0,0,228] >;

D5×Dic7 in GAP, Magma, Sage, TeX

D_5\times {\rm Dic}_7
% in TeX

G:=Group("D5xDic7");
// GroupNames label

G:=SmallGroup(280,8);
// by ID

G=gap.SmallGroup(280,8);
# by ID

G:=PCGroup([5,-2,-2,-2,-5,-7,20,328,6004]);
// Polycyclic

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

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