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G = Dic5×C7⋊C3order 420 = 22·3·5·7

Direct product of Dic5 and C7⋊C3

Aliases: Dic5×C7⋊C3, C355C12, C70.3C6, (C7×Dic5)⋊C3, C72(C3×Dic5), C14.2(C3×D5), C52(C4×C7⋊C3), C2.(D5×C7⋊C3), (C5×C7⋊C3)⋊5C4, C10.(C2×C7⋊C3), (C2×C7⋊C3).2D5, (C10×C7⋊C3).3C2, SmallGroup(420,2)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C35 — Dic5×C7⋊C3
 Chief series C1 — C7 — C35 — C70 — C10×C7⋊C3 — Dic5×C7⋊C3
 Lower central C35 — Dic5×C7⋊C3
 Upper central C1 — C2

Generators and relations for Dic5×C7⋊C3
G = < a,b,c,d | a10=c7=d3=1, b2=a5, bab-1=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c4 >

Smallest permutation representation of Dic5×C7⋊C3
On 140 points
Generators in S140
(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 30 6 25)(2 29 7 24)(3 28 8 23)(4 27 9 22)(5 26 10 21)(11 130 16 125)(12 129 17 124)(13 128 18 123)(14 127 19 122)(15 126 20 121)(31 110 36 105)(32 109 37 104)(33 108 38 103)(34 107 39 102)(35 106 40 101)(41 100 46 95)(42 99 47 94)(43 98 48 93)(44 97 49 92)(45 96 50 91)(51 90 56 85)(52 89 57 84)(53 88 58 83)(54 87 59 82)(55 86 60 81)(61 80 66 75)(62 79 67 74)(63 78 68 73)(64 77 69 72)(65 76 70 71)(111 136 116 131)(112 135 117 140)(113 134 118 139)(114 133 119 138)(115 132 120 137)
(1 109 118 91 121 86 76)(2 110 119 92 122 87 77)(3 101 120 93 123 88 78)(4 102 111 94 124 89 79)(5 103 112 95 125 90 80)(6 104 113 96 126 81 71)(7 105 114 97 127 82 72)(8 106 115 98 128 83 73)(9 107 116 99 129 84 74)(10 108 117 100 130 85 75)(11 56 66 26 33 135 41)(12 57 67 27 34 136 42)(13 58 68 28 35 137 43)(14 59 69 29 36 138 44)(15 60 70 30 37 139 45)(16 51 61 21 38 140 46)(17 52 62 22 39 131 47)(18 53 63 23 40 132 48)(19 54 64 24 31 133 49)(20 55 65 25 32 134 50)
(11 33 135)(12 34 136)(13 35 137)(14 36 138)(15 37 139)(16 38 140)(17 39 131)(18 40 132)(19 31 133)(20 32 134)(41 66 56)(42 67 57)(43 68 58)(44 69 59)(45 70 60)(46 61 51)(47 62 52)(48 63 53)(49 64 54)(50 65 55)(71 81 96)(72 82 97)(73 83 98)(74 84 99)(75 85 100)(76 86 91)(77 87 92)(78 88 93)(79 89 94)(80 90 95)(101 120 123)(102 111 124)(103 112 125)(104 113 126)(105 114 127)(106 115 128)(107 116 129)(108 117 130)(109 118 121)(110 119 122)

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

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

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

40 conjugacy classes

 class 1 2 3A 3B 4A 4B 5A 5B 6A 6B 7A 7B 10A 10B 12A 12B 12C 12D 14A 14B 15A 15B 15C 15D 28A 28B 28C 28D 30A 30B 30C 30D 35A 35B 35C 35D 70A 70B 70C 70D order 1 2 3 3 4 4 5 5 6 6 7 7 10 10 12 12 12 12 14 14 15 15 15 15 28 28 28 28 30 30 30 30 35 35 35 35 70 70 70 70 size 1 1 7 7 5 5 2 2 7 7 3 3 2 2 35 35 35 35 3 3 14 14 14 14 15 15 15 15 14 14 14 14 6 6 6 6 6 6 6 6

40 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 3 3 3 6 6 type + + + - image C1 C2 C3 C4 C6 C12 D5 Dic5 C3×D5 C3×Dic5 C7⋊C3 C2×C7⋊C3 C4×C7⋊C3 D5×C7⋊C3 Dic5×C7⋊C3 kernel Dic5×C7⋊C3 C10×C7⋊C3 C7×Dic5 C5×C7⋊C3 C70 C35 C2×C7⋊C3 C7⋊C3 C14 C7 Dic5 C10 C5 C2 C1 # reps 1 1 2 2 2 4 2 2 4 4 2 2 4 4 4

Matrix representation of Dic5×C7⋊C3 in GL5(𝔽421)

 0 420 0 0 0 1 111 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 287 401 0 0 0 119 134 0 0 0 0 0 420 0 0 0 0 0 420 0 0 0 0 0 420
,
 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 1 0 245 0 0 0 1 244
,
 400 0 0 0 0 0 400 0 0 0 0 0 244 176 0 0 0 420 177 1 0 0 420 1 0

G:=sub<GL(5,GF(421))| [0,1,0,0,0,420,111,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[287,119,0,0,0,401,134,0,0,0,0,0,420,0,0,0,0,0,420,0,0,0,0,0,420],[1,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,1,245,244],[400,0,0,0,0,0,400,0,0,0,0,0,244,420,420,0,0,176,177,1,0,0,0,1,0] >;

Dic5×C7⋊C3 in GAP, Magma, Sage, TeX

{\rm Dic}_5\times C_7\rtimes C_3
% in TeX

G:=Group("Dic5xC7:C3");
// GroupNames label

G:=SmallGroup(420,2);
// by ID

G=gap.SmallGroup(420,2);
# by ID

G:=PCGroup([5,-2,-3,-2,-5,-7,30,963,1509]);
// Polycyclic

G:=Group<a,b,c,d|a^10=c^7=d^3=1,b^2=a^5,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^4>;
// generators/relations

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