direct product, metacyclic, supersoluble, monomial, Z-group
Aliases: Dic5×C7⋊C3, C35⋊5C12, C70.3C6, (C7×Dic5)⋊C3, C7⋊2(C3×Dic5), C14.2(C3×D5), C5⋊2(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
| C35 — Dic5×C7⋊C3 |
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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