direct product, metacyclic, supersoluble, monomial, A-group
Aliases: C2×C10×C7⋊C3, C70⋊4C6, C14⋊2C30, (C2×C70)⋊3C3, C35⋊6(C2×C6), C7⋊2(C2×C30), (C2×C14)⋊3C15, SmallGroup(420,31)
Series: Derived ►Chief ►Lower central ►Upper central
| C7 — C2×C10×C7⋊C3 |
Generators and relations for C2×C10×C7⋊C3
G = < a,b,c,d | a2=b10=c7=d3=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c4 >
Smallest permutation representation of C2×C10×C7⋊C3
►On 140 points
Generators in S140
(1 35)(2 36)(3 37)(4 38)(5 39)(6 40)(7 31)(8 32)(9 33)(10 34)(11 101)(12 102)(13 103)(14 104)(15 105)(16 106)(17 107)(18 108)(19 109)(20 110)(21 79)(22 80)(23 71)(24 72)(25 73)(26 74)(27 75)(28 76)(29 77)(30 78)(41 85)(42 86)(43 87)(44 88)(45 89)(46 90)(47 81)(48 82)(49 83)(50 84)(51 129)(52 130)(53 121)(54 122)(55 123)(56 124)(57 125)(58 126)(59 127)(60 128)(61 116)(62 117)(63 118)(64 119)(65 120)(66 111)(67 112)(68 113)(69 114)(70 115)(91 137)(92 138)(93 139)(94 140)(95 131)(96 132)(97 133)(98 134)(99 135)(100 136)
(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)
(1 117 96 78 110 129 87)(2 118 97 79 101 130 88)(3 119 98 80 102 121 89)(4 120 99 71 103 122 90)(5 111 100 72 104 123 81)(6 112 91 73 105 124 82)(7 113 92 74 106 125 83)(8 114 93 75 107 126 84)(9 115 94 76 108 127 85)(10 116 95 77 109 128 86)(11 52 44 36 63 133 21)(12 53 45 37 64 134 22)(13 54 46 38 65 135 23)(14 55 47 39 66 136 24)(15 56 48 40 67 137 25)(16 57 49 31 68 138 26)(17 58 50 32 69 139 27)(18 59 41 33 70 140 28)(19 60 42 34 61 131 29)(20 51 43 35 62 132 30)
(11 63 133)(12 64 134)(13 65 135)(14 66 136)(15 67 137)(16 68 138)(17 69 139)(18 70 140)(19 61 131)(20 62 132)(21 44 52)(22 45 53)(23 46 54)(24 47 55)(25 48 56)(26 49 57)(27 50 58)(28 41 59)(29 42 60)(30 43 51)(71 90 122)(72 81 123)(73 82 124)(74 83 125)(75 84 126)(76 85 127)(77 86 128)(78 87 129)(79 88 130)(80 89 121)(91 105 112)(92 106 113)(93 107 114)(94 108 115)(95 109 116)(96 110 117)(97 101 118)(98 102 119)(99 103 120)(100 104 111)
G:=sub<Sym(140)| (1,35)(2,36)(3,37)(4,38)(5,39)(6,40)(7,31)(8,32)(9,33)(10,34)(11,101)(12,102)(13,103)(14,104)(15,105)(16,106)(17,107)(18,108)(19,109)(20,110)(21,79)(22,80)(23,71)(24,72)(25,73)(26,74)(27,75)(28,76)(29,77)(30,78)(41,85)(42,86)(43,87)(44,88)(45,89)(46,90)(47,81)(48,82)(49,83)(50,84)(51,129)(52,130)(53,121)(54,122)(55,123)(56,124)(57,125)(58,126)(59,127)(60,128)(61,116)(62,117)(63,118)(64,119)(65,120)(66,111)(67,112)(68,113)(69,114)(70,115)(91,137)(92,138)(93,139)(94,140)(95,131)(96,132)(97,133)(98,134)(99,135)(100,136), (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), (1,117,96,78,110,129,87)(2,118,97,79,101,130,88)(3,119,98,80,102,121,89)(4,120,99,71,103,122,90)(5,111,100,72,104,123,81)(6,112,91,73,105,124,82)(7,113,92,74,106,125,83)(8,114,93,75,107,126,84)(9,115,94,76,108,127,85)(10,116,95,77,109,128,86)(11,52,44,36,63,133,21)(12,53,45,37,64,134,22)(13,54,46,38,65,135,23)(14,55,47,39,66,136,24)(15,56,48,40,67,137,25)(16,57,49,31,68,138,26)(17,58,50,32,69,139,27)(18,59,41,33,70,140,28)(19,60,42,34,61,131,29)(20,51,43,35,62,132,30), (11,63,133)(12,64,134)(13,65,135)(14,66,136)(15,67,137)(16,68,138)(17,69,139)(18,70,140)(19,61,131)(20,62,132)(21,44,52)(22,45,53)(23,46,54)(24,47,55)(25,48,56)(26,49,57)(27,50,58)(28,41,59)(29,42,60)(30,43,51)(71,90,122)(72,81,123)(73,82,124)(74,83,125)(75,84,126)(76,85,127)(77,86,128)(78,87,129)(79,88,130)(80,89,121)(91,105,112)(92,106,113)(93,107,114)(94,108,115)(95,109,116)(96,110,117)(97,101,118)(98,102,119)(99,103,120)(100,104,111)>;
G:=Group( (1,35)(2,36)(3,37)(4,38)(5,39)(6,40)(7,31)(8,32)(9,33)(10,34)(11,101)(12,102)(13,103)(14,104)(15,105)(16,106)(17,107)(18,108)(19,109)(20,110)(21,79)(22,80)(23,71)(24,72)(25,73)(26,74)(27,75)(28,76)(29,77)(30,78)(41,85)(42,86)(43,87)(44,88)(45,89)(46,90)(47,81)(48,82)(49,83)(50,84)(51,129)(52,130)(53,121)(54,122)(55,123)(56,124)(57,125)(58,126)(59,127)(60,128)(61,116)(62,117)(63,118)(64,119)(65,120)(66,111)(67,112)(68,113)(69,114)(70,115)(91,137)(92,138)(93,139)(94,140)(95,131)(96,132)(97,133)(98,134)(99,135)(100,136), (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), (1,117,96,78,110,129,87)(2,118,97,79,101,130,88)(3,119,98,80,102,121,89)(4,120,99,71,103,122,90)(5,111,100,72,104,123,81)(6,112,91,73,105,124,82)(7,113,92,74,106,125,83)(8,114,93,75,107,126,84)(9,115,94,76,108,127,85)(10,116,95,77,109,128,86)(11,52,44,36,63,133,21)(12,53,45,37,64,134,22)(13,54,46,38,65,135,23)(14,55,47,39,66,136,24)(15,56,48,40,67,137,25)(16,57,49,31,68,138,26)(17,58,50,32,69,139,27)(18,59,41,33,70,140,28)(19,60,42,34,61,131,29)(20,51,43,35,62,132,30), (11,63,133)(12,64,134)(13,65,135)(14,66,136)(15,67,137)(16,68,138)(17,69,139)(18,70,140)(19,61,131)(20,62,132)(21,44,52)(22,45,53)(23,46,54)(24,47,55)(25,48,56)(26,49,57)(27,50,58)(28,41,59)(29,42,60)(30,43,51)(71,90,122)(72,81,123)(73,82,124)(74,83,125)(75,84,126)(76,85,127)(77,86,128)(78,87,129)(79,88,130)(80,89,121)(91,105,112)(92,106,113)(93,107,114)(94,108,115)(95,109,116)(96,110,117)(97,101,118)(98,102,119)(99,103,120)(100,104,111) );
G=PermutationGroup([[(1,35),(2,36),(3,37),(4,38),(5,39),(6,40),(7,31),(8,32),(9,33),(10,34),(11,101),(12,102),(13,103),(14,104),(15,105),(16,106),(17,107),(18,108),(19,109),(20,110),(21,79),(22,80),(23,71),(24,72),(25,73),(26,74),(27,75),(28,76),(29,77),(30,78),(41,85),(42,86),(43,87),(44,88),(45,89),(46,90),(47,81),(48,82),(49,83),(50,84),(51,129),(52,130),(53,121),(54,122),(55,123),(56,124),(57,125),(58,126),(59,127),(60,128),(61,116),(62,117),(63,118),(64,119),(65,120),(66,111),(67,112),(68,113),(69,114),(70,115),(91,137),(92,138),(93,139),(94,140),(95,131),(96,132),(97,133),(98,134),(99,135),(100,136)], [(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)], [(1,117,96,78,110,129,87),(2,118,97,79,101,130,88),(3,119,98,80,102,121,89),(4,120,99,71,103,122,90),(5,111,100,72,104,123,81),(6,112,91,73,105,124,82),(7,113,92,74,106,125,83),(8,114,93,75,107,126,84),(9,115,94,76,108,127,85),(10,116,95,77,109,128,86),(11,52,44,36,63,133,21),(12,53,45,37,64,134,22),(13,54,46,38,65,135,23),(14,55,47,39,66,136,24),(15,56,48,40,67,137,25),(16,57,49,31,68,138,26),(17,58,50,32,69,139,27),(18,59,41,33,70,140,28),(19,60,42,34,61,131,29),(20,51,43,35,62,132,30)], [(11,63,133),(12,64,134),(13,65,135),(14,66,136),(15,67,137),(16,68,138),(17,69,139),(18,70,140),(19,61,131),(20,62,132),(21,44,52),(22,45,53),(23,46,54),(24,47,55),(25,48,56),(26,49,57),(27,50,58),(28,41,59),(29,42,60),(30,43,51),(71,90,122),(72,81,123),(73,82,124),(74,83,125),(75,84,126),(76,85,127),(77,86,128),(78,87,129),(79,88,130),(80,89,121),(91,105,112),(92,106,113),(93,107,114),(94,108,115),(95,109,116),(96,110,117),(97,101,118),(98,102,119),(99,103,120),(100,104,111)]])
100 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 5A | 5B | 5C | 5D | 6A | ··· | 6F | 7A | 7B | 10A | ··· | 10L | 14A | ··· | 14F | 15A | ··· | 15H | 30A | ··· | 30X | 35A | ··· | 35H | 70A | ··· | 70X |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 5 | 5 | 5 | 5 | 6 | ··· | 6 | 7 | 7 | 10 | ··· | 10 | 14 | ··· | 14 | 15 | ··· | 15 | 30 | ··· | 30 | 35 | ··· | 35 | 70 | ··· | 70 |
| size | 1 | 1 | 1 | 1 | 7 | 7 | 1 | 1 | 1 | 1 | 7 | ··· | 7 | 3 | 3 | 1 | ··· | 1 | 3 | ··· | 3 | 7 | ··· | 7 | 7 | ··· | 7 | 3 | ··· | 3 | 3 | ··· | 3 |
100 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 3 | 3 | 3 | 3 |
| type | + | + | ||||||||||
| image | C1 | C2 | C3 | C5 | C6 | C10 | C15 | C30 | C7⋊C3 | C2×C7⋊C3 | C5×C7⋊C3 | C10×C7⋊C3 |
| kernel | C2×C10×C7⋊C3 | C10×C7⋊C3 | C2×C70 | C22×C7⋊C3 | C70 | C2×C7⋊C3 | C2×C14 | C14 | C2×C10 | C10 | C22 | C2 |
| # reps | 1 | 3 | 2 | 4 | 6 | 12 | 8 | 24 | 2 | 6 | 8 | 24 |
Matrix representation of C2×C10×C7⋊C3 ►in GL4(𝔽211) generated by
| 1 | 0 | 0 | 0 |
| 0 | 210 | 0 | 0 |
| 0 | 0 | 210 | 0 |
| 0 | 0 | 0 | 210 |
| 210 | 0 | 0 | 0 |
| 0 | 23 | 0 | 0 |
| 0 | 0 | 23 | 0 |
| 0 | 0 | 0 | 23 |
| 1 | 0 | 0 | 0 |
| 0 | 210 | 190 | 1 |
| 0 | 0 | 190 | 1 |
| 0 | 210 | 191 | 1 |
| 196 | 0 | 0 | 0 |
| 0 | 191 | 1 | 21 |
| 0 | 1 | 0 | 0 |
| 0 | 1 | 1 | 20 |
G:=sub<GL(4,GF(211))| [1,0,0,0,0,210,0,0,0,0,210,0,0,0,0,210],[210,0,0,0,0,23,0,0,0,0,23,0,0,0,0,23],[1,0,0,0,0,210,0,210,0,190,190,191,0,1,1,1],[196,0,0,0,0,191,1,1,0,1,0,1,0,21,0,20] >;
C2×C10×C7⋊C3 in GAP, Magma, Sage, TeX
C_2\times C_{10}\times C_7\rtimes C_3 % in TeX
G:=Group("C2xC10xC7:C3"); // GroupNames label
G:=SmallGroup(420,31);
// by ID
G=gap.SmallGroup(420,31);
# by ID
G:=PCGroup([5,-2,-2,-3,-5,-7,764]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^10=c^7=d^3=1,a*b=b*a,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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