direct product, metacyclic, supersoluble, monomial, A-group
Aliases: C2×C7⋊C18, D14⋊C9, C14⋊C18, D7⋊C18, C6.3F7, C42.3C6, C7⋊(C2×C18), C7⋊C9⋊C22, C3.(C2×F7), C21.(C2×C6), (C3×D7).C6, (C6×D7).C3, (C2×C7⋊C9)⋊C2, SmallGroup(252,7)
Series: Derived ►Chief ►Lower central ►Upper central
| C7 — C2×C7⋊C18 |
Generators and relations for C2×C7⋊C18
G = < a,b,c | a2=b7=c18=1, ab=ba, ac=ca, cbc-1=b3 >
Smallest permutation representation of C2×C7⋊C18
►On 126 points
Generators in S126
(1 10)(2 11)(3 12)(4 13)(5 14)(6 15)(7 16)(8 17)(9 18)(19 95)(20 96)(21 97)(22 98)(23 99)(24 100)(25 101)(26 102)(27 103)(28 104)(29 105)(30 106)(31 107)(32 108)(33 91)(34 92)(35 93)(36 94)(37 120)(38 121)(39 122)(40 123)(41 124)(42 125)(43 126)(44 109)(45 110)(46 111)(47 112)(48 113)(49 114)(50 115)(51 116)(52 117)(53 118)(54 119)(55 86)(56 87)(57 88)(58 89)(59 90)(60 73)(61 74)(62 75)(63 76)(64 77)(65 78)(66 79)(67 80)(68 81)(69 82)(70 83)(71 84)(72 85)
(1 74 47 106 21 121 70)(2 107 71 48 122 75 22)(3 49 23 72 76 108 123)(4 55 124 24 91 50 77)(5 25 78 125 51 56 92)(6 126 93 79 57 26 52)(7 80 53 94 27 109 58)(8 95 59 54 110 81 28)(9 37 29 60 82 96 111)(10 61 112 30 97 38 83)(11 31 84 113 39 62 98)(12 114 99 85 63 32 40)(13 86 41 100 33 115 64)(14 101 65 42 116 87 34)(15 43 35 66 88 102 117)(16 67 118 36 103 44 89)(17 19 90 119 45 68 104)(18 120 105 73 69 20 46)
(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)
G:=sub<Sym(126)| (1,10)(2,11)(3,12)(4,13)(5,14)(6,15)(7,16)(8,17)(9,18)(19,95)(20,96)(21,97)(22,98)(23,99)(24,100)(25,101)(26,102)(27,103)(28,104)(29,105)(30,106)(31,107)(32,108)(33,91)(34,92)(35,93)(36,94)(37,120)(38,121)(39,122)(40,123)(41,124)(42,125)(43,126)(44,109)(45,110)(46,111)(47,112)(48,113)(49,114)(50,115)(51,116)(52,117)(53,118)(54,119)(55,86)(56,87)(57,88)(58,89)(59,90)(60,73)(61,74)(62,75)(63,76)(64,77)(65,78)(66,79)(67,80)(68,81)(69,82)(70,83)(71,84)(72,85), (1,74,47,106,21,121,70)(2,107,71,48,122,75,22)(3,49,23,72,76,108,123)(4,55,124,24,91,50,77)(5,25,78,125,51,56,92)(6,126,93,79,57,26,52)(7,80,53,94,27,109,58)(8,95,59,54,110,81,28)(9,37,29,60,82,96,111)(10,61,112,30,97,38,83)(11,31,84,113,39,62,98)(12,114,99,85,63,32,40)(13,86,41,100,33,115,64)(14,101,65,42,116,87,34)(15,43,35,66,88,102,117)(16,67,118,36,103,44,89)(17,19,90,119,45,68,104)(18,120,105,73,69,20,46), (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)>;
G:=Group( (1,10)(2,11)(3,12)(4,13)(5,14)(6,15)(7,16)(8,17)(9,18)(19,95)(20,96)(21,97)(22,98)(23,99)(24,100)(25,101)(26,102)(27,103)(28,104)(29,105)(30,106)(31,107)(32,108)(33,91)(34,92)(35,93)(36,94)(37,120)(38,121)(39,122)(40,123)(41,124)(42,125)(43,126)(44,109)(45,110)(46,111)(47,112)(48,113)(49,114)(50,115)(51,116)(52,117)(53,118)(54,119)(55,86)(56,87)(57,88)(58,89)(59,90)(60,73)(61,74)(62,75)(63,76)(64,77)(65,78)(66,79)(67,80)(68,81)(69,82)(70,83)(71,84)(72,85), (1,74,47,106,21,121,70)(2,107,71,48,122,75,22)(3,49,23,72,76,108,123)(4,55,124,24,91,50,77)(5,25,78,125,51,56,92)(6,126,93,79,57,26,52)(7,80,53,94,27,109,58)(8,95,59,54,110,81,28)(9,37,29,60,82,96,111)(10,61,112,30,97,38,83)(11,31,84,113,39,62,98)(12,114,99,85,63,32,40)(13,86,41,100,33,115,64)(14,101,65,42,116,87,34)(15,43,35,66,88,102,117)(16,67,118,36,103,44,89)(17,19,90,119,45,68,104)(18,120,105,73,69,20,46), (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) );
G=PermutationGroup([[(1,10),(2,11),(3,12),(4,13),(5,14),(6,15),(7,16),(8,17),(9,18),(19,95),(20,96),(21,97),(22,98),(23,99),(24,100),(25,101),(26,102),(27,103),(28,104),(29,105),(30,106),(31,107),(32,108),(33,91),(34,92),(35,93),(36,94),(37,120),(38,121),(39,122),(40,123),(41,124),(42,125),(43,126),(44,109),(45,110),(46,111),(47,112),(48,113),(49,114),(50,115),(51,116),(52,117),(53,118),(54,119),(55,86),(56,87),(57,88),(58,89),(59,90),(60,73),(61,74),(62,75),(63,76),(64,77),(65,78),(66,79),(67,80),(68,81),(69,82),(70,83),(71,84),(72,85)], [(1,74,47,106,21,121,70),(2,107,71,48,122,75,22),(3,49,23,72,76,108,123),(4,55,124,24,91,50,77),(5,25,78,125,51,56,92),(6,126,93,79,57,26,52),(7,80,53,94,27,109,58),(8,95,59,54,110,81,28),(9,37,29,60,82,96,111),(10,61,112,30,97,38,83),(11,31,84,113,39,62,98),(12,114,99,85,63,32,40),(13,86,41,100,33,115,64),(14,101,65,42,116,87,34),(15,43,35,66,88,102,117),(16,67,118,36,103,44,89),(17,19,90,119,45,68,104),(18,120,105,73,69,20,46)], [(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)]])
42 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 6A | 6B | 6C | 6D | 6E | 6F | 7 | 9A | ··· | 9F | 14 | 18A | ··· | 18R | 21A | 21B | 42A | 42B |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 6 | 6 | 6 | 6 | 6 | 6 | 7 | 9 | ··· | 9 | 14 | 18 | ··· | 18 | 21 | 21 | 42 | 42 |
| size | 1 | 1 | 7 | 7 | 1 | 1 | 1 | 1 | 7 | 7 | 7 | 7 | 6 | 7 | ··· | 7 | 6 | 7 | ··· | 7 | 6 | 6 | 6 | 6 |
42 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 6 | 6 | 6 | 6 |
| type | + | + | + | + | + | ||||||||
| image | C1 | C2 | C2 | C3 | C6 | C6 | C9 | C18 | C18 | F7 | C2×F7 | C7⋊C18 | C2×C7⋊C18 |
| kernel | C2×C7⋊C18 | C7⋊C18 | C2×C7⋊C9 | C6×D7 | C3×D7 | C42 | D14 | D7 | C14 | C6 | C3 | C2 | C1 |
| # reps | 1 | 2 | 1 | 2 | 4 | 2 | 6 | 12 | 6 | 1 | 1 | 2 | 2 |
Matrix representation of C2×C7⋊C18 ►in GL8(𝔽127)
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 126 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 126 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 126 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 126 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 126 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 126 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 126 |
| 59 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 124 | 67 | 37 | 0 | 90 |
| 0 | 0 | 3 | 87 | 0 | 40 | 124 | 30 |
| 0 | 0 | 0 | 27 | 37 | 40 | 87 | 90 |
| 0 | 0 | 90 | 87 | 40 | 37 | 27 | 0 |
| 0 | 0 | 30 | 124 | 40 | 0 | 87 | 3 |
| 0 | 0 | 90 | 0 | 37 | 67 | 124 | 3 |
G:=sub<GL(8,GF(127))| [1,0,0,0,0,0,0,0,0,126,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,126,126,126,126,126,126],[59,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,3,3,0,90,30,90,0,0,124,87,27,87,124,0,0,0,67,0,37,40,40,37,0,0,37,40,40,37,0,67,0,0,0,124,87,27,87,124,0,0,90,30,90,0,3,3] >;
C2×C7⋊C18 in GAP, Magma, Sage, TeX
C_2\times C_7\rtimes C_{18} % in TeX
G:=Group("C2xC7:C18"); // GroupNames label
G:=SmallGroup(252,7);
// by ID
G=gap.SmallGroup(252,7);
# by ID
G:=PCGroup([5,-2,-2,-3,-3,-7,57,5404,914]);
// Polycyclic
G:=Group<a,b,c|a^2=b^7=c^18=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^3>;
// generators/relations
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