direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: D7×C18, C126⋊3C2, C14⋊3C18, C63⋊4C22, C42.11C6, C3.(C6×D7), C7⋊3(C2×C18), C6.3(C3×D7), C21.3(C2×C6), (C3×D7).2C6, (C6×D7).2C3, SmallGroup(252,12)
Series: Derived ►Chief ►Lower central ►Upper central
| C7 — D7×C18 |
Generators and relations for D7×C18
G = < a,b,c | a18=b7=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of D7×C18
►On 126 points
Generators in S126
(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)
(1 120 30 83 42 101 67)(2 121 31 84 43 102 68)(3 122 32 85 44 103 69)(4 123 33 86 45 104 70)(5 124 34 87 46 105 71)(6 125 35 88 47 106 72)(7 126 36 89 48 107 55)(8 109 19 90 49 108 56)(9 110 20 73 50 91 57)(10 111 21 74 51 92 58)(11 112 22 75 52 93 59)(12 113 23 76 53 94 60)(13 114 24 77 54 95 61)(14 115 25 78 37 96 62)(15 116 26 79 38 97 63)(16 117 27 80 39 98 64)(17 118 28 81 40 99 65)(18 119 29 82 41 100 66)
(1 58)(2 59)(3 60)(4 61)(5 62)(6 63)(7 64)(8 65)(9 66)(10 67)(11 68)(12 69)(13 70)(14 71)(15 72)(16 55)(17 56)(18 57)(19 40)(20 41)(21 42)(22 43)(23 44)(24 45)(25 46)(26 47)(27 48)(28 49)(29 50)(30 51)(31 52)(32 53)(33 54)(34 37)(35 38)(36 39)(73 82)(74 83)(75 84)(76 85)(77 86)(78 87)(79 88)(80 89)(81 90)(91 119)(92 120)(93 121)(94 122)(95 123)(96 124)(97 125)(98 126)(99 109)(100 110)(101 111)(102 112)(103 113)(104 114)(105 115)(106 116)(107 117)(108 118)
G:=sub<Sym(126)| (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), (1,120,30,83,42,101,67)(2,121,31,84,43,102,68)(3,122,32,85,44,103,69)(4,123,33,86,45,104,70)(5,124,34,87,46,105,71)(6,125,35,88,47,106,72)(7,126,36,89,48,107,55)(8,109,19,90,49,108,56)(9,110,20,73,50,91,57)(10,111,21,74,51,92,58)(11,112,22,75,52,93,59)(12,113,23,76,53,94,60)(13,114,24,77,54,95,61)(14,115,25,78,37,96,62)(15,116,26,79,38,97,63)(16,117,27,80,39,98,64)(17,118,28,81,40,99,65)(18,119,29,82,41,100,66), (1,58)(2,59)(3,60)(4,61)(5,62)(6,63)(7,64)(8,65)(9,66)(10,67)(11,68)(12,69)(13,70)(14,71)(15,72)(16,55)(17,56)(18,57)(19,40)(20,41)(21,42)(22,43)(23,44)(24,45)(25,46)(26,47)(27,48)(28,49)(29,50)(30,51)(31,52)(32,53)(33,54)(34,37)(35,38)(36,39)(73,82)(74,83)(75,84)(76,85)(77,86)(78,87)(79,88)(80,89)(81,90)(91,119)(92,120)(93,121)(94,122)(95,123)(96,124)(97,125)(98,126)(99,109)(100,110)(101,111)(102,112)(103,113)(104,114)(105,115)(106,116)(107,117)(108,118)>;
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), (1,120,30,83,42,101,67)(2,121,31,84,43,102,68)(3,122,32,85,44,103,69)(4,123,33,86,45,104,70)(5,124,34,87,46,105,71)(6,125,35,88,47,106,72)(7,126,36,89,48,107,55)(8,109,19,90,49,108,56)(9,110,20,73,50,91,57)(10,111,21,74,51,92,58)(11,112,22,75,52,93,59)(12,113,23,76,53,94,60)(13,114,24,77,54,95,61)(14,115,25,78,37,96,62)(15,116,26,79,38,97,63)(16,117,27,80,39,98,64)(17,118,28,81,40,99,65)(18,119,29,82,41,100,66), (1,58)(2,59)(3,60)(4,61)(5,62)(6,63)(7,64)(8,65)(9,66)(10,67)(11,68)(12,69)(13,70)(14,71)(15,72)(16,55)(17,56)(18,57)(19,40)(20,41)(21,42)(22,43)(23,44)(24,45)(25,46)(26,47)(27,48)(28,49)(29,50)(30,51)(31,52)(32,53)(33,54)(34,37)(35,38)(36,39)(73,82)(74,83)(75,84)(76,85)(77,86)(78,87)(79,88)(80,89)(81,90)(91,119)(92,120)(93,121)(94,122)(95,123)(96,124)(97,125)(98,126)(99,109)(100,110)(101,111)(102,112)(103,113)(104,114)(105,115)(106,116)(107,117)(108,118) );
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)], [(1,120,30,83,42,101,67),(2,121,31,84,43,102,68),(3,122,32,85,44,103,69),(4,123,33,86,45,104,70),(5,124,34,87,46,105,71),(6,125,35,88,47,106,72),(7,126,36,89,48,107,55),(8,109,19,90,49,108,56),(9,110,20,73,50,91,57),(10,111,21,74,51,92,58),(11,112,22,75,52,93,59),(12,113,23,76,53,94,60),(13,114,24,77,54,95,61),(14,115,25,78,37,96,62),(15,116,26,79,38,97,63),(16,117,27,80,39,98,64),(17,118,28,81,40,99,65),(18,119,29,82,41,100,66)], [(1,58),(2,59),(3,60),(4,61),(5,62),(6,63),(7,64),(8,65),(9,66),(10,67),(11,68),(12,69),(13,70),(14,71),(15,72),(16,55),(17,56),(18,57),(19,40),(20,41),(21,42),(22,43),(23,44),(24,45),(25,46),(26,47),(27,48),(28,49),(29,50),(30,51),(31,52),(32,53),(33,54),(34,37),(35,38),(36,39),(73,82),(74,83),(75,84),(76,85),(77,86),(78,87),(79,88),(80,89),(81,90),(91,119),(92,120),(93,121),(94,122),(95,123),(96,124),(97,125),(98,126),(99,109),(100,110),(101,111),(102,112),(103,113),(104,114),(105,115),(106,116),(107,117),(108,118)]])
90 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 6A | 6B | 6C | 6D | 6E | 6F | 7A | 7B | 7C | 9A | ··· | 9F | 14A | 14B | 14C | 18A | ··· | 18F | 18G | ··· | 18R | 21A | ··· | 21F | 42A | ··· | 42F | 63A | ··· | 63R | 126A | ··· | 126R |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 6 | 6 | 6 | 6 | 6 | 6 | 7 | 7 | 7 | 9 | ··· | 9 | 14 | 14 | 14 | 18 | ··· | 18 | 18 | ··· | 18 | 21 | ··· | 21 | 42 | ··· | 42 | 63 | ··· | 63 | 126 | ··· | 126 |
| size | 1 | 1 | 7 | 7 | 1 | 1 | 1 | 1 | 7 | 7 | 7 | 7 | 2 | 2 | 2 | 1 | ··· | 1 | 2 | 2 | 2 | 1 | ··· | 1 | 7 | ··· | 7 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
90 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | ||||||||||
| image | C1 | C2 | C2 | C3 | C6 | C6 | C9 | C18 | C18 | D7 | D14 | C3×D7 | C6×D7 | C9×D7 | D7×C18 |
| kernel | D7×C18 | C9×D7 | C126 | C6×D7 | C3×D7 | C42 | D14 | D7 | C14 | C18 | C9 | C6 | C3 | C2 | C1 |
| # reps | 1 | 2 | 1 | 2 | 4 | 2 | 6 | 12 | 6 | 3 | 3 | 6 | 6 | 18 | 18 |
Matrix representation of D7×C18 ►in GL3(𝔽127) generated by
| 24 | 0 | 0 |
| 0 | 107 | 0 |
| 0 | 0 | 107 |
| 1 | 0 | 0 |
| 0 | 0 | 1 |
| 0 | 126 | 36 |
| 1 | 0 | 0 |
| 0 | 0 | 1 |
| 0 | 1 | 0 |
G:=sub<GL(3,GF(127))| [24,0,0,0,107,0,0,0,107],[1,0,0,0,0,126,0,1,36],[1,0,0,0,0,1,0,1,0] >;
D7×C18 in GAP, Magma, Sage, TeX
D_7\times C_{18} % in TeX
G:=Group("D7xC18"); // GroupNames label
G:=SmallGroup(252,12);
// by ID
G=gap.SmallGroup(252,12);
# by ID
G:=PCGroup([5,-2,-2,-3,-3,-7,57,5404]);
// Polycyclic
G:=Group<a,b,c|a^18=b^7=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
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