direct product, metacyclic, supersoluble, monomial, A-group
Aliases: S3×C7⋊C9, C21⋊3C18, (S3×C7)⋊C9, C7⋊2(S3×C9), (S3×C21).C3, (C3×C21).5C6, C21.10(C3×S3), C3⋊(C2×C7⋊C9), (C3×C7⋊C9)⋊3C2, C3.5(S3×C7⋊C3), (C3×S3).(C7⋊C3), C32.2(C2×C7⋊C3), SmallGroup(378,16)
Series: Derived ►Chief ►Lower central ►Upper central
| C21 — S3×C7⋊C9 |
Generators and relations for S3×C7⋊C9
G = < a,b,c,d | a3=b2=c7=d9=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c4 >
Smallest permutation representation of S3×C7⋊C9
►On 126 points
Generators in S126
(1 4 7)(2 5 8)(3 6 9)(10 16 13)(11 17 14)(12 18 15)(19 25 22)(20 26 23)(21 27 24)(28 34 31)(29 35 32)(30 36 33)(37 43 40)(38 44 41)(39 45 42)(46 52 49)(47 53 50)(48 54 51)(55 61 58)(56 62 59)(57 63 60)(64 67 70)(65 68 71)(66 69 72)(73 76 79)(74 77 80)(75 78 81)(82 85 88)(83 86 89)(84 87 90)(91 94 97)(92 95 98)(93 96 99)(100 103 106)(101 104 107)(102 105 108)(109 112 115)(110 113 116)(111 114 117)(118 124 121)(119 125 122)(120 126 123)
(1 38)(2 39)(3 40)(4 41)(5 42)(6 43)(7 44)(8 45)(9 37)(10 88)(11 89)(12 90)(13 82)(14 83)(15 84)(16 85)(17 86)(18 87)(19 74)(20 75)(21 76)(22 77)(23 78)(24 79)(25 80)(26 81)(27 73)(28 71)(29 72)(30 64)(31 65)(32 66)(33 67)(34 68)(35 69)(36 70)(46 101)(47 102)(48 103)(49 104)(50 105)(51 106)(52 107)(53 108)(54 100)(55 117)(56 109)(57 110)(58 111)(59 112)(60 113)(61 114)(62 115)(63 116)(91 126)(92 118)(93 119)(94 120)(95 121)(96 122)(97 123)(98 124)(99 125)
(1 97 86 80 69 111 101)(2 70 98 112 87 102 81)(3 88 71 103 99 73 113)(4 91 89 74 72 114 104)(5 64 92 115 90 105 75)(6 82 65 106 93 76 116)(7 94 83 77 66 117 107)(8 67 95 109 84 108 78)(9 85 68 100 96 79 110)(10 28 48 125 27 60 40)(11 19 29 61 49 41 126)(12 50 20 42 30 118 62)(13 31 51 119 21 63 43)(14 22 32 55 52 44 120)(15 53 23 45 33 121 56)(16 34 54 122 24 57 37)(17 25 35 58 46 38 123)(18 47 26 39 36 124 59)
(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)| (1,4,7)(2,5,8)(3,6,9)(10,16,13)(11,17,14)(12,18,15)(19,25,22)(20,26,23)(21,27,24)(28,34,31)(29,35,32)(30,36,33)(37,43,40)(38,44,41)(39,45,42)(46,52,49)(47,53,50)(48,54,51)(55,61,58)(56,62,59)(57,63,60)(64,67,70)(65,68,71)(66,69,72)(73,76,79)(74,77,80)(75,78,81)(82,85,88)(83,86,89)(84,87,90)(91,94,97)(92,95,98)(93,96,99)(100,103,106)(101,104,107)(102,105,108)(109,112,115)(110,113,116)(111,114,117)(118,124,121)(119,125,122)(120,126,123), (1,38)(2,39)(3,40)(4,41)(5,42)(6,43)(7,44)(8,45)(9,37)(10,88)(11,89)(12,90)(13,82)(14,83)(15,84)(16,85)(17,86)(18,87)(19,74)(20,75)(21,76)(22,77)(23,78)(24,79)(25,80)(26,81)(27,73)(28,71)(29,72)(30,64)(31,65)(32,66)(33,67)(34,68)(35,69)(36,70)(46,101)(47,102)(48,103)(49,104)(50,105)(51,106)(52,107)(53,108)(54,100)(55,117)(56,109)(57,110)(58,111)(59,112)(60,113)(61,114)(62,115)(63,116)(91,126)(92,118)(93,119)(94,120)(95,121)(96,122)(97,123)(98,124)(99,125), (1,97,86,80,69,111,101)(2,70,98,112,87,102,81)(3,88,71,103,99,73,113)(4,91,89,74,72,114,104)(5,64,92,115,90,105,75)(6,82,65,106,93,76,116)(7,94,83,77,66,117,107)(8,67,95,109,84,108,78)(9,85,68,100,96,79,110)(10,28,48,125,27,60,40)(11,19,29,61,49,41,126)(12,50,20,42,30,118,62)(13,31,51,119,21,63,43)(14,22,32,55,52,44,120)(15,53,23,45,33,121,56)(16,34,54,122,24,57,37)(17,25,35,58,46,38,123)(18,47,26,39,36,124,59), (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( (1,4,7)(2,5,8)(3,6,9)(10,16,13)(11,17,14)(12,18,15)(19,25,22)(20,26,23)(21,27,24)(28,34,31)(29,35,32)(30,36,33)(37,43,40)(38,44,41)(39,45,42)(46,52,49)(47,53,50)(48,54,51)(55,61,58)(56,62,59)(57,63,60)(64,67,70)(65,68,71)(66,69,72)(73,76,79)(74,77,80)(75,78,81)(82,85,88)(83,86,89)(84,87,90)(91,94,97)(92,95,98)(93,96,99)(100,103,106)(101,104,107)(102,105,108)(109,112,115)(110,113,116)(111,114,117)(118,124,121)(119,125,122)(120,126,123), (1,38)(2,39)(3,40)(4,41)(5,42)(6,43)(7,44)(8,45)(9,37)(10,88)(11,89)(12,90)(13,82)(14,83)(15,84)(16,85)(17,86)(18,87)(19,74)(20,75)(21,76)(22,77)(23,78)(24,79)(25,80)(26,81)(27,73)(28,71)(29,72)(30,64)(31,65)(32,66)(33,67)(34,68)(35,69)(36,70)(46,101)(47,102)(48,103)(49,104)(50,105)(51,106)(52,107)(53,108)(54,100)(55,117)(56,109)(57,110)(58,111)(59,112)(60,113)(61,114)(62,115)(63,116)(91,126)(92,118)(93,119)(94,120)(95,121)(96,122)(97,123)(98,124)(99,125), (1,97,86,80,69,111,101)(2,70,98,112,87,102,81)(3,88,71,103,99,73,113)(4,91,89,74,72,114,104)(5,64,92,115,90,105,75)(6,82,65,106,93,76,116)(7,94,83,77,66,117,107)(8,67,95,109,84,108,78)(9,85,68,100,96,79,110)(10,28,48,125,27,60,40)(11,19,29,61,49,41,126)(12,50,20,42,30,118,62)(13,31,51,119,21,63,43)(14,22,32,55,52,44,120)(15,53,23,45,33,121,56)(16,34,54,122,24,57,37)(17,25,35,58,46,38,123)(18,47,26,39,36,124,59), (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([[(1,4,7),(2,5,8),(3,6,9),(10,16,13),(11,17,14),(12,18,15),(19,25,22),(20,26,23),(21,27,24),(28,34,31),(29,35,32),(30,36,33),(37,43,40),(38,44,41),(39,45,42),(46,52,49),(47,53,50),(48,54,51),(55,61,58),(56,62,59),(57,63,60),(64,67,70),(65,68,71),(66,69,72),(73,76,79),(74,77,80),(75,78,81),(82,85,88),(83,86,89),(84,87,90),(91,94,97),(92,95,98),(93,96,99),(100,103,106),(101,104,107),(102,105,108),(109,112,115),(110,113,116),(111,114,117),(118,124,121),(119,125,122),(120,126,123)], [(1,38),(2,39),(3,40),(4,41),(5,42),(6,43),(7,44),(8,45),(9,37),(10,88),(11,89),(12,90),(13,82),(14,83),(15,84),(16,85),(17,86),(18,87),(19,74),(20,75),(21,76),(22,77),(23,78),(24,79),(25,80),(26,81),(27,73),(28,71),(29,72),(30,64),(31,65),(32,66),(33,67),(34,68),(35,69),(36,70),(46,101),(47,102),(48,103),(49,104),(50,105),(51,106),(52,107),(53,108),(54,100),(55,117),(56,109),(57,110),(58,111),(59,112),(60,113),(61,114),(62,115),(63,116),(91,126),(92,118),(93,119),(94,120),(95,121),(96,122),(97,123),(98,124),(99,125)], [(1,97,86,80,69,111,101),(2,70,98,112,87,102,81),(3,88,71,103,99,73,113),(4,91,89,74,72,114,104),(5,64,92,115,90,105,75),(6,82,65,106,93,76,116),(7,94,83,77,66,117,107),(8,67,95,109,84,108,78),(9,85,68,100,96,79,110),(10,28,48,125,27,60,40),(11,19,29,61,49,41,126),(12,50,20,42,30,118,62),(13,31,51,119,21,63,43),(14,22,32,55,52,44,120),(15,53,23,45,33,121,56),(16,34,54,122,24,57,37),(17,25,35,58,46,38,123),(18,47,26,39,36,124,59)], [(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)]])
45 conjugacy classes
| class | 1 | 2 | 3A | 3B | 3C | 3D | 3E | 6A | 6B | 7A | 7B | 9A | ··· | 9F | 9G | ··· | 9L | 14A | 14B | 18A | ··· | 18F | 21A | 21B | 21C | 21D | 21E | ··· | 21J | 42A | 42B | 42C | 42D |
| order | 1 | 2 | 3 | 3 | 3 | 3 | 3 | 6 | 6 | 7 | 7 | 9 | ··· | 9 | 9 | ··· | 9 | 14 | 14 | 18 | ··· | 18 | 21 | 21 | 21 | 21 | 21 | ··· | 21 | 42 | 42 | 42 | 42 |
| size | 1 | 3 | 1 | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 7 | ··· | 7 | 14 | ··· | 14 | 9 | 9 | 21 | ··· | 21 | 3 | 3 | 3 | 3 | 6 | ··· | 6 | 9 | 9 | 9 | 9 |
45 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 6 | 6 |
| type | + | + | + | ||||||||||||
| image | C1 | C2 | C3 | C6 | C9 | C18 | S3 | C3×S3 | S3×C9 | C7⋊C3 | C2×C7⋊C3 | C7⋊C9 | C2×C7⋊C9 | S3×C7⋊C3 | S3×C7⋊C9 |
| kernel | S3×C7⋊C9 | C3×C7⋊C9 | S3×C21 | C3×C21 | S3×C7 | C21 | C7⋊C9 | C21 | C7 | C3×S3 | C32 | S3 | C3 | C3 | C1 |
| # reps | 1 | 1 | 2 | 2 | 6 | 6 | 1 | 2 | 6 | 2 | 2 | 4 | 4 | 2 | 4 |
Matrix representation of S3×C7⋊C9 ►in GL5(𝔽127)
| 107 | 0 | 0 | 0 | 0 |
| 33 | 19 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 126 | 45 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 | 1 |
| 0 | 0 | 105 | 104 | 104 |
| 0 | 0 | 126 | 0 | 126 |
| 37 | 0 | 0 | 0 | 0 |
| 0 | 37 | 0 | 0 | 0 |
| 0 | 0 | 60 | 9 | 29 |
| 0 | 0 | 28 | 107 | 67 |
| 0 | 0 | 28 | 39 | 87 |
G:=sub<GL(5,GF(127))| [107,33,0,0,0,0,19,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[126,0,0,0,0,45,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,1,105,126,0,0,1,104,0,0,0,1,104,126],[37,0,0,0,0,0,37,0,0,0,0,0,60,28,28,0,0,9,107,39,0,0,29,67,87] >;
S3×C7⋊C9 in GAP, Magma, Sage, TeX
S_3\times C_7\rtimes C_9
% in TeX
G:=Group("S3xC7:C9"); // GroupNames label
G:=SmallGroup(378,16);
// by ID
G=gap.SmallGroup(378,16);
# by ID
G:=PCGroup([5,-2,-3,-3,-3,-7,36,723,1359]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^2=c^7=d^9=1,b*a*b=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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