direct product, metacyclic, supersoluble, monomial, A-group
Aliases: S3×C63, C3⋊C126, C21⋊7C18, C32.2C42, (C3×C9)⋊1C14, (C3×C63)⋊1C2, (C3×S3).C21, C3.4(S3×C21), (S3×C21).2C3, C21.18(C3×S3), (C3×C21).11C6, SmallGroup(378,33)
Series: Derived ►Chief ►Lower central ►Upper central
| C3 — S3×C63 |
Generators and relations for S3×C63
G = < a,b,c | a63=b3=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of S3×C63
►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 22 43)(2 23 44)(3 24 45)(4 25 46)(5 26 47)(6 27 48)(7 28 49)(8 29 50)(9 30 51)(10 31 52)(11 32 53)(12 33 54)(13 34 55)(14 35 56)(15 36 57)(16 37 58)(17 38 59)(18 39 60)(19 40 61)(20 41 62)(21 42 63)(64 106 85)(65 107 86)(66 108 87)(67 109 88)(68 110 89)(69 111 90)(70 112 91)(71 113 92)(72 114 93)(73 115 94)(74 116 95)(75 117 96)(76 118 97)(77 119 98)(78 120 99)(79 121 100)(80 122 101)(81 123 102)(82 124 103)(83 125 104)(84 126 105)
(1 116)(2 117)(3 118)(4 119)(5 120)(6 121)(7 122)(8 123)(9 124)(10 125)(11 126)(12 64)(13 65)(14 66)(15 67)(16 68)(17 69)(18 70)(19 71)(20 72)(21 73)(22 74)(23 75)(24 76)(25 77)(26 78)(27 79)(28 80)(29 81)(30 82)(31 83)(32 84)(33 85)(34 86)(35 87)(36 88)(37 89)(38 90)(39 91)(40 92)(41 93)(42 94)(43 95)(44 96)(45 97)(46 98)(47 99)(48 100)(49 101)(50 102)(51 103)(52 104)(53 105)(54 106)(55 107)(56 108)(57 109)(58 110)(59 111)(60 112)(61 113)(62 114)(63 115)
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), (1,22,43)(2,23,44)(3,24,45)(4,25,46)(5,26,47)(6,27,48)(7,28,49)(8,29,50)(9,30,51)(10,31,52)(11,32,53)(12,33,54)(13,34,55)(14,35,56)(15,36,57)(16,37,58)(17,38,59)(18,39,60)(19,40,61)(20,41,62)(21,42,63)(64,106,85)(65,107,86)(66,108,87)(67,109,88)(68,110,89)(69,111,90)(70,112,91)(71,113,92)(72,114,93)(73,115,94)(74,116,95)(75,117,96)(76,118,97)(77,119,98)(78,120,99)(79,121,100)(80,122,101)(81,123,102)(82,124,103)(83,125,104)(84,126,105), (1,116)(2,117)(3,118)(4,119)(5,120)(6,121)(7,122)(8,123)(9,124)(10,125)(11,126)(12,64)(13,65)(14,66)(15,67)(16,68)(17,69)(18,70)(19,71)(20,72)(21,73)(22,74)(23,75)(24,76)(25,77)(26,78)(27,79)(28,80)(29,81)(30,82)(31,83)(32,84)(33,85)(34,86)(35,87)(36,88)(37,89)(38,90)(39,91)(40,92)(41,93)(42,94)(43,95)(44,96)(45,97)(46,98)(47,99)(48,100)(49,101)(50,102)(51,103)(52,104)(53,105)(54,106)(55,107)(56,108)(57,109)(58,110)(59,111)(60,112)(61,113)(62,114)(63,115)>;
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), (1,22,43)(2,23,44)(3,24,45)(4,25,46)(5,26,47)(6,27,48)(7,28,49)(8,29,50)(9,30,51)(10,31,52)(11,32,53)(12,33,54)(13,34,55)(14,35,56)(15,36,57)(16,37,58)(17,38,59)(18,39,60)(19,40,61)(20,41,62)(21,42,63)(64,106,85)(65,107,86)(66,108,87)(67,109,88)(68,110,89)(69,111,90)(70,112,91)(71,113,92)(72,114,93)(73,115,94)(74,116,95)(75,117,96)(76,118,97)(77,119,98)(78,120,99)(79,121,100)(80,122,101)(81,123,102)(82,124,103)(83,125,104)(84,126,105), (1,116)(2,117)(3,118)(4,119)(5,120)(6,121)(7,122)(8,123)(9,124)(10,125)(11,126)(12,64)(13,65)(14,66)(15,67)(16,68)(17,69)(18,70)(19,71)(20,72)(21,73)(22,74)(23,75)(24,76)(25,77)(26,78)(27,79)(28,80)(29,81)(30,82)(31,83)(32,84)(33,85)(34,86)(35,87)(36,88)(37,89)(38,90)(39,91)(40,92)(41,93)(42,94)(43,95)(44,96)(45,97)(46,98)(47,99)(48,100)(49,101)(50,102)(51,103)(52,104)(53,105)(54,106)(55,107)(56,108)(57,109)(58,110)(59,111)(60,112)(61,113)(62,114)(63,115) );
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)], [(1,22,43),(2,23,44),(3,24,45),(4,25,46),(5,26,47),(6,27,48),(7,28,49),(8,29,50),(9,30,51),(10,31,52),(11,32,53),(12,33,54),(13,34,55),(14,35,56),(15,36,57),(16,37,58),(17,38,59),(18,39,60),(19,40,61),(20,41,62),(21,42,63),(64,106,85),(65,107,86),(66,108,87),(67,109,88),(68,110,89),(69,111,90),(70,112,91),(71,113,92),(72,114,93),(73,115,94),(74,116,95),(75,117,96),(76,118,97),(77,119,98),(78,120,99),(79,121,100),(80,122,101),(81,123,102),(82,124,103),(83,125,104),(84,126,105)], [(1,116),(2,117),(3,118),(4,119),(5,120),(6,121),(7,122),(8,123),(9,124),(10,125),(11,126),(12,64),(13,65),(14,66),(15,67),(16,68),(17,69),(18,70),(19,71),(20,72),(21,73),(22,74),(23,75),(24,76),(25,77),(26,78),(27,79),(28,80),(29,81),(30,82),(31,83),(32,84),(33,85),(34,86),(35,87),(36,88),(37,89),(38,90),(39,91),(40,92),(41,93),(42,94),(43,95),(44,96),(45,97),(46,98),(47,99),(48,100),(49,101),(50,102),(51,103),(52,104),(53,105),(54,106),(55,107),(56,108),(57,109),(58,110),(59,111),(60,112),(61,113),(62,114),(63,115)]])
189 conjugacy classes
| class | 1 | 2 | 3A | 3B | 3C | 3D | 3E | 6A | 6B | 7A | ··· | 7F | 9A | ··· | 9F | 9G | ··· | 9L | 14A | ··· | 14F | 18A | ··· | 18F | 21A | ··· | 21L | 21M | ··· | 21AD | 42A | ··· | 42L | 63A | ··· | 63AJ | 63AK | ··· | 63BT | 126A | ··· | 126AJ |
| order | 1 | 2 | 3 | 3 | 3 | 3 | 3 | 6 | 6 | 7 | ··· | 7 | 9 | ··· | 9 | 9 | ··· | 9 | 14 | ··· | 14 | 18 | ··· | 18 | 21 | ··· | 21 | 21 | ··· | 21 | 42 | ··· | 42 | 63 | ··· | 63 | 63 | ··· | 63 | 126 | ··· | 126 |
| size | 1 | 3 | 1 | 1 | 2 | 2 | 2 | 3 | 3 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 3 | ··· | 3 | 3 | ··· | 3 | 1 | ··· | 1 | 2 | ··· | 2 | 3 | ··· | 3 | 1 | ··· | 1 | 2 | ··· | 2 | 3 | ··· | 3 |
189 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | |||||||||||||||
| image | C1 | C2 | C3 | C6 | C7 | C9 | C14 | C18 | C21 | C42 | C63 | C126 | S3 | C3×S3 | S3×C7 | S3×C9 | S3×C21 | S3×C63 |
| kernel | S3×C63 | C3×C63 | S3×C21 | C3×C21 | S3×C9 | S3×C7 | C3×C9 | C21 | C3×S3 | C32 | S3 | C3 | C63 | C21 | C9 | C7 | C3 | C1 |
| # reps | 1 | 1 | 2 | 2 | 6 | 6 | 6 | 6 | 12 | 12 | 36 | 36 | 1 | 2 | 6 | 6 | 12 | 36 |
Matrix representation of S3×C63 ►in GL2(𝔽127) generated by
| 72 | 0 |
| 0 | 72 |
| 107 | 0 |
| 0 | 19 |
| 0 | 1 |
| 1 | 0 |
G:=sub<GL(2,GF(127))| [72,0,0,72],[107,0,0,19],[0,1,1,0] >;
S3×C63 in GAP, Magma, Sage, TeX
S_3\times C_{63} % in TeX
G:=Group("S3xC63"); // GroupNames label
G:=SmallGroup(378,33);
// by ID
G=gap.SmallGroup(378,33);
# by ID
G:=PCGroup([5,-2,-3,-7,-3,-3,216,6304]);
// Polycyclic
G:=Group<a,b,c|a^63=b^3=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
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