direct product, metacyclic, supersoluble, monomial, A-group
Aliases: C3×D63, C21⋊2D9, C63⋊17C6, C32.2D21, (C3×C9)⋊2D7, C7⋊3(C3×D9), (C3×C63)⋊2C2, C9⋊3(C3×D7), (C3×C21).4S3, C3.1(C3×D21), C21.11(C3×S3), SmallGroup(378,36)
Series: Derived ►Chief ►Lower central ►Upper central
| C63 — C3×D63 |
Generators and relations for C3×D63
G = < a,b,c | a3=b63=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of C3×D63
►On 126 points
Generators in S126
(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 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 72)(2 71)(3 70)(4 69)(5 68)(6 67)(7 66)(8 65)(9 64)(10 126)(11 125)(12 124)(13 123)(14 122)(15 121)(16 120)(17 119)(18 118)(19 117)(20 116)(21 115)(22 114)(23 113)(24 112)(25 111)(26 110)(27 109)(28 108)(29 107)(30 106)(31 105)(32 104)(33 103)(34 102)(35 101)(36 100)(37 99)(38 98)(39 97)(40 96)(41 95)(42 94)(43 93)(44 92)(45 91)(46 90)(47 89)(48 88)(49 87)(50 86)(51 85)(52 84)(53 83)(54 82)(55 81)(56 80)(57 79)(58 78)(59 77)(60 76)(61 75)(62 74)(63 73)
G:=sub<Sym(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), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63)(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), (1,72)(2,71)(3,70)(4,69)(5,68)(6,67)(7,66)(8,65)(9,64)(10,126)(11,125)(12,124)(13,123)(14,122)(15,121)(16,120)(17,119)(18,118)(19,117)(20,116)(21,115)(22,114)(23,113)(24,112)(25,111)(26,110)(27,109)(28,108)(29,107)(30,106)(31,105)(32,104)(33,103)(34,102)(35,101)(36,100)(37,99)(38,98)(39,97)(40,96)(41,95)(42,94)(43,93)(44,92)(45,91)(46,90)(47,89)(48,88)(49,87)(50,86)(51,85)(52,84)(53,83)(54,82)(55,81)(56,80)(57,79)(58,78)(59,77)(60,76)(61,75)(62,74)(63,73)>;
G:=Group( (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), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63)(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), (1,72)(2,71)(3,70)(4,69)(5,68)(6,67)(7,66)(8,65)(9,64)(10,126)(11,125)(12,124)(13,123)(14,122)(15,121)(16,120)(17,119)(18,118)(19,117)(20,116)(21,115)(22,114)(23,113)(24,112)(25,111)(26,110)(27,109)(28,108)(29,107)(30,106)(31,105)(32,104)(33,103)(34,102)(35,101)(36,100)(37,99)(38,98)(39,97)(40,96)(41,95)(42,94)(43,93)(44,92)(45,91)(46,90)(47,89)(48,88)(49,87)(50,86)(51,85)(52,84)(53,83)(54,82)(55,81)(56,80)(57,79)(58,78)(59,77)(60,76)(61,75)(62,74)(63,73) );
G=PermutationGroup([[(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)], [(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63),(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)], [(1,72),(2,71),(3,70),(4,69),(5,68),(6,67),(7,66),(8,65),(9,64),(10,126),(11,125),(12,124),(13,123),(14,122),(15,121),(16,120),(17,119),(18,118),(19,117),(20,116),(21,115),(22,114),(23,113),(24,112),(25,111),(26,110),(27,109),(28,108),(29,107),(30,106),(31,105),(32,104),(33,103),(34,102),(35,101),(36,100),(37,99),(38,98),(39,97),(40,96),(41,95),(42,94),(43,93),(44,92),(45,91),(46,90),(47,89),(48,88),(49,87),(50,86),(51,85),(52,84),(53,83),(54,82),(55,81),(56,80),(57,79),(58,78),(59,77),(60,76),(61,75),(62,74),(63,73)]])
99 conjugacy classes
| class | 1 | 2 | 3A | 3B | 3C | 3D | 3E | 6A | 6B | 7A | 7B | 7C | 9A | ··· | 9I | 21A | ··· | 21X | 63A | ··· | 63BB |
| order | 1 | 2 | 3 | 3 | 3 | 3 | 3 | 6 | 6 | 7 | 7 | 7 | 9 | ··· | 9 | 21 | ··· | 21 | 63 | ··· | 63 |
| size | 1 | 63 | 1 | 1 | 2 | 2 | 2 | 63 | 63 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
99 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | |||||||
| image | C1 | C2 | C3 | C6 | S3 | D7 | D9 | C3×S3 | C3×D7 | D21 | C3×D9 | D63 | C3×D21 | C3×D63 |
| kernel | C3×D63 | C3×C63 | D63 | C63 | C3×C21 | C3×C9 | C21 | C21 | C9 | C32 | C7 | C3 | C3 | C1 |
| # reps | 1 | 1 | 2 | 2 | 1 | 3 | 3 | 2 | 6 | 6 | 6 | 18 | 12 | 36 |
Matrix representation of C3×D63 ►in GL2(𝔽127) generated by
| 19 | 0 |
| 0 | 19 |
| 113 | 0 |
| 0 | 9 |
| 0 | 1 |
| 1 | 0 |
G:=sub<GL(2,GF(127))| [19,0,0,19],[113,0,0,9],[0,1,1,0] >;
C3×D63 in GAP, Magma, Sage, TeX
C_3\times D_{63} % in TeX
G:=Group("C3xD63"); // GroupNames label
G:=SmallGroup(378,36);
// by ID
G=gap.SmallGroup(378,36);
# by ID
G:=PCGroup([5,-2,-3,-3,-7,-3,2072,642,2163,6304]);
// Polycyclic
G:=Group<a,b,c|a^3=b^63=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
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