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 73)(2 72)(3 71)(4 70)(5 69)(6 68)(7 67)(8 66)(9 65)(10 64)(11 126)(12 125)(13 124)(14 123)(15 122)(16 121)(17 120)(18 119)(19 118)(20 117)(21 116)(22 115)(23 114)(24 113)(25 112)(26 111)(27 110)(28 109)(29 108)(30 107)(31 106)(32 105)(33 104)(34 103)(35 102)(36 101)(37 100)(38 99)(39 98)(40 97)(41 96)(42 95)(43 94)(44 93)(45 92)(46 91)(47 90)(48 89)(49 88)(50 87)(51 86)(52 85)(53 84)(54 83)(55 82)(56 81)(57 80)(58 79)(59 78)(60 77)(61 76)(62 75)(63 74)
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), (1,73)(2,72)(3,71)(4,70)(5,69)(6,68)(7,67)(8,66)(9,65)(10,64)(11,126)(12,125)(13,124)(14,123)(15,122)(16,121)(17,120)(18,119)(19,118)(20,117)(21,116)(22,115)(23,114)(24,113)(25,112)(26,111)(27,110)(28,109)(29,108)(30,107)(31,106)(32,105)(33,104)(34,103)(35,102)(36,101)(37,100)(38,99)(39,98)(40,97)(41,96)(42,95)(43,94)(44,93)(45,92)(46,91)(47,90)(48,89)(49,88)(50,87)(51,86)(52,85)(53,84)(54,83)(55,82)(56,81)(57,80)(58,79)(59,78)(60,77)(61,76)(62,75)(63,74)>;
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), (1,73)(2,72)(3,71)(4,70)(5,69)(6,68)(7,67)(8,66)(9,65)(10,64)(11,126)(12,125)(13,124)(14,123)(15,122)(16,121)(17,120)(18,119)(19,118)(20,117)(21,116)(22,115)(23,114)(24,113)(25,112)(26,111)(27,110)(28,109)(29,108)(30,107)(31,106)(32,105)(33,104)(34,103)(35,102)(36,101)(37,100)(38,99)(39,98)(40,97)(41,96)(42,95)(43,94)(44,93)(45,92)(46,91)(47,90)(48,89)(49,88)(50,87)(51,86)(52,85)(53,84)(54,83)(55,82)(56,81)(57,80)(58,79)(59,78)(60,77)(61,76)(62,75)(63,74) );
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)], [(1,73),(2,72),(3,71),(4,70),(5,69),(6,68),(7,67),(8,66),(9,65),(10,64),(11,126),(12,125),(13,124),(14,123),(15,122),(16,121),(17,120),(18,119),(19,118),(20,117),(21,116),(22,115),(23,114),(24,113),(25,112),(26,111),(27,110),(28,109),(29,108),(30,107),(31,106),(32,105),(33,104),(34,103),(35,102),(36,101),(37,100),(38,99),(39,98),(40,97),(41,96),(42,95),(43,94),(44,93),(45,92),(46,91),(47,90),(48,89),(49,88),(50,87),(51,86),(52,85),(53,84),(54,83),(55,82),(56,81),(57,80),(58,79),(59,78),(60,77),(61,76),(62,75),(63,74)])
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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