direct product, metacyclic, supersoluble, monomial, 3-hyperelementary
Aliases: C2×C63⋊C3, C63⋊14C6, C126⋊2C3, C42.2C32, C14⋊13- 1+2, C7⋊C9⋊4C6, C18⋊1(C7⋊C3), C21.8(C3×C6), C7⋊4(C2×3- 1+2), C9⋊3(C2×C7⋊C3), (C2×C7⋊C9)⋊1C3, C3.3(C6×C7⋊C3), C6.2(C3×C7⋊C3), (C6×C7⋊C3).1C3, (C3×C7⋊C3).3C6, SmallGroup(378,24)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×C63⋊C3
G = < a,b,c | a2=b63=c3=1, ab=ba, ac=ca, cbc-1=b25 >
Smallest permutation representation of C2×C63⋊C3
►On 126 points
Generators in S126
(1 70)(2 71)(3 72)(4 73)(5 74)(6 75)(7 76)(8 77)(9 78)(10 79)(11 80)(12 81)(13 82)(14 83)(15 84)(16 85)(17 86)(18 87)(19 88)(20 89)(21 90)(22 91)(23 92)(24 93)(25 94)(26 95)(27 96)(28 97)(29 98)(30 99)(31 100)(32 101)(33 102)(34 103)(35 104)(36 105)(37 106)(38 107)(39 108)(40 109)(41 110)(42 111)(43 112)(44 113)(45 114)(46 115)(47 116)(48 117)(49 118)(50 119)(51 120)(52 121)(53 122)(54 123)(55 124)(56 125)(57 126)(58 64)(59 65)(60 66)(61 67)(62 68)(63 69)
(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)
(2 59 26)(3 54 51)(4 49 13)(5 44 38)(6 39 63)(7 34 25)(8 29 50)(9 24 12)(10 19 37)(11 14 62)(15 57 36)(16 52 61)(17 47 23)(18 42 48)(20 32 35)(21 27 60)(28 55 46)(30 45 33)(31 40 58)(41 53 56)(64 100 109)(65 95 71)(66 90 96)(67 85 121)(68 80 83)(69 75 108)(72 123 120)(73 118 82)(74 113 107)(76 103 94)(77 98 119)(78 93 81)(79 88 106)(84 126 105)(86 116 92)(87 111 117)(89 101 104)(97 124 115)(99 114 102)(110 122 125)
G:=sub<Sym(126)| (1,70)(2,71)(3,72)(4,73)(5,74)(6,75)(7,76)(8,77)(9,78)(10,79)(11,80)(12,81)(13,82)(14,83)(15,84)(16,85)(17,86)(18,87)(19,88)(20,89)(21,90)(22,91)(23,92)(24,93)(25,94)(26,95)(27,96)(28,97)(29,98)(30,99)(31,100)(32,101)(33,102)(34,103)(35,104)(36,105)(37,106)(38,107)(39,108)(40,109)(41,110)(42,111)(43,112)(44,113)(45,114)(46,115)(47,116)(48,117)(49,118)(50,119)(51,120)(52,121)(53,122)(54,123)(55,124)(56,125)(57,126)(58,64)(59,65)(60,66)(61,67)(62,68)(63,69), (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), (2,59,26)(3,54,51)(4,49,13)(5,44,38)(6,39,63)(7,34,25)(8,29,50)(9,24,12)(10,19,37)(11,14,62)(15,57,36)(16,52,61)(17,47,23)(18,42,48)(20,32,35)(21,27,60)(28,55,46)(30,45,33)(31,40,58)(41,53,56)(64,100,109)(65,95,71)(66,90,96)(67,85,121)(68,80,83)(69,75,108)(72,123,120)(73,118,82)(74,113,107)(76,103,94)(77,98,119)(78,93,81)(79,88,106)(84,126,105)(86,116,92)(87,111,117)(89,101,104)(97,124,115)(99,114,102)(110,122,125)>;
G:=Group( (1,70)(2,71)(3,72)(4,73)(5,74)(6,75)(7,76)(8,77)(9,78)(10,79)(11,80)(12,81)(13,82)(14,83)(15,84)(16,85)(17,86)(18,87)(19,88)(20,89)(21,90)(22,91)(23,92)(24,93)(25,94)(26,95)(27,96)(28,97)(29,98)(30,99)(31,100)(32,101)(33,102)(34,103)(35,104)(36,105)(37,106)(38,107)(39,108)(40,109)(41,110)(42,111)(43,112)(44,113)(45,114)(46,115)(47,116)(48,117)(49,118)(50,119)(51,120)(52,121)(53,122)(54,123)(55,124)(56,125)(57,126)(58,64)(59,65)(60,66)(61,67)(62,68)(63,69), (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), (2,59,26)(3,54,51)(4,49,13)(5,44,38)(6,39,63)(7,34,25)(8,29,50)(9,24,12)(10,19,37)(11,14,62)(15,57,36)(16,52,61)(17,47,23)(18,42,48)(20,32,35)(21,27,60)(28,55,46)(30,45,33)(31,40,58)(41,53,56)(64,100,109)(65,95,71)(66,90,96)(67,85,121)(68,80,83)(69,75,108)(72,123,120)(73,118,82)(74,113,107)(76,103,94)(77,98,119)(78,93,81)(79,88,106)(84,126,105)(86,116,92)(87,111,117)(89,101,104)(97,124,115)(99,114,102)(110,122,125) );
G=PermutationGroup([[(1,70),(2,71),(3,72),(4,73),(5,74),(6,75),(7,76),(8,77),(9,78),(10,79),(11,80),(12,81),(13,82),(14,83),(15,84),(16,85),(17,86),(18,87),(19,88),(20,89),(21,90),(22,91),(23,92),(24,93),(25,94),(26,95),(27,96),(28,97),(29,98),(30,99),(31,100),(32,101),(33,102),(34,103),(35,104),(36,105),(37,106),(38,107),(39,108),(40,109),(41,110),(42,111),(43,112),(44,113),(45,114),(46,115),(47,116),(48,117),(49,118),(50,119),(51,120),(52,121),(53,122),(54,123),(55,124),(56,125),(57,126),(58,64),(59,65),(60,66),(61,67),(62,68),(63,69)], [(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)], [(2,59,26),(3,54,51),(4,49,13),(5,44,38),(6,39,63),(7,34,25),(8,29,50),(9,24,12),(10,19,37),(11,14,62),(15,57,36),(16,52,61),(17,47,23),(18,42,48),(20,32,35),(21,27,60),(28,55,46),(30,45,33),(31,40,58),(41,53,56),(64,100,109),(65,95,71),(66,90,96),(67,85,121),(68,80,83),(69,75,108),(72,123,120),(73,118,82),(74,113,107),(76,103,94),(77,98,119),(78,93,81),(79,88,106),(84,126,105),(86,116,92),(87,111,117),(89,101,104),(97,124,115),(99,114,102),(110,122,125)]])
58 conjugacy classes
| class | 1 | 2 | 3A | 3B | 3C | 3D | 6A | 6B | 6C | 6D | 7A | 7B | 9A | 9B | 9C | 9D | 9E | 9F | 14A | 14B | 18A | 18B | 18C | 18D | 18E | 18F | 21A | 21B | 21C | 21D | 42A | 42B | 42C | 42D | 63A | ··· | 63L | 126A | ··· | 126L |
| order | 1 | 2 | 3 | 3 | 3 | 3 | 6 | 6 | 6 | 6 | 7 | 7 | 9 | 9 | 9 | 9 | 9 | 9 | 14 | 14 | 18 | 18 | 18 | 18 | 18 | 18 | 21 | 21 | 21 | 21 | 42 | 42 | 42 | 42 | 63 | ··· | 63 | 126 | ··· | 126 |
| size | 1 | 1 | 1 | 1 | 21 | 21 | 1 | 1 | 21 | 21 | 3 | 3 | 3 | 3 | 21 | 21 | 21 | 21 | 3 | 3 | 3 | 3 | 21 | 21 | 21 | 21 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | ··· | 3 | 3 | ··· | 3 |
58 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 |
| type | + | + | ||||||||||||||
| image | C1 | C2 | C3 | C3 | C3 | C6 | C6 | C6 | C7⋊C3 | 3- 1+2 | C2×C7⋊C3 | C2×3- 1+2 | C3×C7⋊C3 | C6×C7⋊C3 | C63⋊C3 | C2×C63⋊C3 |
| kernel | C2×C63⋊C3 | C63⋊C3 | C2×C7⋊C9 | C126 | C6×C7⋊C3 | C7⋊C9 | C63 | C3×C7⋊C3 | C18 | C14 | C9 | C7 | C6 | C3 | C2 | C1 |
| # reps | 1 | 1 | 4 | 2 | 2 | 4 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 12 | 12 |
Matrix representation of C2×C63⋊C3 ►in GL4(𝔽127) generated by
| 126 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 107 | 0 | 0 | 0 |
| 0 | 22 | 42 | 108 |
| 0 | 108 | 59 | 98 |
| 0 | 98 | 111 | 91 |
| 107 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 104 | 126 | 126 |
| 0 | 0 | 1 | 0 |
G:=sub<GL(4,GF(127))| [126,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1],[107,0,0,0,0,22,108,98,0,42,59,111,0,108,98,91],[107,0,0,0,0,1,104,0,0,0,126,1,0,0,126,0] >;
C2×C63⋊C3 in GAP, Magma, Sage, TeX
C_2\times C_{63}\rtimes C_3 % in TeX
G:=Group("C2xC63:C3"); // GroupNames label
G:=SmallGroup(378,24);
// by ID
G=gap.SmallGroup(378,24);
# by ID
G:=PCGroup([5,-2,-3,-3,-3,-7,187,57,1359]);
// Polycyclic
G:=Group<a,b,c|a^2=b^63=c^3=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^25>;
// generators/relations
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