direct product, metacyclic, supersoluble, monomial, 3-hyperelementary
Aliases: C2×C63⋊3C3, C63⋊15C6, C126⋊3C3, C42.3C32, C14⋊23- 1+2, C7⋊C9⋊5C6, C18⋊2(C7⋊C3), C21.9(C3×C6), C7⋊5(C2×3- 1+2), C9⋊4(C2×C7⋊C3), (C2×C7⋊C9)⋊2C3, C3.4(C6×C7⋊C3), C6.3(C3×C7⋊C3), (C6×C7⋊C3).2C3, (C3×C7⋊C3).4C6, SmallGroup(378,25)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×C63⋊3C3
G = < a,b,c | a2=b63=c3=1, ab=ba, ac=ca, cbc-1=b4 >
Smallest permutation representation of C2×C63⋊3C3
►On 126 points
Generators in S126
(1 69)(2 70)(3 71)(4 72)(5 73)(6 74)(7 75)(8 76)(9 77)(10 78)(11 79)(12 80)(13 81)(14 82)(15 83)(16 84)(17 85)(18 86)(19 87)(20 88)(21 89)(22 90)(23 91)(24 92)(25 93)(26 94)(27 95)(28 96)(29 97)(30 98)(31 99)(32 100)(33 101)(34 102)(35 103)(36 104)(37 105)(38 106)(39 107)(40 108)(41 109)(42 110)(43 111)(44 112)(45 113)(46 114)(47 115)(48 116)(49 117)(50 118)(51 119)(52 120)(53 121)(54 122)(55 123)(56 124)(57 125)(58 126)(59 64)(60 65)(61 66)(62 67)(63 68)
(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 17 5)(3 33 9)(4 49 13)(6 18 21)(7 34 25)(8 50 29)(10 19 37)(11 35 41)(12 51 45)(14 20 53)(15 36 57)(16 52 61)(23 38 26)(24 54 30)(27 39 42)(28 55 46)(31 40 58)(32 56 62)(44 59 47)(48 60 63)(64 115 112)(65 68 116)(66 84 120)(67 100 124)(70 85 73)(71 101 77)(72 117 81)(74 86 89)(75 102 93)(76 118 97)(78 87 105)(79 103 109)(80 119 113)(82 88 121)(83 104 125)(91 106 94)(92 122 98)(95 107 110)(96 123 114)(99 108 126)
G:=sub<Sym(126)| (1,69)(2,70)(3,71)(4,72)(5,73)(6,74)(7,75)(8,76)(9,77)(10,78)(11,79)(12,80)(13,81)(14,82)(15,83)(16,84)(17,85)(18,86)(19,87)(20,88)(21,89)(22,90)(23,91)(24,92)(25,93)(26,94)(27,95)(28,96)(29,97)(30,98)(31,99)(32,100)(33,101)(34,102)(35,103)(36,104)(37,105)(38,106)(39,107)(40,108)(41,109)(42,110)(43,111)(44,112)(45,113)(46,114)(47,115)(48,116)(49,117)(50,118)(51,119)(52,120)(53,121)(54,122)(55,123)(56,124)(57,125)(58,126)(59,64)(60,65)(61,66)(62,67)(63,68), (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), (2,17,5)(3,33,9)(4,49,13)(6,18,21)(7,34,25)(8,50,29)(10,19,37)(11,35,41)(12,51,45)(14,20,53)(15,36,57)(16,52,61)(23,38,26)(24,54,30)(27,39,42)(28,55,46)(31,40,58)(32,56,62)(44,59,47)(48,60,63)(64,115,112)(65,68,116)(66,84,120)(67,100,124)(70,85,73)(71,101,77)(72,117,81)(74,86,89)(75,102,93)(76,118,97)(78,87,105)(79,103,109)(80,119,113)(82,88,121)(83,104,125)(91,106,94)(92,122,98)(95,107,110)(96,123,114)(99,108,126)>;
G:=Group( (1,69)(2,70)(3,71)(4,72)(5,73)(6,74)(7,75)(8,76)(9,77)(10,78)(11,79)(12,80)(13,81)(14,82)(15,83)(16,84)(17,85)(18,86)(19,87)(20,88)(21,89)(22,90)(23,91)(24,92)(25,93)(26,94)(27,95)(28,96)(29,97)(30,98)(31,99)(32,100)(33,101)(34,102)(35,103)(36,104)(37,105)(38,106)(39,107)(40,108)(41,109)(42,110)(43,111)(44,112)(45,113)(46,114)(47,115)(48,116)(49,117)(50,118)(51,119)(52,120)(53,121)(54,122)(55,123)(56,124)(57,125)(58,126)(59,64)(60,65)(61,66)(62,67)(63,68), (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), (2,17,5)(3,33,9)(4,49,13)(6,18,21)(7,34,25)(8,50,29)(10,19,37)(11,35,41)(12,51,45)(14,20,53)(15,36,57)(16,52,61)(23,38,26)(24,54,30)(27,39,42)(28,55,46)(31,40,58)(32,56,62)(44,59,47)(48,60,63)(64,115,112)(65,68,116)(66,84,120)(67,100,124)(70,85,73)(71,101,77)(72,117,81)(74,86,89)(75,102,93)(76,118,97)(78,87,105)(79,103,109)(80,119,113)(82,88,121)(83,104,125)(91,106,94)(92,122,98)(95,107,110)(96,123,114)(99,108,126) );
G=PermutationGroup([[(1,69),(2,70),(3,71),(4,72),(5,73),(6,74),(7,75),(8,76),(9,77),(10,78),(11,79),(12,80),(13,81),(14,82),(15,83),(16,84),(17,85),(18,86),(19,87),(20,88),(21,89),(22,90),(23,91),(24,92),(25,93),(26,94),(27,95),(28,96),(29,97),(30,98),(31,99),(32,100),(33,101),(34,102),(35,103),(36,104),(37,105),(38,106),(39,107),(40,108),(41,109),(42,110),(43,111),(44,112),(45,113),(46,114),(47,115),(48,116),(49,117),(50,118),(51,119),(52,120),(53,121),(54,122),(55,123),(56,124),(57,125),(58,126),(59,64),(60,65),(61,66),(62,67),(63,68)], [(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)], [(2,17,5),(3,33,9),(4,49,13),(6,18,21),(7,34,25),(8,50,29),(10,19,37),(11,35,41),(12,51,45),(14,20,53),(15,36,57),(16,52,61),(23,38,26),(24,54,30),(27,39,42),(28,55,46),(31,40,58),(32,56,62),(44,59,47),(48,60,63),(64,115,112),(65,68,116),(66,84,120),(67,100,124),(70,85,73),(71,101,77),(72,117,81),(74,86,89),(75,102,93),(76,118,97),(78,87,105),(79,103,109),(80,119,113),(82,88,121),(83,104,125),(91,106,94),(92,122,98),(95,107,110),(96,123,114),(99,108,126)]])
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⋊3C3 | C2×C63⋊3C3 |
| kernel | C2×C63⋊3C3 | C63⋊3C3 | 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⋊3C3 ►in GL3(𝔽127) generated by
| 126 | 0 | 0 |
| 0 | 126 | 0 |
| 0 | 0 | 126 |
| 53 | 13 | 2 |
| 2 | 9 | 94 |
| 94 | 93 | 6 |
| 1 | 0 | 0 |
| 104 | 126 | 126 |
| 0 | 1 | 0 |
G:=sub<GL(3,GF(127))| [126,0,0,0,126,0,0,0,126],[53,2,94,13,9,93,2,94,6],[1,104,0,0,126,1,0,126,0] >;
C2×C63⋊3C3 in GAP, Magma, Sage, TeX
C_2\times C_{63}\rtimes_3C_3 % in TeX
G:=Group("C2xC63:3C3"); // GroupNames label
G:=SmallGroup(378,25);
// by ID
G=gap.SmallGroup(378,25);
# by ID
G:=PCGroup([5,-2,-3,-3,-3,-7,187,102,1359]);
// 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^4>;
// generators/relations
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