metacyclic, supersoluble, monomial, Z-group
Aliases: C35⋊3C12, C10.F7, Dic35⋊C3, C70.1C6, C7⋊C3⋊Dic5, C7⋊(C3×Dic5), C5⋊2(C7⋊C12), C2.(C5⋊F7), C14.(C3×D5), (C5×C7⋊C3)⋊3C4, (C2×C7⋊C3).D5, (C10×C7⋊C3).1C2, SmallGroup(420,3)
Series: Derived ►Chief ►Lower central ►Upper central
| C35 — C35⋊3C12 |
Generators and relations for C35⋊3C12
G = < a,b | a35=b12=1, bab-1=a24 >
Smallest permutation representation of C35⋊3C12
►On 140 points
Generators in S140
(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)(106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140)
(1 124 38 101)(2 108 49 100 17 113 39 85 12 123 54 90)(3 127 60 99 33 137 40 104 23 122 70 79)(4 111 36 98 14 126 41 88 34 121 51 103)(5 130 47 97 30 115 42 72 10 120 67 92)(6 114 58 96 11 139 43 91 21 119 48 81)(7 133 69 95 27 128 44 75 32 118 64 105)(8 117 45 94)(9 136 56 93 24 106 46 78 19 116 61 83)(13 107 65 89 18 132 50 84 28 112 55 74)(15 110 52 87)(16 129 63 86 31 134 53 71 26 109 68 76)(20 135 37 82 25 125 57 77 35 140 62 102)(22 138 59 80)(29 131 66 73)
G:=sub<Sym(140)| (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)(106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140), (1,124,38,101)(2,108,49,100,17,113,39,85,12,123,54,90)(3,127,60,99,33,137,40,104,23,122,70,79)(4,111,36,98,14,126,41,88,34,121,51,103)(5,130,47,97,30,115,42,72,10,120,67,92)(6,114,58,96,11,139,43,91,21,119,48,81)(7,133,69,95,27,128,44,75,32,118,64,105)(8,117,45,94)(9,136,56,93,24,106,46,78,19,116,61,83)(13,107,65,89,18,132,50,84,28,112,55,74)(15,110,52,87)(16,129,63,86,31,134,53,71,26,109,68,76)(20,135,37,82,25,125,57,77,35,140,62,102)(22,138,59,80)(29,131,66,73)>;
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)(106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140), (1,124,38,101)(2,108,49,100,17,113,39,85,12,123,54,90)(3,127,60,99,33,137,40,104,23,122,70,79)(4,111,36,98,14,126,41,88,34,121,51,103)(5,130,47,97,30,115,42,72,10,120,67,92)(6,114,58,96,11,139,43,91,21,119,48,81)(7,133,69,95,27,128,44,75,32,118,64,105)(8,117,45,94)(9,136,56,93,24,106,46,78,19,116,61,83)(13,107,65,89,18,132,50,84,28,112,55,74)(15,110,52,87)(16,129,63,86,31,134,53,71,26,109,68,76)(20,135,37,82,25,125,57,77,35,140,62,102)(22,138,59,80)(29,131,66,73) );
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),(106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140)], [(1,124,38,101),(2,108,49,100,17,113,39,85,12,123,54,90),(3,127,60,99,33,137,40,104,23,122,70,79),(4,111,36,98,14,126,41,88,34,121,51,103),(5,130,47,97,30,115,42,72,10,120,67,92),(6,114,58,96,11,139,43,91,21,119,48,81),(7,133,69,95,27,128,44,75,32,118,64,105),(8,117,45,94),(9,136,56,93,24,106,46,78,19,116,61,83),(13,107,65,89,18,132,50,84,28,112,55,74),(15,110,52,87),(16,129,63,86,31,134,53,71,26,109,68,76),(20,135,37,82,25,125,57,77,35,140,62,102),(22,138,59,80),(29,131,66,73)]])
34 conjugacy classes
| class | 1 | 2 | 3A | 3B | 4A | 4B | 5A | 5B | 6A | 6B | 7 | 10A | 10B | 12A | 12B | 12C | 12D | 14 | 15A | 15B | 15C | 15D | 30A | 30B | 30C | 30D | 35A | 35B | 35C | 35D | 70A | 70B | 70C | 70D |
| order | 1 | 2 | 3 | 3 | 4 | 4 | 5 | 5 | 6 | 6 | 7 | 10 | 10 | 12 | 12 | 12 | 12 | 14 | 15 | 15 | 15 | 15 | 30 | 30 | 30 | 30 | 35 | 35 | 35 | 35 | 70 | 70 | 70 | 70 |
| size | 1 | 1 | 7 | 7 | 35 | 35 | 2 | 2 | 7 | 7 | 6 | 2 | 2 | 35 | 35 | 35 | 35 | 6 | 14 | 14 | 14 | 14 | 14 | 14 | 14 | 14 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 |
34 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 6 | 6 | 6 | 6 |
| type | + | + | + | - | + | - | + | - | ||||||
| image | C1 | C2 | C3 | C4 | C6 | C12 | D5 | Dic5 | C3×D5 | C3×Dic5 | F7 | C7⋊C12 | C5⋊F7 | C35⋊3C12 |
| kernel | C35⋊3C12 | C10×C7⋊C3 | Dic35 | C5×C7⋊C3 | C70 | C35 | C2×C7⋊C3 | C7⋊C3 | C14 | C7 | C10 | C5 | C2 | C1 |
| # reps | 1 | 1 | 2 | 2 | 2 | 4 | 2 | 2 | 4 | 4 | 1 | 1 | 4 | 4 |
Matrix representation of C35⋊3C12 ►in GL6(𝔽421)
| 43 | 0 | 0 | 387 | 43 | 43 |
| 378 | 0 | 378 | 378 | 344 | 0 |
| 0 | 378 | 0 | 378 | 378 | 344 |
| 77 | 77 | 34 | 77 | 34 | 34 |
| 387 | 43 | 43 | 0 | 43 | 0 |
| 0 | 387 | 43 | 43 | 0 | 43 |
| 192 | 6 | 6 | 0 | 6 | 0 |
| 6 | 0 | 0 | 192 | 6 | 6 |
| 235 | 235 | 229 | 235 | 229 | 229 |
| 415 | 415 | 186 | 0 | 0 | 415 |
| 0 | 415 | 0 | 415 | 415 | 186 |
| 0 | 192 | 6 | 6 | 0 | 6 |
G:=sub<GL(6,GF(421))| [43,378,0,77,387,0,0,0,378,77,43,387,0,378,0,34,43,43,387,378,378,77,0,43,43,344,378,34,43,0,43,0,344,34,0,43],[192,6,235,415,0,0,6,0,235,415,415,192,6,0,229,186,0,6,0,192,235,0,415,6,6,6,229,0,415,0,0,6,229,415,186,6] >;
C35⋊3C12 in GAP, Magma, Sage, TeX
C_{35}\rtimes_3C_{12} % in TeX
G:=Group("C35:3C12"); // GroupNames label
G:=SmallGroup(420,3);
// by ID
G=gap.SmallGroup(420,3);
# by ID
G:=PCGroup([5,-2,-3,-2,-5,-7,30,963,9004,1509]);
// Polycyclic
G:=Group<a,b|a^35=b^12=1,b*a*b^-1=a^24>;
// generators/relations
Export