metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.5D4, C28.4D6, C21⋊2SD16, D12.2D7, C12.4D14, Dic14⋊2S3, C84.24C22, C21⋊C8⋊8C2, C3⋊2(D4.D7), C4.17(S3×D7), (C7×D12).2C2, C7⋊2(Q8⋊2S3), C6.9(C7⋊D4), (C3×Dic14)⋊4C2, C2.6(C21⋊D4), C14.9(C3⋊D4), SmallGroup(336,33)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42.D4
G = < a,b,c | a28=1, b6=a14, c2=a7, bab-1=a-1, cac-1=a13, cbc-1=a21b5 >
Smallest permutation representation of C42.D4
►On 168 points
Generators in S168
(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)(141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168)
(1 101 56 125 164 77 15 87 42 139 150 63)(2 100 29 124 165 76 16 86 43 138 151 62)(3 99 30 123 166 75 17 85 44 137 152 61)(4 98 31 122 167 74 18 112 45 136 153 60)(5 97 32 121 168 73 19 111 46 135 154 59)(6 96 33 120 141 72 20 110 47 134 155 58)(7 95 34 119 142 71 21 109 48 133 156 57)(8 94 35 118 143 70 22 108 49 132 157 84)(9 93 36 117 144 69 23 107 50 131 158 83)(10 92 37 116 145 68 24 106 51 130 159 82)(11 91 38 115 146 67 25 105 52 129 160 81)(12 90 39 114 147 66 26 104 53 128 161 80)(13 89 40 113 148 65 27 103 54 127 162 79)(14 88 41 140 149 64 28 102 55 126 163 78)
(1 77 8 84 15 63 22 70)(2 62 9 69 16 76 23 83)(3 75 10 82 17 61 24 68)(4 60 11 67 18 74 25 81)(5 73 12 80 19 59 26 66)(6 58 13 65 20 72 27 79)(7 71 14 78 21 57 28 64)(29 138 36 117 43 124 50 131)(30 123 37 130 44 137 51 116)(31 136 38 115 45 122 52 129)(32 121 39 128 46 135 53 114)(33 134 40 113 47 120 54 127)(34 119 41 126 48 133 55 140)(35 132 42 139 49 118 56 125)(85 159 92 166 99 145 106 152)(86 144 93 151 100 158 107 165)(87 157 94 164 101 143 108 150)(88 142 95 149 102 156 109 163)(89 155 96 162 103 141 110 148)(90 168 97 147 104 154 111 161)(91 153 98 160 105 167 112 146)
G:=sub<Sym(168)| (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)(141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168), (1,101,56,125,164,77,15,87,42,139,150,63)(2,100,29,124,165,76,16,86,43,138,151,62)(3,99,30,123,166,75,17,85,44,137,152,61)(4,98,31,122,167,74,18,112,45,136,153,60)(5,97,32,121,168,73,19,111,46,135,154,59)(6,96,33,120,141,72,20,110,47,134,155,58)(7,95,34,119,142,71,21,109,48,133,156,57)(8,94,35,118,143,70,22,108,49,132,157,84)(9,93,36,117,144,69,23,107,50,131,158,83)(10,92,37,116,145,68,24,106,51,130,159,82)(11,91,38,115,146,67,25,105,52,129,160,81)(12,90,39,114,147,66,26,104,53,128,161,80)(13,89,40,113,148,65,27,103,54,127,162,79)(14,88,41,140,149,64,28,102,55,126,163,78), (1,77,8,84,15,63,22,70)(2,62,9,69,16,76,23,83)(3,75,10,82,17,61,24,68)(4,60,11,67,18,74,25,81)(5,73,12,80,19,59,26,66)(6,58,13,65,20,72,27,79)(7,71,14,78,21,57,28,64)(29,138,36,117,43,124,50,131)(30,123,37,130,44,137,51,116)(31,136,38,115,45,122,52,129)(32,121,39,128,46,135,53,114)(33,134,40,113,47,120,54,127)(34,119,41,126,48,133,55,140)(35,132,42,139,49,118,56,125)(85,159,92,166,99,145,106,152)(86,144,93,151,100,158,107,165)(87,157,94,164,101,143,108,150)(88,142,95,149,102,156,109,163)(89,155,96,162,103,141,110,148)(90,168,97,147,104,154,111,161)(91,153,98,160,105,167,112,146)>;
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)(141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168), (1,101,56,125,164,77,15,87,42,139,150,63)(2,100,29,124,165,76,16,86,43,138,151,62)(3,99,30,123,166,75,17,85,44,137,152,61)(4,98,31,122,167,74,18,112,45,136,153,60)(5,97,32,121,168,73,19,111,46,135,154,59)(6,96,33,120,141,72,20,110,47,134,155,58)(7,95,34,119,142,71,21,109,48,133,156,57)(8,94,35,118,143,70,22,108,49,132,157,84)(9,93,36,117,144,69,23,107,50,131,158,83)(10,92,37,116,145,68,24,106,51,130,159,82)(11,91,38,115,146,67,25,105,52,129,160,81)(12,90,39,114,147,66,26,104,53,128,161,80)(13,89,40,113,148,65,27,103,54,127,162,79)(14,88,41,140,149,64,28,102,55,126,163,78), (1,77,8,84,15,63,22,70)(2,62,9,69,16,76,23,83)(3,75,10,82,17,61,24,68)(4,60,11,67,18,74,25,81)(5,73,12,80,19,59,26,66)(6,58,13,65,20,72,27,79)(7,71,14,78,21,57,28,64)(29,138,36,117,43,124,50,131)(30,123,37,130,44,137,51,116)(31,136,38,115,45,122,52,129)(32,121,39,128,46,135,53,114)(33,134,40,113,47,120,54,127)(34,119,41,126,48,133,55,140)(35,132,42,139,49,118,56,125)(85,159,92,166,99,145,106,152)(86,144,93,151,100,158,107,165)(87,157,94,164,101,143,108,150)(88,142,95,149,102,156,109,163)(89,155,96,162,103,141,110,148)(90,168,97,147,104,154,111,161)(91,153,98,160,105,167,112,146) );
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),(141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168)], [(1,101,56,125,164,77,15,87,42,139,150,63),(2,100,29,124,165,76,16,86,43,138,151,62),(3,99,30,123,166,75,17,85,44,137,152,61),(4,98,31,122,167,74,18,112,45,136,153,60),(5,97,32,121,168,73,19,111,46,135,154,59),(6,96,33,120,141,72,20,110,47,134,155,58),(7,95,34,119,142,71,21,109,48,133,156,57),(8,94,35,118,143,70,22,108,49,132,157,84),(9,93,36,117,144,69,23,107,50,131,158,83),(10,92,37,116,145,68,24,106,51,130,159,82),(11,91,38,115,146,67,25,105,52,129,160,81),(12,90,39,114,147,66,26,104,53,128,161,80),(13,89,40,113,148,65,27,103,54,127,162,79),(14,88,41,140,149,64,28,102,55,126,163,78)], [(1,77,8,84,15,63,22,70),(2,62,9,69,16,76,23,83),(3,75,10,82,17,61,24,68),(4,60,11,67,18,74,25,81),(5,73,12,80,19,59,26,66),(6,58,13,65,20,72,27,79),(7,71,14,78,21,57,28,64),(29,138,36,117,43,124,50,131),(30,123,37,130,44,137,51,116),(31,136,38,115,45,122,52,129),(32,121,39,128,46,135,53,114),(33,134,40,113,47,120,54,127),(34,119,41,126,48,133,55,140),(35,132,42,139,49,118,56,125),(85,159,92,166,99,145,106,152),(86,144,93,151,100,158,107,165),(87,157,94,164,101,143,108,150),(88,142,95,149,102,156,109,163),(89,155,96,162,103,141,110,148),(90,168,97,147,104,154,111,161),(91,153,98,160,105,167,112,146)]])
39 conjugacy classes
| class | 1 | 2A | 2B | 3 | 4A | 4B | 6 | 7A | 7B | 7C | 8A | 8B | 12A | 12B | 12C | 14A | 14B | 14C | 14D | ··· | 14I | 21A | 21B | 21C | 28A | 28B | 28C | 42A | 42B | 42C | 84A | ··· | 84F |
| order | 1 | 2 | 2 | 3 | 4 | 4 | 6 | 7 | 7 | 7 | 8 | 8 | 12 | 12 | 12 | 14 | 14 | 14 | 14 | ··· | 14 | 21 | 21 | 21 | 28 | 28 | 28 | 42 | 42 | 42 | 84 | ··· | 84 |
| size | 1 | 1 | 12 | 2 | 2 | 28 | 2 | 2 | 2 | 2 | 42 | 42 | 4 | 28 | 28 | 2 | 2 | 2 | 12 | ··· | 12 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
39 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | - | - | ||||
| image | C1 | C2 | C2 | C2 | S3 | D4 | D6 | D7 | SD16 | C3⋊D4 | D14 | C7⋊D4 | Q8⋊2S3 | S3×D7 | D4.D7 | C21⋊D4 | C42.D4 |
| kernel | C42.D4 | C21⋊C8 | C3×Dic14 | C7×D12 | Dic14 | C42 | C28 | D12 | C21 | C14 | C12 | C6 | C7 | C4 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 3 | 2 | 2 | 3 | 6 | 1 | 3 | 3 | 3 | 6 |
Matrix representation of C42.D4 ►in GL6(𝔽337)
| 329 | 0 | 0 | 0 | 0 | 0 |
| 18 | 42 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 307 | 3 |
| 0 | 0 | 0 | 0 | 149 | 30 |
| 55 | 3 | 0 | 0 | 0 | 0 |
| 3 | 282 | 0 | 0 | 0 | 0 |
| 0 | 0 | 336 | 336 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 5 | 43 |
| 0 | 0 | 0 | 0 | 211 | 332 |
| 282 | 334 | 0 | 0 | 0 | 0 |
| 110 | 55 | 0 | 0 | 0 | 0 |
| 0 | 0 | 336 | 336 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 191 | 294 |
| 0 | 0 | 0 | 0 | 111 | 5 |
G:=sub<GL(6,GF(337))| [329,18,0,0,0,0,0,42,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,307,149,0,0,0,0,3,30],[55,3,0,0,0,0,3,282,0,0,0,0,0,0,336,1,0,0,0,0,336,0,0,0,0,0,0,0,5,211,0,0,0,0,43,332],[282,110,0,0,0,0,334,55,0,0,0,0,0,0,336,0,0,0,0,0,336,1,0,0,0,0,0,0,191,111,0,0,0,0,294,5] >;
C42.D4 in GAP, Magma, Sage, TeX
C_{42}.D_4 % in TeX
G:=Group("C42.D4"); // GroupNames label
G:=SmallGroup(336,33);
// by ID
G=gap.SmallGroup(336,33);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-3,-7,48,73,218,116,50,490,10373]);
// Polycyclic
G:=Group<a,b,c|a^28=1,b^6=a^14,c^2=a^7,b*a*b^-1=a^-1,c*a*c^-1=a^13,c*b*c^-1=a^21*b^5>;
// generators/relations
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