metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C28.32D6, D14.Dic3, C21⋊1M4(2), C12.32D14, Dic7.Dic3, C84.32C22, C3⋊C8⋊4D7, C3⋊3(C8⋊D7), C42.4(C2×C4), (C6×D7).2C4, (C4×D7).2S3, C21⋊C8⋊11C2, C4.25(S3×D7), C6.11(C4×D7), (C12×D7).3C2, C7⋊1(C4.Dic3), C2.3(Dic3×D7), (C3×Dic7).2C4, C14.2(C2×Dic3), (C7×C3⋊C8)⋊6C2, SmallGroup(336,26)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C28.32D6
G = < a,b,c | a28=b6=1, c2=a7, bab-1=cac-1=a13, cbc-1=a14b-1 >
Smallest permutation representation of C28.32D6
►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 65 94)(2 78 95 14 66 107)(3 63 96 27 67 92)(4 76 97 12 68 105)(5 61 98 25 69 90)(6 74 99 10 70 103)(7 59 100 23 71 88)(8 72 101)(9 57 102 21 73 86)(11 83 104 19 75 112)(13 81 106 17 77 110)(15 79 108)(16 64 109 28 80 93)(18 62 111 26 82 91)(20 60 85 24 84 89)(22 58 87)(29 136 155 39 126 165)(30 121 156 52 127 150)(31 134 157 37 128 163)(32 119 158 50 129 148)(33 132 159 35 130 161)(34 117 160 48 131 146)(36 115 162 46 133 144)(38 113 164 44 135 142)(40 139 166 42 137 168)(41 124 167 55 138 153)(43 122 141 53 140 151)(45 120 143 51 114 149)(47 118 145 49 116 147)(54 125 152 56 123 154)
(1 153 8 160 15 167 22 146)(2 166 9 145 16 152 23 159)(3 151 10 158 17 165 24 144)(4 164 11 143 18 150 25 157)(5 149 12 156 19 163 26 142)(6 162 13 141 20 148 27 155)(7 147 14 154 21 161 28 168)(29 99 36 106 43 85 50 92)(30 112 37 91 44 98 51 105)(31 97 38 104 45 111 52 90)(32 110 39 89 46 96 53 103)(33 95 40 102 47 109 54 88)(34 108 41 87 48 94 55 101)(35 93 42 100 49 107 56 86)(57 132 64 139 71 118 78 125)(58 117 65 124 72 131 79 138)(59 130 66 137 73 116 80 123)(60 115 67 122 74 129 81 136)(61 128 68 135 75 114 82 121)(62 113 69 120 76 127 83 134)(63 126 70 133 77 140 84 119)
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), (1,65,94)(2,78,95,14,66,107)(3,63,96,27,67,92)(4,76,97,12,68,105)(5,61,98,25,69,90)(6,74,99,10,70,103)(7,59,100,23,71,88)(8,72,101)(9,57,102,21,73,86)(11,83,104,19,75,112)(13,81,106,17,77,110)(15,79,108)(16,64,109,28,80,93)(18,62,111,26,82,91)(20,60,85,24,84,89)(22,58,87)(29,136,155,39,126,165)(30,121,156,52,127,150)(31,134,157,37,128,163)(32,119,158,50,129,148)(33,132,159,35,130,161)(34,117,160,48,131,146)(36,115,162,46,133,144)(38,113,164,44,135,142)(40,139,166,42,137,168)(41,124,167,55,138,153)(43,122,141,53,140,151)(45,120,143,51,114,149)(47,118,145,49,116,147)(54,125,152,56,123,154), (1,153,8,160,15,167,22,146)(2,166,9,145,16,152,23,159)(3,151,10,158,17,165,24,144)(4,164,11,143,18,150,25,157)(5,149,12,156,19,163,26,142)(6,162,13,141,20,148,27,155)(7,147,14,154,21,161,28,168)(29,99,36,106,43,85,50,92)(30,112,37,91,44,98,51,105)(31,97,38,104,45,111,52,90)(32,110,39,89,46,96,53,103)(33,95,40,102,47,109,54,88)(34,108,41,87,48,94,55,101)(35,93,42,100,49,107,56,86)(57,132,64,139,71,118,78,125)(58,117,65,124,72,131,79,138)(59,130,66,137,73,116,80,123)(60,115,67,122,74,129,81,136)(61,128,68,135,75,114,82,121)(62,113,69,120,76,127,83,134)(63,126,70,133,77,140,84,119)>;
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), (1,65,94)(2,78,95,14,66,107)(3,63,96,27,67,92)(4,76,97,12,68,105)(5,61,98,25,69,90)(6,74,99,10,70,103)(7,59,100,23,71,88)(8,72,101)(9,57,102,21,73,86)(11,83,104,19,75,112)(13,81,106,17,77,110)(15,79,108)(16,64,109,28,80,93)(18,62,111,26,82,91)(20,60,85,24,84,89)(22,58,87)(29,136,155,39,126,165)(30,121,156,52,127,150)(31,134,157,37,128,163)(32,119,158,50,129,148)(33,132,159,35,130,161)(34,117,160,48,131,146)(36,115,162,46,133,144)(38,113,164,44,135,142)(40,139,166,42,137,168)(41,124,167,55,138,153)(43,122,141,53,140,151)(45,120,143,51,114,149)(47,118,145,49,116,147)(54,125,152,56,123,154), (1,153,8,160,15,167,22,146)(2,166,9,145,16,152,23,159)(3,151,10,158,17,165,24,144)(4,164,11,143,18,150,25,157)(5,149,12,156,19,163,26,142)(6,162,13,141,20,148,27,155)(7,147,14,154,21,161,28,168)(29,99,36,106,43,85,50,92)(30,112,37,91,44,98,51,105)(31,97,38,104,45,111,52,90)(32,110,39,89,46,96,53,103)(33,95,40,102,47,109,54,88)(34,108,41,87,48,94,55,101)(35,93,42,100,49,107,56,86)(57,132,64,139,71,118,78,125)(58,117,65,124,72,131,79,138)(59,130,66,137,73,116,80,123)(60,115,67,122,74,129,81,136)(61,128,68,135,75,114,82,121)(62,113,69,120,76,127,83,134)(63,126,70,133,77,140,84,119) );
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)], [(1,65,94),(2,78,95,14,66,107),(3,63,96,27,67,92),(4,76,97,12,68,105),(5,61,98,25,69,90),(6,74,99,10,70,103),(7,59,100,23,71,88),(8,72,101),(9,57,102,21,73,86),(11,83,104,19,75,112),(13,81,106,17,77,110),(15,79,108),(16,64,109,28,80,93),(18,62,111,26,82,91),(20,60,85,24,84,89),(22,58,87),(29,136,155,39,126,165),(30,121,156,52,127,150),(31,134,157,37,128,163),(32,119,158,50,129,148),(33,132,159,35,130,161),(34,117,160,48,131,146),(36,115,162,46,133,144),(38,113,164,44,135,142),(40,139,166,42,137,168),(41,124,167,55,138,153),(43,122,141,53,140,151),(45,120,143,51,114,149),(47,118,145,49,116,147),(54,125,152,56,123,154)], [(1,153,8,160,15,167,22,146),(2,166,9,145,16,152,23,159),(3,151,10,158,17,165,24,144),(4,164,11,143,18,150,25,157),(5,149,12,156,19,163,26,142),(6,162,13,141,20,148,27,155),(7,147,14,154,21,161,28,168),(29,99,36,106,43,85,50,92),(30,112,37,91,44,98,51,105),(31,97,38,104,45,111,52,90),(32,110,39,89,46,96,53,103),(33,95,40,102,47,109,54,88),(34,108,41,87,48,94,55,101),(35,93,42,100,49,107,56,86),(57,132,64,139,71,118,78,125),(58,117,65,124,72,131,79,138),(59,130,66,137,73,116,80,123),(60,115,67,122,74,129,81,136),(61,128,68,135,75,114,82,121),(62,113,69,120,76,127,83,134),(63,126,70,133,77,140,84,119)]])
54 conjugacy classes
| class | 1 | 2A | 2B | 3 | 4A | 4B | 4C | 6A | 6B | 6C | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 14A | 14B | 14C | 21A | 21B | 21C | 28A | ··· | 28F | 42A | 42B | 42C | 56A | ··· | 56L | 84A | ··· | 84F |
| order | 1 | 2 | 2 | 3 | 4 | 4 | 4 | 6 | 6 | 6 | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 14 | 14 | 14 | 21 | 21 | 21 | 28 | ··· | 28 | 42 | 42 | 42 | 56 | ··· | 56 | 84 | ··· | 84 |
| size | 1 | 1 | 14 | 2 | 1 | 1 | 14 | 2 | 14 | 14 | 2 | 2 | 2 | 6 | 6 | 42 | 42 | 2 | 2 | 14 | 14 | 2 | 2 | 2 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | 4 | 4 | 6 | ··· | 6 | 4 | ··· | 4 |
54 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | - | + | - | + | + | + | - | |||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | S3 | Dic3 | D6 | Dic3 | D7 | M4(2) | D14 | C4.Dic3 | C4×D7 | C8⋊D7 | S3×D7 | Dic3×D7 | C28.32D6 |
| kernel | C28.32D6 | C7×C3⋊C8 | C21⋊C8 | C12×D7 | C3×Dic7 | C6×D7 | C4×D7 | Dic7 | C28 | D14 | C3⋊C8 | C21 | C12 | C7 | C6 | C3 | C4 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 3 | 2 | 3 | 4 | 6 | 12 | 3 | 3 | 6 |
Matrix representation of C28.32D6 ►in GL4(𝔽337) generated by
| 189 | 0 | 0 | 0 |
| 0 | 189 | 0 | 0 |
| 0 | 0 | 109 | 227 |
| 0 | 0 | 252 | 194 |
| 208 | 0 | 0 | 0 |
| 0 | 209 | 0 | 0 |
| 0 | 0 | 109 | 1 |
| 0 | 0 | 252 | 228 |
| 0 | 128 | 0 | 0 |
| 117 | 0 | 0 | 0 |
| 0 | 0 | 109 | 1 |
| 0 | 0 | 252 | 228 |
G:=sub<GL(4,GF(337))| [189,0,0,0,0,189,0,0,0,0,109,252,0,0,227,194],[208,0,0,0,0,209,0,0,0,0,109,252,0,0,1,228],[0,117,0,0,128,0,0,0,0,0,109,252,0,0,1,228] >;
C28.32D6 in GAP, Magma, Sage, TeX
C_{28}._{32}D_6 % in TeX
G:=Group("C28.32D6"); // GroupNames label
G:=SmallGroup(336,26);
// by ID
G=gap.SmallGroup(336,26);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-3,-7,121,31,50,490,10373]);
// Polycyclic
G:=Group<a,b,c|a^28=b^6=1,c^2=a^7,b*a*b^-1=c*a*c^-1=a^13,c*b*c^-1=a^14*b^-1>;
// generators/relations
Export