metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.6D4, C6.7D28, C21⋊3SD16, D28.1S3, C28.23D6, C12.5D14, Dic42⋊5C2, C84.9C22, C3⋊C8⋊2D7, C4.2(S3×D7), C3⋊3(C56⋊C2), C7⋊1(D4.S3), (C3×D28).1C2, C14.2(C3⋊D4), C2.5(C3⋊D28), (C7×C3⋊C8)⋊2C2, SmallGroup(336,34)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C6.D28
G = < a,b,c | a6=1, b28=c2=a3, bab-1=cac-1=a-1, cbc-1=b27 >
Smallest permutation representation of C6.D28
►On 168 points
Generators in S168
(1 131 110 29 159 82)(2 83 160 30 111 132)(3 133 112 31 161 84)(4 85 162 32 57 134)(5 135 58 33 163 86)(6 87 164 34 59 136)(7 137 60 35 165 88)(8 89 166 36 61 138)(9 139 62 37 167 90)(10 91 168 38 63 140)(11 141 64 39 113 92)(12 93 114 40 65 142)(13 143 66 41 115 94)(14 95 116 42 67 144)(15 145 68 43 117 96)(16 97 118 44 69 146)(17 147 70 45 119 98)(18 99 120 46 71 148)(19 149 72 47 121 100)(20 101 122 48 73 150)(21 151 74 49 123 102)(22 103 124 50 75 152)(23 153 76 51 125 104)(24 105 126 52 77 154)(25 155 78 53 127 106)(26 107 128 54 79 156)(27 157 80 55 129 108)(28 109 130 56 81 158)
(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 28 29 56)(2 55 30 27)(3 26 31 54)(4 53 32 25)(5 24 33 52)(6 51 34 23)(7 22 35 50)(8 49 36 21)(9 20 37 48)(10 47 38 19)(11 18 39 46)(12 45 40 17)(13 16 41 44)(14 43 42 15)(57 106 85 78)(58 77 86 105)(59 104 87 76)(60 75 88 103)(61 102 89 74)(62 73 90 101)(63 100 91 72)(64 71 92 99)(65 98 93 70)(66 69 94 97)(67 96 95 68)(79 84 107 112)(80 111 108 83)(81 82 109 110)(113 120 141 148)(114 147 142 119)(115 118 143 146)(116 145 144 117)(121 168 149 140)(122 139 150 167)(123 166 151 138)(124 137 152 165)(125 164 153 136)(126 135 154 163)(127 162 155 134)(128 133 156 161)(129 160 157 132)(130 131 158 159)
G:=sub<Sym(168)| (1,131,110,29,159,82)(2,83,160,30,111,132)(3,133,112,31,161,84)(4,85,162,32,57,134)(5,135,58,33,163,86)(6,87,164,34,59,136)(7,137,60,35,165,88)(8,89,166,36,61,138)(9,139,62,37,167,90)(10,91,168,38,63,140)(11,141,64,39,113,92)(12,93,114,40,65,142)(13,143,66,41,115,94)(14,95,116,42,67,144)(15,145,68,43,117,96)(16,97,118,44,69,146)(17,147,70,45,119,98)(18,99,120,46,71,148)(19,149,72,47,121,100)(20,101,122,48,73,150)(21,151,74,49,123,102)(22,103,124,50,75,152)(23,153,76,51,125,104)(24,105,126,52,77,154)(25,155,78,53,127,106)(26,107,128,54,79,156)(27,157,80,55,129,108)(28,109,130,56,81,158), (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), (1,28,29,56)(2,55,30,27)(3,26,31,54)(4,53,32,25)(5,24,33,52)(6,51,34,23)(7,22,35,50)(8,49,36,21)(9,20,37,48)(10,47,38,19)(11,18,39,46)(12,45,40,17)(13,16,41,44)(14,43,42,15)(57,106,85,78)(58,77,86,105)(59,104,87,76)(60,75,88,103)(61,102,89,74)(62,73,90,101)(63,100,91,72)(64,71,92,99)(65,98,93,70)(66,69,94,97)(67,96,95,68)(79,84,107,112)(80,111,108,83)(81,82,109,110)(113,120,141,148)(114,147,142,119)(115,118,143,146)(116,145,144,117)(121,168,149,140)(122,139,150,167)(123,166,151,138)(124,137,152,165)(125,164,153,136)(126,135,154,163)(127,162,155,134)(128,133,156,161)(129,160,157,132)(130,131,158,159)>;
G:=Group( (1,131,110,29,159,82)(2,83,160,30,111,132)(3,133,112,31,161,84)(4,85,162,32,57,134)(5,135,58,33,163,86)(6,87,164,34,59,136)(7,137,60,35,165,88)(8,89,166,36,61,138)(9,139,62,37,167,90)(10,91,168,38,63,140)(11,141,64,39,113,92)(12,93,114,40,65,142)(13,143,66,41,115,94)(14,95,116,42,67,144)(15,145,68,43,117,96)(16,97,118,44,69,146)(17,147,70,45,119,98)(18,99,120,46,71,148)(19,149,72,47,121,100)(20,101,122,48,73,150)(21,151,74,49,123,102)(22,103,124,50,75,152)(23,153,76,51,125,104)(24,105,126,52,77,154)(25,155,78,53,127,106)(26,107,128,54,79,156)(27,157,80,55,129,108)(28,109,130,56,81,158), (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), (1,28,29,56)(2,55,30,27)(3,26,31,54)(4,53,32,25)(5,24,33,52)(6,51,34,23)(7,22,35,50)(8,49,36,21)(9,20,37,48)(10,47,38,19)(11,18,39,46)(12,45,40,17)(13,16,41,44)(14,43,42,15)(57,106,85,78)(58,77,86,105)(59,104,87,76)(60,75,88,103)(61,102,89,74)(62,73,90,101)(63,100,91,72)(64,71,92,99)(65,98,93,70)(66,69,94,97)(67,96,95,68)(79,84,107,112)(80,111,108,83)(81,82,109,110)(113,120,141,148)(114,147,142,119)(115,118,143,146)(116,145,144,117)(121,168,149,140)(122,139,150,167)(123,166,151,138)(124,137,152,165)(125,164,153,136)(126,135,154,163)(127,162,155,134)(128,133,156,161)(129,160,157,132)(130,131,158,159) );
G=PermutationGroup([[(1,131,110,29,159,82),(2,83,160,30,111,132),(3,133,112,31,161,84),(4,85,162,32,57,134),(5,135,58,33,163,86),(6,87,164,34,59,136),(7,137,60,35,165,88),(8,89,166,36,61,138),(9,139,62,37,167,90),(10,91,168,38,63,140),(11,141,64,39,113,92),(12,93,114,40,65,142),(13,143,66,41,115,94),(14,95,116,42,67,144),(15,145,68,43,117,96),(16,97,118,44,69,146),(17,147,70,45,119,98),(18,99,120,46,71,148),(19,149,72,47,121,100),(20,101,122,48,73,150),(21,151,74,49,123,102),(22,103,124,50,75,152),(23,153,76,51,125,104),(24,105,126,52,77,154),(25,155,78,53,127,106),(26,107,128,54,79,156),(27,157,80,55,129,108),(28,109,130,56,81,158)], [(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)], [(1,28,29,56),(2,55,30,27),(3,26,31,54),(4,53,32,25),(5,24,33,52),(6,51,34,23),(7,22,35,50),(8,49,36,21),(9,20,37,48),(10,47,38,19),(11,18,39,46),(12,45,40,17),(13,16,41,44),(14,43,42,15),(57,106,85,78),(58,77,86,105),(59,104,87,76),(60,75,88,103),(61,102,89,74),(62,73,90,101),(63,100,91,72),(64,71,92,99),(65,98,93,70),(66,69,94,97),(67,96,95,68),(79,84,107,112),(80,111,108,83),(81,82,109,110),(113,120,141,148),(114,147,142,119),(115,118,143,146),(116,145,144,117),(121,168,149,140),(122,139,150,167),(123,166,151,138),(124,137,152,165),(125,164,153,136),(126,135,154,163),(127,162,155,134),(128,133,156,161),(129,160,157,132),(130,131,158,159)]])
48 conjugacy classes
| class | 1 | 2A | 2B | 3 | 4A | 4B | 6A | 6B | 6C | 7A | 7B | 7C | 8A | 8B | 12 | 14A | 14B | 14C | 21A | 21B | 21C | 28A | ··· | 28F | 42A | 42B | 42C | 56A | ··· | 56L | 84A | ··· | 84F |
| order | 1 | 2 | 2 | 3 | 4 | 4 | 6 | 6 | 6 | 7 | 7 | 7 | 8 | 8 | 12 | 14 | 14 | 14 | 21 | 21 | 21 | 28 | ··· | 28 | 42 | 42 | 42 | 56 | ··· | 56 | 84 | ··· | 84 |
| size | 1 | 1 | 28 | 2 | 2 | 84 | 2 | 28 | 28 | 2 | 2 | 2 | 6 | 6 | 4 | 2 | 2 | 2 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | 4 | 4 | 6 | ··· | 6 | 4 | ··· | 4 |
48 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | - | + | + | - | |||
| image | C1 | C2 | C2 | C2 | S3 | D4 | D6 | D7 | SD16 | C3⋊D4 | D14 | D28 | C56⋊C2 | D4.S3 | S3×D7 | C3⋊D28 | C6.D28 |
| kernel | C6.D28 | C7×C3⋊C8 | C3×D28 | Dic42 | D28 | C42 | C28 | C3⋊C8 | C21 | C14 | C12 | C6 | C3 | C7 | C4 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 3 | 2 | 2 | 3 | 6 | 12 | 1 | 3 | 3 | 6 |
Matrix representation of C6.D28 ►in GL6(𝔽337)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 336 | 0 | 0 | 0 |
| 0 | 0 | 0 | 336 | 0 | 0 |
| 0 | 0 | 0 | 0 | 208 | 0 |
| 0 | 0 | 0 | 0 | 178 | 128 |
| 109 | 143 | 0 | 0 | 0 | 0 |
| 194 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 290 | 0 | 0 |
| 0 | 0 | 43 | 196 | 0 | 0 |
| 0 | 0 | 0 | 0 | 165 | 331 |
| 0 | 0 | 0 | 0 | 269 | 172 |
| 109 | 143 | 0 | 0 | 0 | 0 |
| 228 | 228 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 290 | 0 | 0 |
| 0 | 0 | 294 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 165 | 331 |
| 0 | 0 | 0 | 0 | 44 | 172 |
G:=sub<GL(6,GF(337))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,336,0,0,0,0,0,0,336,0,0,0,0,0,0,208,178,0,0,0,0,0,128],[109,194,0,0,0,0,143,1,0,0,0,0,0,0,0,43,0,0,0,0,290,196,0,0,0,0,0,0,165,269,0,0,0,0,331,172],[109,228,0,0,0,0,143,228,0,0,0,0,0,0,0,294,0,0,0,0,290,0,0,0,0,0,0,0,165,44,0,0,0,0,331,172] >;
C6.D28 in GAP, Magma, Sage, TeX
C_6.D_{28} % in TeX
G:=Group("C6.D28"); // GroupNames label
G:=SmallGroup(336,34);
// by ID
G=gap.SmallGroup(336,34);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-3,-7,73,31,218,50,490,10373]);
// Polycyclic
G:=Group<a,b,c|a^6=1,b^28=c^2=a^3,b*a*b^-1=c*a*c^-1=a^-1,c*b*c^-1=b^27>;
// generators/relations
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