direct product, metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C7×C8⋊S3, C56⋊7S3, D6.C28, C24⋊5C14, C168⋊13C2, C28.57D6, Dic3.C28, C21⋊6M4(2), C84.74C22, C3⋊C8⋊4C14, C8⋊3(S3×C7), C2.3(S3×C28), C6.2(C2×C28), C3⋊1(C7×M4(2)), (S3×C28).5C2, (C4×S3).2C14, (S3×C14).3C4, C14.16(C4×S3), C4.13(S3×C14), C42.25(C2×C4), C12.13(C2×C14), (C7×Dic3).3C4, (C7×C3⋊C8)⋊11C2, SmallGroup(336,75)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C7×C8⋊S3
G = < a,b,c,d | a7=b8=c3=d2=1, ab=ba, ac=ca, ad=da, bc=cb, dbd=b5, dcd=c-1 >
Smallest permutation representation of C7×C8⋊S3
►On 168 points
Generators in S168
(1 15 48 156 107 65 27)(2 16 41 157 108 66 28)(3 9 42 158 109 67 29)(4 10 43 159 110 68 30)(5 11 44 160 111 69 31)(6 12 45 153 112 70 32)(7 13 46 154 105 71 25)(8 14 47 155 106 72 26)(17 55 82 129 80 40 64)(18 56 83 130 73 33 57)(19 49 84 131 74 34 58)(20 50 85 132 75 35 59)(21 51 86 133 76 36 60)(22 52 87 134 77 37 61)(23 53 88 135 78 38 62)(24 54 81 136 79 39 63)(89 122 163 114 147 98 139)(90 123 164 115 148 99 140)(91 124 165 116 149 100 141)(92 125 166 117 150 101 142)(93 126 167 118 151 102 143)(94 127 168 119 152 103 144)(95 128 161 120 145 104 137)(96 121 162 113 146 97 138)
(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)
(1 143 18)(2 144 19)(3 137 20)(4 138 21)(5 139 22)(6 140 23)(7 141 24)(8 142 17)(9 95 50)(10 96 51)(11 89 52)(12 90 53)(13 91 54)(14 92 55)(15 93 56)(16 94 49)(25 100 63)(26 101 64)(27 102 57)(28 103 58)(29 104 59)(30 97 60)(31 98 61)(32 99 62)(33 65 151)(34 66 152)(35 67 145)(36 68 146)(37 69 147)(38 70 148)(39 71 149)(40 72 150)(41 127 84)(42 128 85)(43 121 86)(44 122 87)(45 123 88)(46 124 81)(47 125 82)(48 126 83)(73 107 118)(74 108 119)(75 109 120)(76 110 113)(77 111 114)(78 112 115)(79 105 116)(80 106 117)(129 155 166)(130 156 167)(131 157 168)(132 158 161)(133 159 162)(134 160 163)(135 153 164)(136 154 165)
(2 6)(4 8)(10 14)(12 16)(17 138)(18 143)(19 140)(20 137)(21 142)(22 139)(23 144)(24 141)(26 30)(28 32)(33 151)(34 148)(35 145)(36 150)(37 147)(38 152)(39 149)(40 146)(41 45)(43 47)(49 90)(50 95)(51 92)(52 89)(53 94)(54 91)(55 96)(56 93)(57 102)(58 99)(59 104)(60 101)(61 98)(62 103)(63 100)(64 97)(66 70)(68 72)(73 118)(74 115)(75 120)(76 117)(77 114)(78 119)(79 116)(80 113)(81 124)(82 121)(83 126)(84 123)(85 128)(86 125)(87 122)(88 127)(106 110)(108 112)(129 162)(130 167)(131 164)(132 161)(133 166)(134 163)(135 168)(136 165)(153 157)(155 159)
G:=sub<Sym(168)| (1,15,48,156,107,65,27)(2,16,41,157,108,66,28)(3,9,42,158,109,67,29)(4,10,43,159,110,68,30)(5,11,44,160,111,69,31)(6,12,45,153,112,70,32)(7,13,46,154,105,71,25)(8,14,47,155,106,72,26)(17,55,82,129,80,40,64)(18,56,83,130,73,33,57)(19,49,84,131,74,34,58)(20,50,85,132,75,35,59)(21,51,86,133,76,36,60)(22,52,87,134,77,37,61)(23,53,88,135,78,38,62)(24,54,81,136,79,39,63)(89,122,163,114,147,98,139)(90,123,164,115,148,99,140)(91,124,165,116,149,100,141)(92,125,166,117,150,101,142)(93,126,167,118,151,102,143)(94,127,168,119,152,103,144)(95,128,161,120,145,104,137)(96,121,162,113,146,97,138), (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), (1,143,18)(2,144,19)(3,137,20)(4,138,21)(5,139,22)(6,140,23)(7,141,24)(8,142,17)(9,95,50)(10,96,51)(11,89,52)(12,90,53)(13,91,54)(14,92,55)(15,93,56)(16,94,49)(25,100,63)(26,101,64)(27,102,57)(28,103,58)(29,104,59)(30,97,60)(31,98,61)(32,99,62)(33,65,151)(34,66,152)(35,67,145)(36,68,146)(37,69,147)(38,70,148)(39,71,149)(40,72,150)(41,127,84)(42,128,85)(43,121,86)(44,122,87)(45,123,88)(46,124,81)(47,125,82)(48,126,83)(73,107,118)(74,108,119)(75,109,120)(76,110,113)(77,111,114)(78,112,115)(79,105,116)(80,106,117)(129,155,166)(130,156,167)(131,157,168)(132,158,161)(133,159,162)(134,160,163)(135,153,164)(136,154,165), (2,6)(4,8)(10,14)(12,16)(17,138)(18,143)(19,140)(20,137)(21,142)(22,139)(23,144)(24,141)(26,30)(28,32)(33,151)(34,148)(35,145)(36,150)(37,147)(38,152)(39,149)(40,146)(41,45)(43,47)(49,90)(50,95)(51,92)(52,89)(53,94)(54,91)(55,96)(56,93)(57,102)(58,99)(59,104)(60,101)(61,98)(62,103)(63,100)(64,97)(66,70)(68,72)(73,118)(74,115)(75,120)(76,117)(77,114)(78,119)(79,116)(80,113)(81,124)(82,121)(83,126)(84,123)(85,128)(86,125)(87,122)(88,127)(106,110)(108,112)(129,162)(130,167)(131,164)(132,161)(133,166)(134,163)(135,168)(136,165)(153,157)(155,159)>;
G:=Group( (1,15,48,156,107,65,27)(2,16,41,157,108,66,28)(3,9,42,158,109,67,29)(4,10,43,159,110,68,30)(5,11,44,160,111,69,31)(6,12,45,153,112,70,32)(7,13,46,154,105,71,25)(8,14,47,155,106,72,26)(17,55,82,129,80,40,64)(18,56,83,130,73,33,57)(19,49,84,131,74,34,58)(20,50,85,132,75,35,59)(21,51,86,133,76,36,60)(22,52,87,134,77,37,61)(23,53,88,135,78,38,62)(24,54,81,136,79,39,63)(89,122,163,114,147,98,139)(90,123,164,115,148,99,140)(91,124,165,116,149,100,141)(92,125,166,117,150,101,142)(93,126,167,118,151,102,143)(94,127,168,119,152,103,144)(95,128,161,120,145,104,137)(96,121,162,113,146,97,138), (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), (1,143,18)(2,144,19)(3,137,20)(4,138,21)(5,139,22)(6,140,23)(7,141,24)(8,142,17)(9,95,50)(10,96,51)(11,89,52)(12,90,53)(13,91,54)(14,92,55)(15,93,56)(16,94,49)(25,100,63)(26,101,64)(27,102,57)(28,103,58)(29,104,59)(30,97,60)(31,98,61)(32,99,62)(33,65,151)(34,66,152)(35,67,145)(36,68,146)(37,69,147)(38,70,148)(39,71,149)(40,72,150)(41,127,84)(42,128,85)(43,121,86)(44,122,87)(45,123,88)(46,124,81)(47,125,82)(48,126,83)(73,107,118)(74,108,119)(75,109,120)(76,110,113)(77,111,114)(78,112,115)(79,105,116)(80,106,117)(129,155,166)(130,156,167)(131,157,168)(132,158,161)(133,159,162)(134,160,163)(135,153,164)(136,154,165), (2,6)(4,8)(10,14)(12,16)(17,138)(18,143)(19,140)(20,137)(21,142)(22,139)(23,144)(24,141)(26,30)(28,32)(33,151)(34,148)(35,145)(36,150)(37,147)(38,152)(39,149)(40,146)(41,45)(43,47)(49,90)(50,95)(51,92)(52,89)(53,94)(54,91)(55,96)(56,93)(57,102)(58,99)(59,104)(60,101)(61,98)(62,103)(63,100)(64,97)(66,70)(68,72)(73,118)(74,115)(75,120)(76,117)(77,114)(78,119)(79,116)(80,113)(81,124)(82,121)(83,126)(84,123)(85,128)(86,125)(87,122)(88,127)(106,110)(108,112)(129,162)(130,167)(131,164)(132,161)(133,166)(134,163)(135,168)(136,165)(153,157)(155,159) );
G=PermutationGroup([[(1,15,48,156,107,65,27),(2,16,41,157,108,66,28),(3,9,42,158,109,67,29),(4,10,43,159,110,68,30),(5,11,44,160,111,69,31),(6,12,45,153,112,70,32),(7,13,46,154,105,71,25),(8,14,47,155,106,72,26),(17,55,82,129,80,40,64),(18,56,83,130,73,33,57),(19,49,84,131,74,34,58),(20,50,85,132,75,35,59),(21,51,86,133,76,36,60),(22,52,87,134,77,37,61),(23,53,88,135,78,38,62),(24,54,81,136,79,39,63),(89,122,163,114,147,98,139),(90,123,164,115,148,99,140),(91,124,165,116,149,100,141),(92,125,166,117,150,101,142),(93,126,167,118,151,102,143),(94,127,168,119,152,103,144),(95,128,161,120,145,104,137),(96,121,162,113,146,97,138)], [(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)], [(1,143,18),(2,144,19),(3,137,20),(4,138,21),(5,139,22),(6,140,23),(7,141,24),(8,142,17),(9,95,50),(10,96,51),(11,89,52),(12,90,53),(13,91,54),(14,92,55),(15,93,56),(16,94,49),(25,100,63),(26,101,64),(27,102,57),(28,103,58),(29,104,59),(30,97,60),(31,98,61),(32,99,62),(33,65,151),(34,66,152),(35,67,145),(36,68,146),(37,69,147),(38,70,148),(39,71,149),(40,72,150),(41,127,84),(42,128,85),(43,121,86),(44,122,87),(45,123,88),(46,124,81),(47,125,82),(48,126,83),(73,107,118),(74,108,119),(75,109,120),(76,110,113),(77,111,114),(78,112,115),(79,105,116),(80,106,117),(129,155,166),(130,156,167),(131,157,168),(132,158,161),(133,159,162),(134,160,163),(135,153,164),(136,154,165)], [(2,6),(4,8),(10,14),(12,16),(17,138),(18,143),(19,140),(20,137),(21,142),(22,139),(23,144),(24,141),(26,30),(28,32),(33,151),(34,148),(35,145),(36,150),(37,147),(38,152),(39,149),(40,146),(41,45),(43,47),(49,90),(50,95),(51,92),(52,89),(53,94),(54,91),(55,96),(56,93),(57,102),(58,99),(59,104),(60,101),(61,98),(62,103),(63,100),(64,97),(66,70),(68,72),(73,118),(74,115),(75,120),(76,117),(77,114),(78,119),(79,116),(80,113),(81,124),(82,121),(83,126),(84,123),(85,128),(86,125),(87,122),(88,127),(106,110),(108,112),(129,162),(130,167),(131,164),(132,161),(133,166),(134,163),(135,168),(136,165),(153,157),(155,159)]])
126 conjugacy classes
| class | 1 | 2A | 2B | 3 | 4A | 4B | 4C | 6 | 7A | ··· | 7F | 8A | 8B | 8C | 8D | 12A | 12B | 14A | ··· | 14F | 14G | ··· | 14L | 21A | ··· | 21F | 24A | 24B | 24C | 24D | 28A | ··· | 28L | 28M | ··· | 28R | 42A | ··· | 42F | 56A | ··· | 56L | 56M | ··· | 56X | 84A | ··· | 84L | 168A | ··· | 168X |
| order | 1 | 2 | 2 | 3 | 4 | 4 | 4 | 6 | 7 | ··· | 7 | 8 | 8 | 8 | 8 | 12 | 12 | 14 | ··· | 14 | 14 | ··· | 14 | 21 | ··· | 21 | 24 | 24 | 24 | 24 | 28 | ··· | 28 | 28 | ··· | 28 | 42 | ··· | 42 | 56 | ··· | 56 | 56 | ··· | 56 | 84 | ··· | 84 | 168 | ··· | 168 |
| size | 1 | 1 | 6 | 2 | 1 | 1 | 6 | 2 | 1 | ··· | 1 | 2 | 2 | 6 | 6 | 2 | 2 | 1 | ··· | 1 | 6 | ··· | 6 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 6 | ··· | 6 | 2 | ··· | 2 | 2 | ··· | 2 | 6 | ··· | 6 | 2 | ··· | 2 | 2 | ··· | 2 |
126 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | ||||||||||||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C7 | C14 | C14 | C14 | C28 | C28 | S3 | D6 | M4(2) | C4×S3 | S3×C7 | C8⋊S3 | S3×C14 | C7×M4(2) | S3×C28 | C7×C8⋊S3 |
| kernel | C7×C8⋊S3 | C7×C3⋊C8 | C168 | S3×C28 | C7×Dic3 | S3×C14 | C8⋊S3 | C3⋊C8 | C24 | C4×S3 | Dic3 | D6 | C56 | C28 | C21 | C14 | C8 | C7 | C4 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 6 | 6 | 6 | 6 | 12 | 12 | 1 | 1 | 2 | 2 | 6 | 4 | 6 | 12 | 12 | 24 |
Matrix representation of C7×C8⋊S3 ►in GL2(𝔽29) generated by
| 20 | 0 |
| 0 | 20 |
| 23 | 4 |
| 17 | 6 |
| 27 | 1 |
| 26 | 1 |
| 1 | 28 |
| 0 | 28 |
G:=sub<GL(2,GF(29))| [20,0,0,20],[23,17,4,6],[27,26,1,1],[1,0,28,28] >;
C7×C8⋊S3 in GAP, Magma, Sage, TeX
C_7\times C_8\rtimes S_3
% in TeX
G:=Group("C7xC8:S3"); // GroupNames label
G:=SmallGroup(336,75);
// by ID
G=gap.SmallGroup(336,75);
# by ID
G:=PCGroup([6,-2,-2,-7,-2,-2,-3,697,175,69,8069]);
// Polycyclic
G:=Group<a,b,c,d|a^7=b^8=c^3=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d=b^5,d*c*d=c^-1>;
// generators/relations
Export