direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: C7×C3⋊C8, C3⋊C56, C21⋊3C8, C6.C28, C84.6C2, C42.3C4, C28.4S3, C12.2C14, C14.2Dic3, C4.2(S3×C7), C2.(C7×Dic3), SmallGroup(168,3)
Series: Derived ►Chief ►Lower central ►Upper central
| C3 — C7×C3⋊C8 |
Generators and relations for C7×C3⋊C8
G = < a,b,c | a7=b3=c8=1, ab=ba, ac=ca, cbc-1=b-1 >
Smallest permutation representation of C7×C3⋊C8
►Regular action on 168 points
Generators in S168
(1 85 48 75 125 65 106)(2 86 41 76 126 66 107)(3 87 42 77 127 67 108)(4 88 43 78 128 68 109)(5 81 44 79 121 69 110)(6 82 45 80 122 70 111)(7 83 46 73 123 71 112)(8 84 47 74 124 72 105)(9 36 158 32 61 21 133)(10 37 159 25 62 22 134)(11 38 160 26 63 23 135)(12 39 153 27 64 24 136)(13 40 154 28 57 17 129)(14 33 155 29 58 18 130)(15 34 156 30 59 19 131)(16 35 157 31 60 20 132)(49 161 120 145 104 137 95)(50 162 113 146 97 138 96)(51 163 114 147 98 139 89)(52 164 115 148 99 140 90)(53 165 116 149 100 141 91)(54 166 117 150 101 142 92)(55 167 118 151 102 143 93)(56 168 119 152 103 144 94)
(1 17 143)(2 144 18)(3 19 137)(4 138 20)(5 21 139)(6 140 22)(7 23 141)(8 142 24)(9 51 44)(10 45 52)(11 53 46)(12 47 54)(13 55 48)(14 41 56)(15 49 42)(16 43 50)(25 70 148)(26 149 71)(27 72 150)(28 151 65)(29 66 152)(30 145 67)(31 68 146)(32 147 69)(33 76 168)(34 161 77)(35 78 162)(36 163 79)(37 80 164)(38 165 73)(39 74 166)(40 167 75)(57 102 106)(58 107 103)(59 104 108)(60 109 97)(61 98 110)(62 111 99)(63 100 112)(64 105 101)(81 133 89)(82 90 134)(83 135 91)(84 92 136)(85 129 93)(86 94 130)(87 131 95)(88 96 132)(113 157 128)(114 121 158)(115 159 122)(116 123 160)(117 153 124)(118 125 154)(119 155 126)(120 127 156)
(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)
G:=sub<Sym(168)| (1,85,48,75,125,65,106)(2,86,41,76,126,66,107)(3,87,42,77,127,67,108)(4,88,43,78,128,68,109)(5,81,44,79,121,69,110)(6,82,45,80,122,70,111)(7,83,46,73,123,71,112)(8,84,47,74,124,72,105)(9,36,158,32,61,21,133)(10,37,159,25,62,22,134)(11,38,160,26,63,23,135)(12,39,153,27,64,24,136)(13,40,154,28,57,17,129)(14,33,155,29,58,18,130)(15,34,156,30,59,19,131)(16,35,157,31,60,20,132)(49,161,120,145,104,137,95)(50,162,113,146,97,138,96)(51,163,114,147,98,139,89)(52,164,115,148,99,140,90)(53,165,116,149,100,141,91)(54,166,117,150,101,142,92)(55,167,118,151,102,143,93)(56,168,119,152,103,144,94), (1,17,143)(2,144,18)(3,19,137)(4,138,20)(5,21,139)(6,140,22)(7,23,141)(8,142,24)(9,51,44)(10,45,52)(11,53,46)(12,47,54)(13,55,48)(14,41,56)(15,49,42)(16,43,50)(25,70,148)(26,149,71)(27,72,150)(28,151,65)(29,66,152)(30,145,67)(31,68,146)(32,147,69)(33,76,168)(34,161,77)(35,78,162)(36,163,79)(37,80,164)(38,165,73)(39,74,166)(40,167,75)(57,102,106)(58,107,103)(59,104,108)(60,109,97)(61,98,110)(62,111,99)(63,100,112)(64,105,101)(81,133,89)(82,90,134)(83,135,91)(84,92,136)(85,129,93)(86,94,130)(87,131,95)(88,96,132)(113,157,128)(114,121,158)(115,159,122)(116,123,160)(117,153,124)(118,125,154)(119,155,126)(120,127,156), (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)>;
G:=Group( (1,85,48,75,125,65,106)(2,86,41,76,126,66,107)(3,87,42,77,127,67,108)(4,88,43,78,128,68,109)(5,81,44,79,121,69,110)(6,82,45,80,122,70,111)(7,83,46,73,123,71,112)(8,84,47,74,124,72,105)(9,36,158,32,61,21,133)(10,37,159,25,62,22,134)(11,38,160,26,63,23,135)(12,39,153,27,64,24,136)(13,40,154,28,57,17,129)(14,33,155,29,58,18,130)(15,34,156,30,59,19,131)(16,35,157,31,60,20,132)(49,161,120,145,104,137,95)(50,162,113,146,97,138,96)(51,163,114,147,98,139,89)(52,164,115,148,99,140,90)(53,165,116,149,100,141,91)(54,166,117,150,101,142,92)(55,167,118,151,102,143,93)(56,168,119,152,103,144,94), (1,17,143)(2,144,18)(3,19,137)(4,138,20)(5,21,139)(6,140,22)(7,23,141)(8,142,24)(9,51,44)(10,45,52)(11,53,46)(12,47,54)(13,55,48)(14,41,56)(15,49,42)(16,43,50)(25,70,148)(26,149,71)(27,72,150)(28,151,65)(29,66,152)(30,145,67)(31,68,146)(32,147,69)(33,76,168)(34,161,77)(35,78,162)(36,163,79)(37,80,164)(38,165,73)(39,74,166)(40,167,75)(57,102,106)(58,107,103)(59,104,108)(60,109,97)(61,98,110)(62,111,99)(63,100,112)(64,105,101)(81,133,89)(82,90,134)(83,135,91)(84,92,136)(85,129,93)(86,94,130)(87,131,95)(88,96,132)(113,157,128)(114,121,158)(115,159,122)(116,123,160)(117,153,124)(118,125,154)(119,155,126)(120,127,156), (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) );
G=PermutationGroup([[(1,85,48,75,125,65,106),(2,86,41,76,126,66,107),(3,87,42,77,127,67,108),(4,88,43,78,128,68,109),(5,81,44,79,121,69,110),(6,82,45,80,122,70,111),(7,83,46,73,123,71,112),(8,84,47,74,124,72,105),(9,36,158,32,61,21,133),(10,37,159,25,62,22,134),(11,38,160,26,63,23,135),(12,39,153,27,64,24,136),(13,40,154,28,57,17,129),(14,33,155,29,58,18,130),(15,34,156,30,59,19,131),(16,35,157,31,60,20,132),(49,161,120,145,104,137,95),(50,162,113,146,97,138,96),(51,163,114,147,98,139,89),(52,164,115,148,99,140,90),(53,165,116,149,100,141,91),(54,166,117,150,101,142,92),(55,167,118,151,102,143,93),(56,168,119,152,103,144,94)], [(1,17,143),(2,144,18),(3,19,137),(4,138,20),(5,21,139),(6,140,22),(7,23,141),(8,142,24),(9,51,44),(10,45,52),(11,53,46),(12,47,54),(13,55,48),(14,41,56),(15,49,42),(16,43,50),(25,70,148),(26,149,71),(27,72,150),(28,151,65),(29,66,152),(30,145,67),(31,68,146),(32,147,69),(33,76,168),(34,161,77),(35,78,162),(36,163,79),(37,80,164),(38,165,73),(39,74,166),(40,167,75),(57,102,106),(58,107,103),(59,104,108),(60,109,97),(61,98,110),(62,111,99),(63,100,112),(64,105,101),(81,133,89),(82,90,134),(83,135,91),(84,92,136),(85,129,93),(86,94,130),(87,131,95),(88,96,132),(113,157,128),(114,121,158),(115,159,122),(116,123,160),(117,153,124),(118,125,154),(119,155,126),(120,127,156)], [(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)]])
C7×C3⋊C8 is a maximal subgroup of
D21⋊C8 C28.32D6 D42.C4 C3⋊D56 C6.D28 C21⋊SD16 C3⋊Dic28 S3×C56
84 conjugacy classes
| class | 1 | 2 | 3 | 4A | 4B | 6 | 7A | ··· | 7F | 8A | 8B | 8C | 8D | 12A | 12B | 14A | ··· | 14F | 21A | ··· | 21F | 28A | ··· | 28L | 42A | ··· | 42F | 56A | ··· | 56X | 84A | ··· | 84L |
| order | 1 | 2 | 3 | 4 | 4 | 6 | 7 | ··· | 7 | 8 | 8 | 8 | 8 | 12 | 12 | 14 | ··· | 14 | 21 | ··· | 21 | 28 | ··· | 28 | 42 | ··· | 42 | 56 | ··· | 56 | 84 | ··· | 84 |
| size | 1 | 1 | 2 | 1 | 1 | 2 | 1 | ··· | 1 | 3 | 3 | 3 | 3 | 2 | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 3 | ··· | 3 | 2 | ··· | 2 |
84 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | - | ||||||||||
| image | C1 | C2 | C4 | C7 | C8 | C14 | C28 | C56 | S3 | Dic3 | C3⋊C8 | S3×C7 | C7×Dic3 | C7×C3⋊C8 |
| kernel | C7×C3⋊C8 | C84 | C42 | C3⋊C8 | C21 | C12 | C6 | C3 | C28 | C14 | C7 | C4 | C2 | C1 |
| # reps | 1 | 1 | 2 | 6 | 4 | 6 | 12 | 24 | 1 | 1 | 2 | 6 | 6 | 12 |
Matrix representation of C7×C3⋊C8 ►in GL2(𝔽29) generated by
| 16 | 0 |
| 0 | 16 |
| 14 | 2 |
| 25 | 14 |
| 0 | 8 |
| 16 | 0 |
G:=sub<GL(2,GF(29))| [16,0,0,16],[14,25,2,14],[0,16,8,0] >;
C7×C3⋊C8 in GAP, Magma, Sage, TeX
C_7\times C_3\rtimes C_8
% in TeX
G:=Group("C7xC3:C8"); // GroupNames label
G:=SmallGroup(168,3);
// by ID
G=gap.SmallGroup(168,3);
# by ID
G:=PCGroup([5,-2,-7,-2,-2,-3,70,42,2804]);
// Polycyclic
G:=Group<a,b,c|a^7=b^3=c^8=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
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