direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C8×D21, C56⋊3S3, C24⋊4D7, C168⋊4C2, D42.4C4, C4.12D42, C28.47D6, C12.48D14, C84.54C22, Dic21.4C4, C7⋊2(S3×C8), C3⋊2(C8×D7), C21⋊4(C2×C8), C6.5(C4×D7), C14.5(C4×S3), C2.1(C4×D21), C21⋊C8⋊13C2, C42.14(C2×C4), (C4×D21).6C2, SmallGroup(336,90)
Series: Derived ►Chief ►Lower central ►Upper central
| C21 — C8×D21 |
Generators and relations for C8×D21
G = < a,b,c | a8=b21=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of C8×D21
►On 168 points
Generators in S168
(1 148 76 118 31 144 52 93)(2 149 77 119 32 145 53 94)(3 150 78 120 33 146 54 95)(4 151 79 121 34 147 55 96)(5 152 80 122 35 127 56 97)(6 153 81 123 36 128 57 98)(7 154 82 124 37 129 58 99)(8 155 83 125 38 130 59 100)(9 156 84 126 39 131 60 101)(10 157 64 106 40 132 61 102)(11 158 65 107 41 133 62 103)(12 159 66 108 42 134 63 104)(13 160 67 109 22 135 43 105)(14 161 68 110 23 136 44 85)(15 162 69 111 24 137 45 86)(16 163 70 112 25 138 46 87)(17 164 71 113 26 139 47 88)(18 165 72 114 27 140 48 89)(19 166 73 115 28 141 49 90)(20 167 74 116 29 142 50 91)(21 168 75 117 30 143 51 92)
(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 21)(2 20)(3 19)(4 18)(5 17)(6 16)(7 15)(8 14)(9 13)(10 12)(22 39)(23 38)(24 37)(25 36)(26 35)(27 34)(28 33)(29 32)(30 31)(40 42)(43 60)(44 59)(45 58)(46 57)(47 56)(48 55)(49 54)(50 53)(51 52)(61 63)(64 66)(67 84)(68 83)(69 82)(70 81)(71 80)(72 79)(73 78)(74 77)(75 76)(85 100)(86 99)(87 98)(88 97)(89 96)(90 95)(91 94)(92 93)(101 105)(102 104)(106 108)(109 126)(110 125)(111 124)(112 123)(113 122)(114 121)(115 120)(116 119)(117 118)(127 139)(128 138)(129 137)(130 136)(131 135)(132 134)(140 147)(141 146)(142 145)(143 144)(148 168)(149 167)(150 166)(151 165)(152 164)(153 163)(154 162)(155 161)(156 160)(157 159)
G:=sub<Sym(168)| (1,148,76,118,31,144,52,93)(2,149,77,119,32,145,53,94)(3,150,78,120,33,146,54,95)(4,151,79,121,34,147,55,96)(5,152,80,122,35,127,56,97)(6,153,81,123,36,128,57,98)(7,154,82,124,37,129,58,99)(8,155,83,125,38,130,59,100)(9,156,84,126,39,131,60,101)(10,157,64,106,40,132,61,102)(11,158,65,107,41,133,62,103)(12,159,66,108,42,134,63,104)(13,160,67,109,22,135,43,105)(14,161,68,110,23,136,44,85)(15,162,69,111,24,137,45,86)(16,163,70,112,25,138,46,87)(17,164,71,113,26,139,47,88)(18,165,72,114,27,140,48,89)(19,166,73,115,28,141,49,90)(20,167,74,116,29,142,50,91)(21,168,75,117,30,143,51,92), (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), (1,21)(2,20)(3,19)(4,18)(5,17)(6,16)(7,15)(8,14)(9,13)(10,12)(22,39)(23,38)(24,37)(25,36)(26,35)(27,34)(28,33)(29,32)(30,31)(40,42)(43,60)(44,59)(45,58)(46,57)(47,56)(48,55)(49,54)(50,53)(51,52)(61,63)(64,66)(67,84)(68,83)(69,82)(70,81)(71,80)(72,79)(73,78)(74,77)(75,76)(85,100)(86,99)(87,98)(88,97)(89,96)(90,95)(91,94)(92,93)(101,105)(102,104)(106,108)(109,126)(110,125)(111,124)(112,123)(113,122)(114,121)(115,120)(116,119)(117,118)(127,139)(128,138)(129,137)(130,136)(131,135)(132,134)(140,147)(141,146)(142,145)(143,144)(148,168)(149,167)(150,166)(151,165)(152,164)(153,163)(154,162)(155,161)(156,160)(157,159)>;
G:=Group( (1,148,76,118,31,144,52,93)(2,149,77,119,32,145,53,94)(3,150,78,120,33,146,54,95)(4,151,79,121,34,147,55,96)(5,152,80,122,35,127,56,97)(6,153,81,123,36,128,57,98)(7,154,82,124,37,129,58,99)(8,155,83,125,38,130,59,100)(9,156,84,126,39,131,60,101)(10,157,64,106,40,132,61,102)(11,158,65,107,41,133,62,103)(12,159,66,108,42,134,63,104)(13,160,67,109,22,135,43,105)(14,161,68,110,23,136,44,85)(15,162,69,111,24,137,45,86)(16,163,70,112,25,138,46,87)(17,164,71,113,26,139,47,88)(18,165,72,114,27,140,48,89)(19,166,73,115,28,141,49,90)(20,167,74,116,29,142,50,91)(21,168,75,117,30,143,51,92), (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), (1,21)(2,20)(3,19)(4,18)(5,17)(6,16)(7,15)(8,14)(9,13)(10,12)(22,39)(23,38)(24,37)(25,36)(26,35)(27,34)(28,33)(29,32)(30,31)(40,42)(43,60)(44,59)(45,58)(46,57)(47,56)(48,55)(49,54)(50,53)(51,52)(61,63)(64,66)(67,84)(68,83)(69,82)(70,81)(71,80)(72,79)(73,78)(74,77)(75,76)(85,100)(86,99)(87,98)(88,97)(89,96)(90,95)(91,94)(92,93)(101,105)(102,104)(106,108)(109,126)(110,125)(111,124)(112,123)(113,122)(114,121)(115,120)(116,119)(117,118)(127,139)(128,138)(129,137)(130,136)(131,135)(132,134)(140,147)(141,146)(142,145)(143,144)(148,168)(149,167)(150,166)(151,165)(152,164)(153,163)(154,162)(155,161)(156,160)(157,159) );
G=PermutationGroup([[(1,148,76,118,31,144,52,93),(2,149,77,119,32,145,53,94),(3,150,78,120,33,146,54,95),(4,151,79,121,34,147,55,96),(5,152,80,122,35,127,56,97),(6,153,81,123,36,128,57,98),(7,154,82,124,37,129,58,99),(8,155,83,125,38,130,59,100),(9,156,84,126,39,131,60,101),(10,157,64,106,40,132,61,102),(11,158,65,107,41,133,62,103),(12,159,66,108,42,134,63,104),(13,160,67,109,22,135,43,105),(14,161,68,110,23,136,44,85),(15,162,69,111,24,137,45,86),(16,163,70,112,25,138,46,87),(17,164,71,113,26,139,47,88),(18,165,72,114,27,140,48,89),(19,166,73,115,28,141,49,90),(20,167,74,116,29,142,50,91),(21,168,75,117,30,143,51,92)], [(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)], [(1,21),(2,20),(3,19),(4,18),(5,17),(6,16),(7,15),(8,14),(9,13),(10,12),(22,39),(23,38),(24,37),(25,36),(26,35),(27,34),(28,33),(29,32),(30,31),(40,42),(43,60),(44,59),(45,58),(46,57),(47,56),(48,55),(49,54),(50,53),(51,52),(61,63),(64,66),(67,84),(68,83),(69,82),(70,81),(71,80),(72,79),(73,78),(74,77),(75,76),(85,100),(86,99),(87,98),(88,97),(89,96),(90,95),(91,94),(92,93),(101,105),(102,104),(106,108),(109,126),(110,125),(111,124),(112,123),(113,122),(114,121),(115,120),(116,119),(117,118),(127,139),(128,138),(129,137),(130,136),(131,135),(132,134),(140,147),(141,146),(142,145),(143,144),(148,168),(149,167),(150,166),(151,165),(152,164),(153,163),(154,162),(155,161),(156,160),(157,159)]])
96 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 6 | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 12A | 12B | 14A | 14B | 14C | 21A | ··· | 21F | 24A | 24B | 24C | 24D | 28A | ··· | 28F | 42A | ··· | 42F | 56A | ··· | 56L | 84A | ··· | 84L | 168A | ··· | 168X |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 6 | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 12 | 12 | 14 | 14 | 14 | 21 | ··· | 21 | 24 | 24 | 24 | 24 | 28 | ··· | 28 | 42 | ··· | 42 | 56 | ··· | 56 | 84 | ··· | 84 | 168 | ··· | 168 |
| size | 1 | 1 | 21 | 21 | 2 | 1 | 1 | 21 | 21 | 2 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 21 | 21 | 21 | 21 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
96 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | |||||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C8 | S3 | D6 | D7 | C4×S3 | D14 | D21 | S3×C8 | C4×D7 | D42 | C8×D7 | C4×D21 | C8×D21 |
| kernel | C8×D21 | C21⋊C8 | C168 | C4×D21 | Dic21 | D42 | D21 | C56 | C28 | C24 | C14 | C12 | C8 | C7 | C6 | C4 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 8 | 1 | 1 | 3 | 2 | 3 | 6 | 4 | 6 | 6 | 12 | 12 | 24 |
Matrix representation of C8×D21 ►in GL2(𝔽41) generated by
| 38 | 0 |
| 0 | 38 |
| 40 | 11 |
| 17 | 17 |
| 24 | 12 |
| 17 | 17 |
G:=sub<GL(2,GF(41))| [38,0,0,38],[40,17,11,17],[24,17,12,17] >;
C8×D21 in GAP, Magma, Sage, TeX
C_8\times D_{21} % in TeX
G:=Group("C8xD21"); // GroupNames label
G:=SmallGroup(336,90);
// by ID
G=gap.SmallGroup(336,90);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-3,-7,31,50,964,10373]);
// Polycyclic
G:=Group<a,b,c|a^8=b^21=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export