metabelian, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: D21⋊C8, D42.2C4, C28.31D6, C12.31D14, C84.31C22, Dic21.2C4, C7⋊C8⋊6S3, C7⋊1(S3×C8), C3⋊C8⋊6D7, C3⋊1(C8×D7), C21⋊3(C2×C8), C6.1(C4×D7), C14.1(C4×S3), C42.3(C2×C4), C4.24(S3×D7), (C4×D21).4C2, C2.1(D21⋊C4), (C3×C7⋊C8)⋊5C2, (C7×C3⋊C8)⋊5C2, SmallGroup(336,25)
Series: Derived ►Chief ►Lower central ►Upper central
| C21 — D21⋊C8 |
Generators and relations for D21⋊C8
G = < a,b,c | a21=b2=c8=1, bab=a-1, cac-1=a8, cbc-1=a7b >
Smallest permutation representation of D21⋊C8
►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 39)(2 38)(3 37)(4 36)(5 35)(6 34)(7 33)(8 32)(9 31)(10 30)(11 29)(12 28)(13 27)(14 26)(15 25)(16 24)(17 23)(18 22)(19 42)(20 41)(21 40)(43 69)(44 68)(45 67)(46 66)(47 65)(48 64)(49 84)(50 83)(51 82)(52 81)(53 80)(54 79)(55 78)(56 77)(57 76)(58 75)(59 74)(60 73)(61 72)(62 71)(63 70)(85 112)(86 111)(87 110)(88 109)(89 108)(90 107)(91 106)(92 126)(93 125)(94 124)(95 123)(96 122)(97 121)(98 120)(99 119)(100 118)(101 117)(102 116)(103 115)(104 114)(105 113)(127 163)(128 162)(129 161)(130 160)(131 159)(132 158)(133 157)(134 156)(135 155)(136 154)(137 153)(138 152)(139 151)(140 150)(141 149)(142 148)(143 168)(144 167)(145 166)(146 165)(147 164)
(1 164 73 109 40 127 61 89)(2 151 74 117 41 135 62 97)(3 159 75 125 42 143 63 105)(4 167 76 112 22 130 43 92)(5 154 77 120 23 138 44 100)(6 162 78 107 24 146 45 87)(7 149 79 115 25 133 46 95)(8 157 80 123 26 141 47 103)(9 165 81 110 27 128 48 90)(10 152 82 118 28 136 49 98)(11 160 83 126 29 144 50 85)(12 168 84 113 30 131 51 93)(13 155 64 121 31 139 52 101)(14 163 65 108 32 147 53 88)(15 150 66 116 33 134 54 96)(16 158 67 124 34 142 55 104)(17 166 68 111 35 129 56 91)(18 153 69 119 36 137 57 99)(19 161 70 106 37 145 58 86)(20 148 71 114 38 132 59 94)(21 156 72 122 39 140 60 102)
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), (1,39)(2,38)(3,37)(4,36)(5,35)(6,34)(7,33)(8,32)(9,31)(10,30)(11,29)(12,28)(13,27)(14,26)(15,25)(16,24)(17,23)(18,22)(19,42)(20,41)(21,40)(43,69)(44,68)(45,67)(46,66)(47,65)(48,64)(49,84)(50,83)(51,82)(52,81)(53,80)(54,79)(55,78)(56,77)(57,76)(58,75)(59,74)(60,73)(61,72)(62,71)(63,70)(85,112)(86,111)(87,110)(88,109)(89,108)(90,107)(91,106)(92,126)(93,125)(94,124)(95,123)(96,122)(97,121)(98,120)(99,119)(100,118)(101,117)(102,116)(103,115)(104,114)(105,113)(127,163)(128,162)(129,161)(130,160)(131,159)(132,158)(133,157)(134,156)(135,155)(136,154)(137,153)(138,152)(139,151)(140,150)(141,149)(142,148)(143,168)(144,167)(145,166)(146,165)(147,164), (1,164,73,109,40,127,61,89)(2,151,74,117,41,135,62,97)(3,159,75,125,42,143,63,105)(4,167,76,112,22,130,43,92)(5,154,77,120,23,138,44,100)(6,162,78,107,24,146,45,87)(7,149,79,115,25,133,46,95)(8,157,80,123,26,141,47,103)(9,165,81,110,27,128,48,90)(10,152,82,118,28,136,49,98)(11,160,83,126,29,144,50,85)(12,168,84,113,30,131,51,93)(13,155,64,121,31,139,52,101)(14,163,65,108,32,147,53,88)(15,150,66,116,33,134,54,96)(16,158,67,124,34,142,55,104)(17,166,68,111,35,129,56,91)(18,153,69,119,36,137,57,99)(19,161,70,106,37,145,58,86)(20,148,71,114,38,132,59,94)(21,156,72,122,39,140,60,102)>;
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), (1,39)(2,38)(3,37)(4,36)(5,35)(6,34)(7,33)(8,32)(9,31)(10,30)(11,29)(12,28)(13,27)(14,26)(15,25)(16,24)(17,23)(18,22)(19,42)(20,41)(21,40)(43,69)(44,68)(45,67)(46,66)(47,65)(48,64)(49,84)(50,83)(51,82)(52,81)(53,80)(54,79)(55,78)(56,77)(57,76)(58,75)(59,74)(60,73)(61,72)(62,71)(63,70)(85,112)(86,111)(87,110)(88,109)(89,108)(90,107)(91,106)(92,126)(93,125)(94,124)(95,123)(96,122)(97,121)(98,120)(99,119)(100,118)(101,117)(102,116)(103,115)(104,114)(105,113)(127,163)(128,162)(129,161)(130,160)(131,159)(132,158)(133,157)(134,156)(135,155)(136,154)(137,153)(138,152)(139,151)(140,150)(141,149)(142,148)(143,168)(144,167)(145,166)(146,165)(147,164), (1,164,73,109,40,127,61,89)(2,151,74,117,41,135,62,97)(3,159,75,125,42,143,63,105)(4,167,76,112,22,130,43,92)(5,154,77,120,23,138,44,100)(6,162,78,107,24,146,45,87)(7,149,79,115,25,133,46,95)(8,157,80,123,26,141,47,103)(9,165,81,110,27,128,48,90)(10,152,82,118,28,136,49,98)(11,160,83,126,29,144,50,85)(12,168,84,113,30,131,51,93)(13,155,64,121,31,139,52,101)(14,163,65,108,32,147,53,88)(15,150,66,116,33,134,54,96)(16,158,67,124,34,142,55,104)(17,166,68,111,35,129,56,91)(18,153,69,119,36,137,57,99)(19,161,70,106,37,145,58,86)(20,148,71,114,38,132,59,94)(21,156,72,122,39,140,60,102) );
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)], [(1,39),(2,38),(3,37),(4,36),(5,35),(6,34),(7,33),(8,32),(9,31),(10,30),(11,29),(12,28),(13,27),(14,26),(15,25),(16,24),(17,23),(18,22),(19,42),(20,41),(21,40),(43,69),(44,68),(45,67),(46,66),(47,65),(48,64),(49,84),(50,83),(51,82),(52,81),(53,80),(54,79),(55,78),(56,77),(57,76),(58,75),(59,74),(60,73),(61,72),(62,71),(63,70),(85,112),(86,111),(87,110),(88,109),(89,108),(90,107),(91,106),(92,126),(93,125),(94,124),(95,123),(96,122),(97,121),(98,120),(99,119),(100,118),(101,117),(102,116),(103,115),(104,114),(105,113),(127,163),(128,162),(129,161),(130,160),(131,159),(132,158),(133,157),(134,156),(135,155),(136,154),(137,153),(138,152),(139,151),(140,150),(141,149),(142,148),(143,168),(144,167),(145,166),(146,165),(147,164)], [(1,164,73,109,40,127,61,89),(2,151,74,117,41,135,62,97),(3,159,75,125,42,143,63,105),(4,167,76,112,22,130,43,92),(5,154,77,120,23,138,44,100),(6,162,78,107,24,146,45,87),(7,149,79,115,25,133,46,95),(8,157,80,123,26,141,47,103),(9,165,81,110,27,128,48,90),(10,152,82,118,28,136,49,98),(11,160,83,126,29,144,50,85),(12,168,84,113,30,131,51,93),(13,155,64,121,31,139,52,101),(14,163,65,108,32,147,53,88),(15,150,66,116,33,134,54,96),(16,158,67,124,34,142,55,104),(17,166,68,111,35,129,56,91),(18,153,69,119,36,137,57,99),(19,161,70,106,37,145,58,86),(20,148,71,114,38,132,59,94),(21,156,72,122,39,140,60,102)]])
60 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 | 21B | 21C | 24A | 24B | 24C | 24D | 28A | ··· | 28F | 42A | 42B | 42C | 56A | ··· | 56L | 84A | ··· | 84F |
| 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 | 21 | 24 | 24 | 24 | 24 | 28 | ··· | 28 | 42 | 42 | 42 | 56 | ··· | 56 | 84 | ··· | 84 |
| size | 1 | 1 | 21 | 21 | 2 | 1 | 1 | 21 | 21 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 7 | 7 | 7 | 7 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 14 | 14 | 14 | 14 | 2 | ··· | 2 | 4 | 4 | 4 | 6 | ··· | 6 | 4 | ··· | 4 |
60 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | ||||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C8 | S3 | D6 | D7 | C4×S3 | D14 | S3×C8 | C4×D7 | C8×D7 | S3×D7 | D21⋊C4 | D21⋊C8 |
| kernel | D21⋊C8 | C7×C3⋊C8 | C3×C7⋊C8 | C4×D21 | Dic21 | D42 | D21 | C7⋊C8 | C28 | C3⋊C8 | C14 | C12 | C7 | C6 | C3 | C4 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 8 | 1 | 1 | 3 | 2 | 3 | 4 | 6 | 12 | 3 | 3 | 6 |
Matrix representation of D21⋊C8 ►in GL4(𝔽337) generated by
| 144 | 34 | 0 | 0 |
| 303 | 336 | 0 | 0 |
| 0 | 0 | 335 | 196 |
| 0 | 0 | 294 | 1 |
| 193 | 303 | 0 | 0 |
| 144 | 144 | 0 | 0 |
| 0 | 0 | 336 | 0 |
| 0 | 0 | 294 | 1 |
| 85 | 0 | 0 | 0 |
| 0 | 85 | 0 | 0 |
| 0 | 0 | 1 | 141 |
| 0 | 0 | 0 | 336 |
G:=sub<GL(4,GF(337))| [144,303,0,0,34,336,0,0,0,0,335,294,0,0,196,1],[193,144,0,0,303,144,0,0,0,0,336,294,0,0,0,1],[85,0,0,0,0,85,0,0,0,0,1,0,0,0,141,336] >;
D21⋊C8 in GAP, Magma, Sage, TeX
D_{21}\rtimes C_8 % in TeX
G:=Group("D21:C8"); // GroupNames label
G:=SmallGroup(336,25);
// by ID
G=gap.SmallGroup(336,25);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-3,-7,24,31,50,490,10373]);
// Polycyclic
G:=Group<a,b,c|a^21=b^2=c^8=1,b*a*b=a^-1,c*a*c^-1=a^8,c*b*c^-1=a^7*b>;
// generators/relations
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