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G = C8⋊D21order 336 = 24·3·7

2nd semidirect product of C8 and D21 acting via D21/C21=C2

metacyclic, supersoluble, monomial, 2-hyperelementary

Aliases: C562S3, C82D21, C242D7, C1682C2, C4.8D42, C2.3D84, C6.1D28, C217SD16, D84.1C2, C42.19D4, C28.43D6, C14.1D12, Dic421C2, C12.43D14, C84.50C22, C71(C24⋊C2), C31(C56⋊C2), SmallGroup(336,92)

Series: Derived Chief Lower central Upper central

C1C84 — C8⋊D21
C1C7C21C42C84D84 — C8⋊D21
C21C42C84 — C8⋊D21
C1C2C4C8

Generators and relations for C8⋊D21
 G = < a,b,c | a24=b7=c2=1, ab=ba, cac=a11, cbc=b-1 >

84C2
42C22
42C4
28S3
12D7
21Q8
21D4
14Dic3
14D6
6D14
6Dic7
4D21
21SD16
7Dic6
7D12
3D28
3Dic14
2D42
2Dic21
7C24⋊C2
3C56⋊C2

Smallest permutation representation of C8⋊D21
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)
(1 109 143 91 59 26 157)(2 110 144 92 60 27 158)(3 111 121 93 61 28 159)(4 112 122 94 62 29 160)(5 113 123 95 63 30 161)(6 114 124 96 64 31 162)(7 115 125 73 65 32 163)(8 116 126 74 66 33 164)(9 117 127 75 67 34 165)(10 118 128 76 68 35 166)(11 119 129 77 69 36 167)(12 120 130 78 70 37 168)(13 97 131 79 71 38 145)(14 98 132 80 72 39 146)(15 99 133 81 49 40 147)(16 100 134 82 50 41 148)(17 101 135 83 51 42 149)(18 102 136 84 52 43 150)(19 103 137 85 53 44 151)(20 104 138 86 54 45 152)(21 105 139 87 55 46 153)(22 106 140 88 56 47 154)(23 107 141 89 57 48 155)(24 108 142 90 58 25 156)
(1 157)(2 168)(3 155)(4 166)(5 153)(6 164)(7 151)(8 162)(9 149)(10 160)(11 147)(12 158)(13 145)(14 156)(15 167)(16 154)(17 165)(18 152)(19 163)(20 150)(21 161)(22 148)(23 159)(24 146)(25 98)(26 109)(27 120)(28 107)(29 118)(30 105)(31 116)(32 103)(33 114)(34 101)(35 112)(36 99)(37 110)(38 97)(39 108)(40 119)(41 106)(42 117)(43 104)(44 115)(45 102)(46 113)(47 100)(48 111)(49 129)(50 140)(51 127)(52 138)(53 125)(54 136)(55 123)(56 134)(57 121)(58 132)(59 143)(60 130)(61 141)(62 128)(63 139)(64 126)(65 137)(66 124)(67 135)(68 122)(69 133)(70 144)(71 131)(72 142)(73 85)(74 96)(75 83)(76 94)(77 81)(78 92)(80 90)(82 88)(84 86)(87 95)(89 93)

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), (1,109,143,91,59,26,157)(2,110,144,92,60,27,158)(3,111,121,93,61,28,159)(4,112,122,94,62,29,160)(5,113,123,95,63,30,161)(6,114,124,96,64,31,162)(7,115,125,73,65,32,163)(8,116,126,74,66,33,164)(9,117,127,75,67,34,165)(10,118,128,76,68,35,166)(11,119,129,77,69,36,167)(12,120,130,78,70,37,168)(13,97,131,79,71,38,145)(14,98,132,80,72,39,146)(15,99,133,81,49,40,147)(16,100,134,82,50,41,148)(17,101,135,83,51,42,149)(18,102,136,84,52,43,150)(19,103,137,85,53,44,151)(20,104,138,86,54,45,152)(21,105,139,87,55,46,153)(22,106,140,88,56,47,154)(23,107,141,89,57,48,155)(24,108,142,90,58,25,156), (1,157)(2,168)(3,155)(4,166)(5,153)(6,164)(7,151)(8,162)(9,149)(10,160)(11,147)(12,158)(13,145)(14,156)(15,167)(16,154)(17,165)(18,152)(19,163)(20,150)(21,161)(22,148)(23,159)(24,146)(25,98)(26,109)(27,120)(28,107)(29,118)(30,105)(31,116)(32,103)(33,114)(34,101)(35,112)(36,99)(37,110)(38,97)(39,108)(40,119)(41,106)(42,117)(43,104)(44,115)(45,102)(46,113)(47,100)(48,111)(49,129)(50,140)(51,127)(52,138)(53,125)(54,136)(55,123)(56,134)(57,121)(58,132)(59,143)(60,130)(61,141)(62,128)(63,139)(64,126)(65,137)(66,124)(67,135)(68,122)(69,133)(70,144)(71,131)(72,142)(73,85)(74,96)(75,83)(76,94)(77,81)(78,92)(80,90)(82,88)(84,86)(87,95)(89,93)>;

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), (1,109,143,91,59,26,157)(2,110,144,92,60,27,158)(3,111,121,93,61,28,159)(4,112,122,94,62,29,160)(5,113,123,95,63,30,161)(6,114,124,96,64,31,162)(7,115,125,73,65,32,163)(8,116,126,74,66,33,164)(9,117,127,75,67,34,165)(10,118,128,76,68,35,166)(11,119,129,77,69,36,167)(12,120,130,78,70,37,168)(13,97,131,79,71,38,145)(14,98,132,80,72,39,146)(15,99,133,81,49,40,147)(16,100,134,82,50,41,148)(17,101,135,83,51,42,149)(18,102,136,84,52,43,150)(19,103,137,85,53,44,151)(20,104,138,86,54,45,152)(21,105,139,87,55,46,153)(22,106,140,88,56,47,154)(23,107,141,89,57,48,155)(24,108,142,90,58,25,156), (1,157)(2,168)(3,155)(4,166)(5,153)(6,164)(7,151)(8,162)(9,149)(10,160)(11,147)(12,158)(13,145)(14,156)(15,167)(16,154)(17,165)(18,152)(19,163)(20,150)(21,161)(22,148)(23,159)(24,146)(25,98)(26,109)(27,120)(28,107)(29,118)(30,105)(31,116)(32,103)(33,114)(34,101)(35,112)(36,99)(37,110)(38,97)(39,108)(40,119)(41,106)(42,117)(43,104)(44,115)(45,102)(46,113)(47,100)(48,111)(49,129)(50,140)(51,127)(52,138)(53,125)(54,136)(55,123)(56,134)(57,121)(58,132)(59,143)(60,130)(61,141)(62,128)(63,139)(64,126)(65,137)(66,124)(67,135)(68,122)(69,133)(70,144)(71,131)(72,142)(73,85)(74,96)(75,83)(76,94)(77,81)(78,92)(80,90)(82,88)(84,86)(87,95)(89,93) );

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)], [(1,109,143,91,59,26,157),(2,110,144,92,60,27,158),(3,111,121,93,61,28,159),(4,112,122,94,62,29,160),(5,113,123,95,63,30,161),(6,114,124,96,64,31,162),(7,115,125,73,65,32,163),(8,116,126,74,66,33,164),(9,117,127,75,67,34,165),(10,118,128,76,68,35,166),(11,119,129,77,69,36,167),(12,120,130,78,70,37,168),(13,97,131,79,71,38,145),(14,98,132,80,72,39,146),(15,99,133,81,49,40,147),(16,100,134,82,50,41,148),(17,101,135,83,51,42,149),(18,102,136,84,52,43,150),(19,103,137,85,53,44,151),(20,104,138,86,54,45,152),(21,105,139,87,55,46,153),(22,106,140,88,56,47,154),(23,107,141,89,57,48,155),(24,108,142,90,58,25,156)], [(1,157),(2,168),(3,155),(4,166),(5,153),(6,164),(7,151),(8,162),(9,149),(10,160),(11,147),(12,158),(13,145),(14,156),(15,167),(16,154),(17,165),(18,152),(19,163),(20,150),(21,161),(22,148),(23,159),(24,146),(25,98),(26,109),(27,120),(28,107),(29,118),(30,105),(31,116),(32,103),(33,114),(34,101),(35,112),(36,99),(37,110),(38,97),(39,108),(40,119),(41,106),(42,117),(43,104),(44,115),(45,102),(46,113),(47,100),(48,111),(49,129),(50,140),(51,127),(52,138),(53,125),(54,136),(55,123),(56,134),(57,121),(58,132),(59,143),(60,130),(61,141),(62,128),(63,139),(64,126),(65,137),(66,124),(67,135),(68,122),(69,133),(70,144),(71,131),(72,142),(73,85),(74,96),(75,83),(76,94),(77,81),(78,92),(80,90),(82,88),(84,86),(87,95),(89,93)])

87 conjugacy classes

class 1 2A2B 3 4A4B 6 7A7B7C8A8B12A12B14A14B14C21A···21F24A24B24C24D28A···28F42A···42F56A···56L84A···84L168A···168X
order122344677788121214141421···212424242428···2842···4256···5684···84168···168
size11842284222222222222···222222···22···22···22···22···2

87 irreducible representations

dim111122222222222222
type++++++++++++++
imageC1C2C2C2S3D4D6D7SD16D12D14D21C24⋊C2D28D42C56⋊C2D84C8⋊D21
kernelC8⋊D21C168Dic42D84C56C42C28C24C21C14C12C8C7C6C4C3C2C1
# reps111111132236466121224

Matrix representation of C8⋊D21 in GL2(𝔽337) generated by

158209
12852
,
01
336143
,
01
10
G:=sub<GL(2,GF(337))| [158,128,209,52],[0,336,1,143],[0,1,1,0] >;

C8⋊D21 in GAP, Magma, Sage, TeX

C_8\rtimes D_{21}
% in TeX

G:=Group("C8:D21");
// GroupNames label

G:=SmallGroup(336,92);
// by ID

G=gap.SmallGroup(336,92);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-3,-7,73,31,218,50,964,10373]);
// Polycyclic

G:=Group<a,b,c|a^24=b^7=c^2=1,a*b=b*a,c*a*c=a^11,c*b*c=b^-1>;
// generators/relations

Export

Subgroup lattice of C8⋊D21 in TeX

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