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## G = C2×C4×D21order 336 = 24·3·7

### Direct product of C2×C4 and D21

Series: Derived Chief Lower central Upper central

 Derived series C1 — C21 — C2×C4×D21
 Chief series C1 — C7 — C21 — C42 — D42 — C22×D21 — C2×C4×D21
 Lower central C21 — C2×C4×D21
 Upper central C1 — C2×C4

Generators and relations for C2×C4×D21
G = < a,b,c,d | a2=b4=c21=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

Subgroups: 640 in 108 conjugacy classes, 51 normal (19 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C7, C2×C4, C2×C4, C23, Dic3, C12, D6, C2×C6, D7, C14, C14, C22×C4, C21, C4×S3, C2×Dic3, C2×C12, C22×S3, Dic7, C28, D14, C2×C14, D21, C42, C42, S3×C2×C4, C4×D7, C2×Dic7, C2×C28, C22×D7, Dic21, C84, D42, C2×C42, C2×C4×D7, C4×D21, C2×Dic21, C2×C84, C22×D21, C2×C4×D21
Quotients: C1, C2, C4, C22, S3, C2×C4, C23, D6, D7, C22×C4, C4×S3, C22×S3, D14, D21, S3×C2×C4, C4×D7, C22×D7, D42, C2×C4×D7, C4×D21, C22×D21, C2×C4×D21

Smallest permutation representation of C2×C4×D21
On 168 points
Generators in S168
(1 93)(2 94)(3 95)(4 96)(5 97)(6 98)(7 99)(8 100)(9 101)(10 102)(11 103)(12 104)(13 105)(14 85)(15 86)(16 87)(17 88)(18 89)(19 90)(20 91)(21 92)(22 115)(23 116)(24 117)(25 118)(26 119)(27 120)(28 121)(29 122)(30 123)(31 124)(32 125)(33 126)(34 106)(35 107)(36 108)(37 109)(38 110)(39 111)(40 112)(41 113)(42 114)(43 140)(44 141)(45 142)(46 143)(47 144)(48 145)(49 146)(50 147)(51 127)(52 128)(53 129)(54 130)(55 131)(56 132)(57 133)(58 134)(59 135)(60 136)(61 137)(62 138)(63 139)(64 163)(65 164)(66 165)(67 166)(68 167)(69 168)(70 148)(71 149)(72 150)(73 151)(74 152)(75 153)(76 154)(77 155)(78 156)(79 157)(80 158)(81 159)(82 160)(83 161)(84 162)
(1 65 32 61)(2 66 33 62)(3 67 34 63)(4 68 35 43)(5 69 36 44)(6 70 37 45)(7 71 38 46)(8 72 39 47)(9 73 40 48)(10 74 41 49)(11 75 42 50)(12 76 22 51)(13 77 23 52)(14 78 24 53)(15 79 25 54)(16 80 26 55)(17 81 27 56)(18 82 28 57)(19 83 29 58)(20 84 30 59)(21 64 31 60)(85 156 117 129)(86 157 118 130)(87 158 119 131)(88 159 120 132)(89 160 121 133)(90 161 122 134)(91 162 123 135)(92 163 124 136)(93 164 125 137)(94 165 126 138)(95 166 106 139)(96 167 107 140)(97 168 108 141)(98 148 109 142)(99 149 110 143)(100 150 111 144)(101 151 112 145)(102 152 113 146)(103 153 114 147)(104 154 115 127)(105 155 116 128)
(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 31)(2 30)(3 29)(4 28)(5 27)(6 26)(7 25)(8 24)(9 23)(10 22)(11 42)(12 41)(13 40)(14 39)(15 38)(16 37)(17 36)(18 35)(19 34)(20 33)(21 32)(43 82)(44 81)(45 80)(46 79)(47 78)(48 77)(49 76)(50 75)(51 74)(52 73)(53 72)(54 71)(55 70)(56 69)(57 68)(58 67)(59 66)(60 65)(61 64)(62 84)(63 83)(85 111)(86 110)(87 109)(88 108)(89 107)(90 106)(91 126)(92 125)(93 124)(94 123)(95 122)(96 121)(97 120)(98 119)(99 118)(100 117)(101 116)(102 115)(103 114)(104 113)(105 112)(127 152)(128 151)(129 150)(130 149)(131 148)(132 168)(133 167)(134 166)(135 165)(136 164)(137 163)(138 162)(139 161)(140 160)(141 159)(142 158)(143 157)(144 156)(145 155)(146 154)(147 153)

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

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

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

96 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3 4A 4B 4C 4D 4E 4F 4G 4H 6A 6B 6C 7A 7B 7C 12A 12B 12C 12D 14A ··· 14I 21A ··· 21F 28A ··· 28L 42A ··· 42R 84A ··· 84X order 1 2 2 2 2 2 2 2 3 4 4 4 4 4 4 4 4 6 6 6 7 7 7 12 12 12 12 14 ··· 14 21 ··· 21 28 ··· 28 42 ··· 42 84 ··· 84 size 1 1 1 1 21 21 21 21 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

96 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 type + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C4 S3 D6 D6 D7 C4×S3 D14 D14 D21 C4×D7 D42 D42 C4×D21 kernel C2×C4×D21 C4×D21 C2×Dic21 C2×C84 C22×D21 D42 C2×C28 C28 C2×C14 C2×C12 C14 C12 C2×C6 C2×C4 C6 C4 C22 C2 # reps 1 4 1 1 1 8 1 2 1 3 4 6 3 6 12 12 6 24

Matrix representation of C2×C4×D21 in GL4(𝔽337) generated by

 1 0 0 0 0 1 0 0 0 0 336 0 0 0 0 336
,
 189 0 0 0 0 189 0 0 0 0 148 0 0 0 0 148
,
 76 110 0 0 117 227 0 0 0 0 70 323 0 0 14 262
,
 226 33 0 0 76 111 0 0 0 0 304 110 0 0 33 33
G:=sub<GL(4,GF(337))| [1,0,0,0,0,1,0,0,0,0,336,0,0,0,0,336],[189,0,0,0,0,189,0,0,0,0,148,0,0,0,0,148],[76,117,0,0,110,227,0,0,0,0,70,14,0,0,323,262],[226,76,0,0,33,111,0,0,0,0,304,33,0,0,110,33] >;

C2×C4×D21 in GAP, Magma, Sage, TeX

C_2\times C_4\times D_{21}
% in TeX

G:=Group("C2xC4xD21");
// GroupNames label

G:=SmallGroup(336,195);
// by ID

G=gap.SmallGroup(336,195);
# by ID

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

G:=Group<a,b,c,d|a^2=b^4=c^21=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations

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