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## G = C23×D21order 336 = 24·3·7

### Direct product of C23 and D21

Aliases: C23×D21, C212C24, C422C23, (C2×C6)⋊9D14, C72(S3×C23), C32(C23×D7), (C2×C14)⋊12D6, C62(C22×D7), (C22×C6)⋊3D7, C142(C22×S3), (C22×C42)⋊3C2, (C22×C14)⋊5S3, (C2×C42)⋊10C22, SmallGroup(336,227)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C21 — C23×D21
 Chief series C1 — C7 — C21 — D21 — D42 — C22×D21 — C23×D21
 Lower central C21 — C23×D21
 Upper central C1 — C23

Generators and relations for C23×D21
G = < a,b,c,d,e | a2=b2=c2=d21=e2=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede=d-1 >

Subgroups: 1696 in 268 conjugacy classes, 115 normal (9 characteristic)
C1, C2, C2, C3, C22, C22, S3, C6, C7, C23, C23, D6, C2×C6, D7, C14, C24, C21, C22×S3, C22×C6, D14, C2×C14, D21, C42, S3×C23, C22×D7, C22×C14, D42, C2×C42, C23×D7, C22×D21, C22×C42, C23×D21
Quotients: C1, C2, C22, S3, C23, D6, D7, C24, C22×S3, D14, D21, S3×C23, C22×D7, D42, C23×D7, C22×D21, C23×D21

Smallest permutation representation of C23×D21
On 168 points
Generators in S168
(1 161)(2 162)(3 163)(4 164)(5 165)(6 166)(7 167)(8 168)(9 148)(10 149)(11 150)(12 151)(13 152)(14 153)(15 154)(16 155)(17 156)(18 157)(19 158)(20 159)(21 160)(22 130)(23 131)(24 132)(25 133)(26 134)(27 135)(28 136)(29 137)(30 138)(31 139)(32 140)(33 141)(34 142)(35 143)(36 144)(37 145)(38 146)(39 147)(40 127)(41 128)(42 129)(43 114)(44 115)(45 116)(46 117)(47 118)(48 119)(49 120)(50 121)(51 122)(52 123)(53 124)(54 125)(55 126)(56 106)(57 107)(58 108)(59 109)(60 110)(61 111)(62 112)(63 113)(64 99)(65 100)(66 101)(67 102)(68 103)(69 104)(70 105)(71 85)(72 86)(73 87)(74 88)(75 89)(76 90)(77 91)(78 92)(79 93)(80 94)(81 95)(82 96)(83 97)(84 98)
(1 54)(2 55)(3 56)(4 57)(5 58)(6 59)(7 60)(8 61)(9 62)(10 63)(11 43)(12 44)(13 45)(14 46)(15 47)(16 48)(17 49)(18 50)(19 51)(20 52)(21 53)(22 83)(23 84)(24 64)(25 65)(26 66)(27 67)(28 68)(29 69)(30 70)(31 71)(32 72)(33 73)(34 74)(35 75)(36 76)(37 77)(38 78)(39 79)(40 80)(41 81)(42 82)(85 139)(86 140)(87 141)(88 142)(89 143)(90 144)(91 145)(92 146)(93 147)(94 127)(95 128)(96 129)(97 130)(98 131)(99 132)(100 133)(101 134)(102 135)(103 136)(104 137)(105 138)(106 163)(107 164)(108 165)(109 166)(110 167)(111 168)(112 148)(113 149)(114 150)(115 151)(116 152)(117 153)(118 154)(119 155)(120 156)(121 157)(122 158)(123 159)(124 160)(125 161)(126 162)
(1 23)(2 24)(3 25)(4 26)(5 27)(6 28)(7 29)(8 30)(9 31)(10 32)(11 33)(12 34)(13 35)(14 36)(15 37)(16 38)(17 39)(18 40)(19 41)(20 42)(21 22)(43 73)(44 74)(45 75)(46 76)(47 77)(48 78)(49 79)(50 80)(51 81)(52 82)(53 83)(54 84)(55 64)(56 65)(57 66)(58 67)(59 68)(60 69)(61 70)(62 71)(63 72)(85 112)(86 113)(87 114)(88 115)(89 116)(90 117)(91 118)(92 119)(93 120)(94 121)(95 122)(96 123)(97 124)(98 125)(99 126)(100 106)(101 107)(102 108)(103 109)(104 110)(105 111)(127 157)(128 158)(129 159)(130 160)(131 161)(132 162)(133 163)(134 164)(135 165)(136 166)(137 167)(138 168)(139 148)(140 149)(141 150)(142 151)(143 152)(144 153)(145 154)(146 155)(147 156)
(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)
(1 124)(2 123)(3 122)(4 121)(5 120)(6 119)(7 118)(8 117)(9 116)(10 115)(11 114)(12 113)(13 112)(14 111)(15 110)(16 109)(17 108)(18 107)(19 106)(20 126)(21 125)(22 98)(23 97)(24 96)(25 95)(26 94)(27 93)(28 92)(29 91)(30 90)(31 89)(32 88)(33 87)(34 86)(35 85)(36 105)(37 104)(38 103)(39 102)(40 101)(41 100)(42 99)(43 150)(44 149)(45 148)(46 168)(47 167)(48 166)(49 165)(50 164)(51 163)(52 162)(53 161)(54 160)(55 159)(56 158)(57 157)(58 156)(59 155)(60 154)(61 153)(62 152)(63 151)(64 129)(65 128)(66 127)(67 147)(68 146)(69 145)(70 144)(71 143)(72 142)(73 141)(74 140)(75 139)(76 138)(77 137)(78 136)(79 135)(80 134)(81 133)(82 132)(83 131)(84 130)

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

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

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

96 conjugacy classes

 class 1 2A ··· 2G 2H ··· 2O 3 6A ··· 6G 7A 7B 7C 14A ··· 14U 21A ··· 21F 42A ··· 42AP order 1 2 ··· 2 2 ··· 2 3 6 ··· 6 7 7 7 14 ··· 14 21 ··· 21 42 ··· 42 size 1 1 ··· 1 21 ··· 21 2 2 ··· 2 2 2 2 2 ··· 2 2 ··· 2 2 ··· 2

96 irreducible representations

 dim 1 1 1 2 2 2 2 2 2 type + + + + + + + + + image C1 C2 C2 S3 D6 D7 D14 D21 D42 kernel C23×D21 C22×D21 C22×C42 C22×C14 C2×C14 C22×C6 C2×C6 C23 C22 # reps 1 14 1 1 7 3 21 6 42

Matrix representation of C23×D21 in GL6(𝔽43)

 42 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 42 0 0 0 0 0 0 42 0 0 0 0 0 0 42 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 42 0 0 0 0 0 0 42 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 20 35 0 0 0 0 8 42 0 0 0 0 0 0 12 18 0 0 0 0 24 11
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 23 8 0 0 0 0 20 20 0 0 0 0 0 0 27 7 0 0 0 0 25 16

G:=sub<GL(6,GF(43))| [42,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,42,0,0,0,0,0,0,42,0,0,0,0,0,0,42,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,42,0,0,0,0,0,0,42,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,20,8,0,0,0,0,35,42,0,0,0,0,0,0,12,24,0,0,0,0,18,11],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,23,20,0,0,0,0,8,20,0,0,0,0,0,0,27,25,0,0,0,0,7,16] >;

C23×D21 in GAP, Magma, Sage, TeX

C_2^3\times D_{21}
% in TeX

G:=Group("C2^3xD21");
// GroupNames label

G:=SmallGroup(336,227);
// by ID

G=gap.SmallGroup(336,227);
# by ID

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

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

׿
×
𝔽