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G = C23xD21order 336 = 24·3·7

Direct product of C23 and D21

direct product, metabelian, supersoluble, monomial, A-group, 2-hyperelementary

Aliases: C23xD21, C21:2C24, C42:2C23, (C2xC6):9D14, C7:2(S3xC23), C3:2(C23xD7), (C2xC14):12D6, C6:2(C22xD7), (C22xC6):3D7, C14:2(C22xS3), (C22xC42):3C2, (C22xC14):5S3, (C2xC42):10C22, SmallGroup(336,227)

Series: Derived Chief Lower central Upper central

C1C21 — C23xD21
C1C7C21D21D42C22xD21 — C23xD21
C21 — C23xD21
C1C23

Generators and relations for C23xD21
 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, C2xC6, D7, C14, C24, C21, C22xS3, C22xC6, D14, C2xC14, D21, C42, S3xC23, C22xD7, C22xC14, D42, C2xC42, C23xD7, C22xD21, C22xC42, C23xD21
Quotients: C1, C2, C22, S3, C23, D6, D7, C24, C22xS3, D14, D21, S3xC23, C22xD7, D42, C23xD7, C22xD21, C23xD21

Smallest permutation representation of C23xD21
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···2G2H···2O 3 6A···6G7A7B7C14A···14U21A···21F42A···42AP
order12···22···236···677714···1421···2142···42
size11···121···2122···22222···22···22···2

96 irreducible representations

dim111222222
type+++++++++
imageC1C2C2S3D6D7D14D21D42
kernelC23xD21C22xD21C22xC42C22xC14C2xC14C22xC6C2xC6C23C22
# reps114117321642

Matrix representation of C23xD21 in GL6(F43)

4200000
010000
001000
000100
000010
000001
,
100000
0420000
0042000
0004200
000010
000001
,
100000
010000
0042000
0004200
000010
000001
,
100000
010000
00203500
0084200
00001218
00002411
,
100000
010000
0023800
00202000
0000277
00002516

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] >;

C23xD21 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

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