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## G = C3×D4⋊D7order 336 = 24·3·7

### Direct product of C3 and D4⋊D7

Series: Derived Chief Lower central Upper central

 Derived series C1 — C28 — C3×D4⋊D7
 Chief series C1 — C7 — C14 — C28 — C84 — C3×D28 — C3×D4⋊D7
 Lower central C7 — C14 — C28 — C3×D4⋊D7
 Upper central C1 — C6 — C12 — C3×D4

Generators and relations for C3×D4⋊D7
G = < a,b,c,d,e | a3=b4=c2=d7=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=ebe=b-1, bd=db, cd=dc, ece=bc, ede=d-1 >

Smallest permutation representation of C3×D4⋊D7
On 168 points
Generators in S168
(1 57 29)(2 58 30)(3 59 31)(4 60 32)(5 61 33)(6 62 34)(7 63 35)(8 64 36)(9 65 37)(10 66 38)(11 67 39)(12 68 40)(13 69 41)(14 70 42)(15 71 43)(16 72 44)(17 73 45)(18 74 46)(19 75 47)(20 76 48)(21 77 49)(22 78 50)(23 79 51)(24 80 52)(25 81 53)(26 82 54)(27 83 55)(28 84 56)(85 141 113)(86 142 114)(87 143 115)(88 144 116)(89 145 117)(90 146 118)(91 147 119)(92 148 120)(93 149 121)(94 150 122)(95 151 123)(96 152 124)(97 153 125)(98 154 126)(99 155 127)(100 156 128)(101 157 129)(102 158 130)(103 159 131)(104 160 132)(105 161 133)(106 162 134)(107 163 135)(108 164 136)(109 165 137)(110 166 138)(111 167 139)(112 168 140)
(1 22 8 15)(2 23 9 16)(3 24 10 17)(4 25 11 18)(5 26 12 19)(6 27 13 20)(7 28 14 21)(29 50 36 43)(30 51 37 44)(31 52 38 45)(32 53 39 46)(33 54 40 47)(34 55 41 48)(35 56 42 49)(57 78 64 71)(58 79 65 72)(59 80 66 73)(60 81 67 74)(61 82 68 75)(62 83 69 76)(63 84 70 77)(85 99 92 106)(86 100 93 107)(87 101 94 108)(88 102 95 109)(89 103 96 110)(90 104 97 111)(91 105 98 112)(113 127 120 134)(114 128 121 135)(115 129 122 136)(116 130 123 137)(117 131 124 138)(118 132 125 139)(119 133 126 140)(141 155 148 162)(142 156 149 163)(143 157 150 164)(144 158 151 165)(145 159 152 166)(146 160 153 167)(147 161 154 168)
(1 106)(2 107)(3 108)(4 109)(5 110)(6 111)(7 112)(8 99)(9 100)(10 101)(11 102)(12 103)(13 104)(14 105)(15 85)(16 86)(17 87)(18 88)(19 89)(20 90)(21 91)(22 92)(23 93)(24 94)(25 95)(26 96)(27 97)(28 98)(29 134)(30 135)(31 136)(32 137)(33 138)(34 139)(35 140)(36 127)(37 128)(38 129)(39 130)(40 131)(41 132)(42 133)(43 113)(44 114)(45 115)(46 116)(47 117)(48 118)(49 119)(50 120)(51 121)(52 122)(53 123)(54 124)(55 125)(56 126)(57 162)(58 163)(59 164)(60 165)(61 166)(62 167)(63 168)(64 155)(65 156)(66 157)(67 158)(68 159)(69 160)(70 161)(71 141)(72 142)(73 143)(74 144)(75 145)(76 146)(77 147)(78 148)(79 149)(80 150)(81 151)(82 152)(83 153)(84 154)
(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 7)(2 6)(3 5)(8 14)(9 13)(10 12)(15 28)(16 27)(17 26)(18 25)(19 24)(20 23)(21 22)(29 35)(30 34)(31 33)(36 42)(37 41)(38 40)(43 56)(44 55)(45 54)(46 53)(47 52)(48 51)(49 50)(57 63)(58 62)(59 61)(64 70)(65 69)(66 68)(71 84)(72 83)(73 82)(74 81)(75 80)(76 79)(77 78)(85 105)(86 104)(87 103)(88 102)(89 101)(90 100)(91 99)(92 112)(93 111)(94 110)(95 109)(96 108)(97 107)(98 106)(113 133)(114 132)(115 131)(116 130)(117 129)(118 128)(119 127)(120 140)(121 139)(122 138)(123 137)(124 136)(125 135)(126 134)(141 161)(142 160)(143 159)(144 158)(145 157)(146 156)(147 155)(148 168)(149 167)(150 166)(151 165)(152 164)(153 163)(154 162)

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

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

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

66 conjugacy classes

 class 1 2A 2B 2C 3A 3B 4 6A 6B 6C 6D 6E 6F 7A 7B 7C 8A 8B 12A 12B 14A 14B 14C 14D ··· 14I 21A ··· 21F 24A 24B 24C 24D 28A 28B 28C 42A ··· 42F 42G ··· 42R 84A ··· 84F order 1 2 2 2 3 3 4 6 6 6 6 6 6 7 7 7 8 8 12 12 14 14 14 14 ··· 14 21 ··· 21 24 24 24 24 28 28 28 42 ··· 42 42 ··· 42 84 ··· 84 size 1 1 4 28 1 1 2 1 1 4 4 28 28 2 2 2 14 14 2 2 2 2 2 4 ··· 4 2 ··· 2 14 14 14 14 4 4 4 2 ··· 2 4 ··· 4 4 ··· 4

66 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 4 4 type + + + + + + + + + image C1 C2 C2 C2 C3 C6 C6 C6 D4 D7 D8 C3×D4 D14 C3×D7 C3×D8 C7⋊D4 C6×D7 C3×C7⋊D4 D4⋊D7 C3×D4⋊D7 kernel C3×D4⋊D7 C3×C7⋊C8 C3×D28 D4×C21 D4⋊D7 C7⋊C8 D28 C7×D4 C42 C3×D4 C21 C14 C12 D4 C7 C6 C4 C2 C3 C1 # reps 1 1 1 1 2 2 2 2 1 3 2 2 3 6 4 6 6 12 3 6

Matrix representation of C3×D4⋊D7 in GL4(𝔽337) generated by

 208 0 0 0 0 208 0 0 0 0 128 0 0 0 0 128
,
 336 0 0 0 0 336 0 0 0 0 1 141 0 0 141 336
,
 242 274 0 0 63 95 0 0 0 0 311 189 0 0 189 26
,
 227 336 0 0 1 0 0 0 0 0 1 0 0 0 0 1
,
 227 336 0 0 304 110 0 0 0 0 1 141 0 0 0 336
G:=sub<GL(4,GF(337))| [208,0,0,0,0,208,0,0,0,0,128,0,0,0,0,128],[336,0,0,0,0,336,0,0,0,0,1,141,0,0,141,336],[242,63,0,0,274,95,0,0,0,0,311,189,0,0,189,26],[227,1,0,0,336,0,0,0,0,0,1,0,0,0,0,1],[227,304,0,0,336,110,0,0,0,0,1,0,0,0,141,336] >;

C3×D4⋊D7 in GAP, Magma, Sage, TeX

C_3\times D_4\rtimes D_7
% in TeX

G:=Group("C3xD4:D7");
// GroupNames label

G:=SmallGroup(336,69);
// by ID

G=gap.SmallGroup(336,69);
# by ID

G:=PCGroup([6,-2,-2,-3,-2,-2,-7,169,867,441,69,10373]);
// Polycyclic

G:=Group<a,b,c,d,e|a^3=b^4=c^2=d^7=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=e*b*e=b^-1,b*d=d*b,c*d=d*c,e*c*e=b*c,e*d*e=d^-1>;
// generators/relations

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