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G = C3×D4⋊D7order 336 = 24·3·7

Direct product of C3 and D4⋊D7

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

Aliases: C3×D4⋊D7, C218D8, D288C6, C42.40D4, C12.36D14, C84.36C22, C7⋊C88C6, D4⋊(C3×D7), C75(C3×D8), (C7×D4)⋊7C6, (C3×D4)⋊4D7, C4.1(C6×D7), (C3×D28)⋊8C2, (D4×C21)⋊4C2, C28.22(C2×C6), C14.23(C3×D4), C6.23(C7⋊D4), (C3×C7⋊C8)⋊8C2, C2.4(C3×C7⋊D4), SmallGroup(336,69)

Series: Derived Chief Lower central Upper central

C1C28 — C3×D4⋊D7
C1C7C14C28C84C3×D28 — C3×D4⋊D7
C7C14C28 — C3×D4⋊D7
C1C6C12C3×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 >

4C2
28C2
2C22
14C22
4C6
28C6
4C14
4D7
7D4
7C8
2C2×C6
14C2×C6
2D14
2C2×C14
4C42
4C3×D7
7D8
7C3×D4
7C24
2C6×D7
2C2×C42
7C3×D8

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 2A2B2C3A3B 4 6A6B6C6D6E6F7A7B7C8A8B12A12B14A14B14C14D···14I21A···21F24A24B24C24D28A28B28C42A···42F42G···42R84A···84F
order122233466666677788121214141414···1421···212424242428282842···4242···4284···84
size11428112114428282221414222224···42···2141414144442···24···44···4

66 irreducible representations

dim11111111222222222244
type+++++++++
imageC1C2C2C2C3C6C6C6D4D7D8C3×D4D14C3×D7C3×D8C7⋊D4C6×D7C3×C7⋊D4D4⋊D7C3×D4⋊D7
kernelC3×D4⋊D7C3×C7⋊C8C3×D28D4×C21D4⋊D7C7⋊C8D28C7×D4C42C3×D4C21C14C12D4C7C6C4C2C3C1
# reps111122221322364661236

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

208000
020800
001280
000128
,
336000
033600
001141
00141336
,
24227400
639500
00311189
0018926
,
22733600
1000
0010
0001
,
22733600
30411000
001141
000336
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

Export

Subgroup lattice of C3×D4⋊D7 in TeX

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