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## G = Dic3.D14order 336 = 24·3·7

### 6th non-split extension by Dic3 of D14 acting via D14/D7=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C42 — Dic3.D14
 Chief series C1 — C7 — C21 — C42 — C3×Dic7 — S3×Dic7 — Dic3.D14
 Lower central C21 — C42 — Dic3.D14
 Upper central C1 — C2 — C22

Generators and relations for Dic3.D14
G = < a,b,c,d | a42=c2=d2=1, b2=a21, bab-1=a13, cac=a29, ad=da, bc=cb, bd=db, dcd=a21c >

Subgroups: 428 in 80 conjugacy classes, 32 normal (all characteristic)
C1, C2, C2, C3, C4, C22, C22, S3, C6, C6, C7, C2×C4, D4, Q8, Dic3, Dic3, C12, D6, D6, C2×C6, D7, C14, C14, C4○D4, C21, Dic6, C4×S3, D12, C3⋊D4, C3⋊D4, C2×C12, Dic7, Dic7, C28, D14, C2×C14, C2×C14, S3×C7, D21, C42, C42, C4○D12, Dic14, C4×D7, C2×Dic7, C2×Dic7, C7⋊D4, C7×D4, C7×Dic3, C3×Dic7, Dic21, S3×C14, D42, C2×C42, D42D7, S3×Dic7, D21⋊C4, C7⋊D12, C21⋊Q8, C6×Dic7, C7×C3⋊D4, C217D4, Dic3.D14
Quotients: C1, C2, C22, S3, C23, D6, D7, C4○D4, C22×S3, D14, C4○D12, C22×D7, S3×D7, D42D7, C2×S3×D7, Dic3.D14

Smallest permutation representation of Dic3.D14
On 168 points
Generators in S168
(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)
(1 134 22 155)(2 147 23 168)(3 160 24 139)(4 131 25 152)(5 144 26 165)(6 157 27 136)(7 128 28 149)(8 141 29 162)(9 154 30 133)(10 167 31 146)(11 138 32 159)(12 151 33 130)(13 164 34 143)(14 135 35 156)(15 148 36 127)(16 161 37 140)(17 132 38 153)(18 145 39 166)(19 158 40 137)(20 129 41 150)(21 142 42 163)(43 99 64 120)(44 112 65 91)(45 125 66 104)(46 96 67 117)(47 109 68 88)(48 122 69 101)(49 93 70 114)(50 106 71 85)(51 119 72 98)(52 90 73 111)(53 103 74 124)(54 116 75 95)(55 87 76 108)(56 100 77 121)(57 113 78 92)(58 126 79 105)(59 97 80 118)(60 110 81 89)(61 123 82 102)(62 94 83 115)(63 107 84 86)
(1 99)(2 86)(3 115)(4 102)(5 89)(6 118)(7 105)(8 92)(9 121)(10 108)(11 95)(12 124)(13 111)(14 98)(15 85)(16 114)(17 101)(18 88)(19 117)(20 104)(21 91)(22 120)(23 107)(24 94)(25 123)(26 110)(27 97)(28 126)(29 113)(30 100)(31 87)(32 116)(33 103)(34 90)(35 119)(36 106)(37 93)(38 122)(39 109)(40 96)(41 125)(42 112)(43 155)(44 142)(45 129)(46 158)(47 145)(48 132)(49 161)(50 148)(51 135)(52 164)(53 151)(54 138)(55 167)(56 154)(57 141)(58 128)(59 157)(60 144)(61 131)(62 160)(63 147)(64 134)(65 163)(66 150)(67 137)(68 166)(69 153)(70 140)(71 127)(72 156)(73 143)(74 130)(75 159)(76 146)(77 133)(78 162)(79 149)(80 136)(81 165)(82 152)(83 139)(84 168)
(1 64)(2 65)(3 66)(4 67)(5 68)(6 69)(7 70)(8 71)(9 72)(10 73)(11 74)(12 75)(13 76)(14 77)(15 78)(16 79)(17 80)(18 81)(19 82)(20 83)(21 84)(22 43)(23 44)(24 45)(25 46)(26 47)(27 48)(28 49)(29 50)(30 51)(31 52)(32 53)(33 54)(34 55)(35 56)(36 57)(37 58)(38 59)(39 60)(40 61)(41 62)(42 63)(85 141)(86 142)(87 143)(88 144)(89 145)(90 146)(91 147)(92 148)(93 149)(94 150)(95 151)(96 152)(97 153)(98 154)(99 155)(100 156)(101 157)(102 158)(103 159)(104 160)(105 161)(106 162)(107 163)(108 164)(109 165)(110 166)(111 167)(112 168)(113 127)(114 128)(115 129)(116 130)(117 131)(118 132)(119 133)(120 134)(121 135)(122 136)(123 137)(124 138)(125 139)(126 140)

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

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

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

45 conjugacy classes

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

45 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + + + + + + + - + image C1 C2 C2 C2 C2 C2 C2 C2 S3 D6 D6 D7 C4○D4 D14 D14 D14 C4○D12 S3×D7 D4⋊2D7 C2×S3×D7 Dic3.D14 kernel Dic3.D14 S3×Dic7 D21⋊C4 C7⋊D12 C21⋊Q8 C6×Dic7 C7×C3⋊D4 C21⋊7D4 C2×Dic7 Dic7 C2×C14 C3⋊D4 C21 Dic3 D6 C2×C6 C7 C22 C3 C2 C1 # reps 1 1 1 1 1 1 1 1 1 2 1 3 2 3 3 3 4 3 3 3 6

Matrix representation of Dic3.D14 in GL4(𝔽337) generated by

 0 1 0 0 336 1 0 0 0 0 228 336 0 0 229 336
,
 189 0 0 0 0 189 0 0 0 0 228 336 0 0 85 109
,
 322 322 0 0 307 15 0 0 0 0 1 0 0 0 0 1
,
 198 278 0 0 59 139 0 0 0 0 336 0 0 0 0 336
G:=sub<GL(4,GF(337))| [0,336,0,0,1,1,0,0,0,0,228,229,0,0,336,336],[189,0,0,0,0,189,0,0,0,0,228,85,0,0,336,109],[322,307,0,0,322,15,0,0,0,0,1,0,0,0,0,1],[198,59,0,0,278,139,0,0,0,0,336,0,0,0,0,336] >;

Dic3.D14 in GAP, Magma, Sage, TeX

{\rm Dic}_3.D_{14}
% in TeX

G:=Group("Dic3.D14");
// GroupNames label

G:=SmallGroup(336,155);
// by ID

G=gap.SmallGroup(336,155);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-3,-7,48,116,490,10373]);
// Polycyclic

G:=Group<a,b,c,d|a^42=c^2=d^2=1,b^2=a^21,b*a*b^-1=a^13,c*a*c=a^29,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=a^21*c>;
// generators/relations

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