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## G = D14⋊Dic3order 336 = 24·3·7

### The semidirect product of D14 and Dic3 acting via Dic3/C6=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C42 — D14⋊Dic3
 Chief series C1 — C7 — C21 — C42 — C2×C42 — C2×C6×D7 — D14⋊Dic3
 Lower central C21 — C42 — D14⋊Dic3
 Upper central C1 — C22

Generators and relations for D14⋊Dic3
G = < a,b,c,d | a14=b2=c6=1, d2=c3, bab=a-1, ac=ca, ad=da, bc=cb, dbd-1=a7b, dcd-1=c-1 >

Subgroups: 348 in 68 conjugacy classes, 30 normal (26 characteristic)
C1, C2, C2, C3, C4, C22, C22, C6, C6, C7, C2×C4, C23, Dic3, C2×C6, C2×C6, D7, C14, C22⋊C4, C21, C2×Dic3, C2×Dic3, C22×C6, Dic7, C28, D14, D14, C2×C14, C3×D7, C42, C6.D4, C2×Dic7, C2×C28, C22×D7, C7×Dic3, Dic21, C6×D7, C6×D7, C2×C42, D14⋊C4, Dic3×C14, C2×Dic21, C2×C6×D7, D14⋊Dic3
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, Dic3, D6, D7, C22⋊C4, C2×Dic3, C3⋊D4, D14, C6.D4, C4×D7, D28, C7⋊D4, S3×D7, D14⋊C4, Dic3×D7, C21⋊D4, C3⋊D28, D14⋊Dic3

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

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

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

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

54 conjugacy classes

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

54 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + - + + + + + - - + image C1 C2 C2 C2 C4 S3 D4 Dic3 D6 D7 C3⋊D4 D14 C4×D7 D28 C7⋊D4 S3×D7 Dic3×D7 C21⋊D4 C3⋊D28 kernel D14⋊Dic3 Dic3×C14 C2×Dic21 C2×C6×D7 C6×D7 C22×D7 C42 D14 C2×C14 C2×Dic3 C14 C2×C6 C6 C6 C6 C22 C2 C2 C2 # reps 1 1 1 1 4 1 2 2 1 3 4 3 6 6 6 3 3 3 3

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

 34 228 0 0 303 0 0 0 0 0 336 0 0 0 0 336
,
 336 0 0 0 145 1 0 0 0 0 336 0 0 0 279 1
,
 336 0 0 0 0 336 0 0 0 0 208 0 0 0 298 128
,
 195 177 0 0 284 142 0 0 0 0 148 53 0 0 159 189
G:=sub<GL(4,GF(337))| [34,303,0,0,228,0,0,0,0,0,336,0,0,0,0,336],[336,145,0,0,0,1,0,0,0,0,336,279,0,0,0,1],[336,0,0,0,0,336,0,0,0,0,208,298,0,0,0,128],[195,284,0,0,177,142,0,0,0,0,148,159,0,0,53,189] >;

D14⋊Dic3 in GAP, Magma, Sage, TeX

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

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

G:=SmallGroup(336,42);
// by ID

G=gap.SmallGroup(336,42);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-3,-7,121,31,490,10373]);
// Polycyclic

G:=Group<a,b,c,d|a^14=b^2=c^6=1,d^2=c^3,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=a^7*b,d*c*d^-1=c^-1>;
// generators/relations

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