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## G = S3×Dic14order 336 = 24·3·7

### Direct product of S3 and Dic14

Series: Derived Chief Lower central Upper central

 Derived series C1 — C42 — S3×Dic14
 Chief series C1 — C7 — C21 — C42 — C3×Dic7 — S3×Dic7 — S3×Dic14
 Lower central C21 — C42 — S3×Dic14
 Upper central C1 — C2 — C4

Generators and relations for S3×Dic14
G = < a,b,c,d | a3=b2=c28=1, d2=c14, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 356 in 76 conjugacy classes, 36 normal (22 characteristic)
C1, C2, C2, C3, C4, C4, C22, S3, C6, C7, C2×C4, Q8, Dic3, Dic3, C12, C12, D6, C14, C14, C2×Q8, C21, Dic6, C4×S3, C4×S3, C3×Q8, Dic7, Dic7, C28, C28, C2×C14, S3×C7, C42, S3×Q8, Dic14, Dic14, C2×Dic7, C2×C28, C7×Dic3, C3×Dic7, Dic21, C84, S3×C14, C2×Dic14, S3×Dic7, C21⋊Q8, C3×Dic14, S3×C28, Dic42, S3×Dic14
Quotients: C1, C2, C22, S3, Q8, C23, D6, D7, C2×Q8, C22×S3, D14, S3×Q8, Dic14, C22×D7, S3×D7, C2×Dic14, C2×S3×D7, S3×Dic14

Smallest permutation representation of S3×Dic14
On 168 points
Generators in S168
(1 161 39)(2 162 40)(3 163 41)(4 164 42)(5 165 43)(6 166 44)(7 167 45)(8 168 46)(9 141 47)(10 142 48)(11 143 49)(12 144 50)(13 145 51)(14 146 52)(15 147 53)(16 148 54)(17 149 55)(18 150 56)(19 151 29)(20 152 30)(21 153 31)(22 154 32)(23 155 33)(24 156 34)(25 157 35)(26 158 36)(27 159 37)(28 160 38)(57 124 110)(58 125 111)(59 126 112)(60 127 85)(61 128 86)(62 129 87)(63 130 88)(64 131 89)(65 132 90)(66 133 91)(67 134 92)(68 135 93)(69 136 94)(70 137 95)(71 138 96)(72 139 97)(73 140 98)(74 113 99)(75 114 100)(76 115 101)(77 116 102)(78 117 103)(79 118 104)(80 119 105)(81 120 106)(82 121 107)(83 122 108)(84 123 109)
(29 151)(30 152)(31 153)(32 154)(33 155)(34 156)(35 157)(36 158)(37 159)(38 160)(39 161)(40 162)(41 163)(42 164)(43 165)(44 166)(45 167)(46 168)(47 141)(48 142)(49 143)(50 144)(51 145)(52 146)(53 147)(54 148)(55 149)(56 150)(57 110)(58 111)(59 112)(60 85)(61 86)(62 87)(63 88)(64 89)(65 90)(66 91)(67 92)(68 93)(69 94)(70 95)(71 96)(72 97)(73 98)(74 99)(75 100)(76 101)(77 102)(78 103)(79 104)(80 105)(81 106)(82 107)(83 108)(84 109)
(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)
(1 138 15 124)(2 137 16 123)(3 136 17 122)(4 135 18 121)(5 134 19 120)(6 133 20 119)(7 132 21 118)(8 131 22 117)(9 130 23 116)(10 129 24 115)(11 128 25 114)(12 127 26 113)(13 126 27 140)(14 125 28 139)(29 81 43 67)(30 80 44 66)(31 79 45 65)(32 78 46 64)(33 77 47 63)(34 76 48 62)(35 75 49 61)(36 74 50 60)(37 73 51 59)(38 72 52 58)(39 71 53 57)(40 70 54 84)(41 69 55 83)(42 68 56 82)(85 158 99 144)(86 157 100 143)(87 156 101 142)(88 155 102 141)(89 154 103 168)(90 153 104 167)(91 152 105 166)(92 151 106 165)(93 150 107 164)(94 149 108 163)(95 148 109 162)(96 147 110 161)(97 146 111 160)(98 145 112 159)

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

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

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

51 conjugacy classes

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

51 irreducible representations

 dim 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 S3 Q8 D6 D6 D7 D14 D14 D14 Dic14 S3×Q8 S3×D7 C2×S3×D7 S3×Dic14 kernel S3×Dic14 S3×Dic7 C21⋊Q8 C3×Dic14 S3×C28 Dic42 Dic14 S3×C7 Dic7 C28 C4×S3 Dic3 C12 D6 S3 C7 C4 C2 C1 # reps 1 2 2 1 1 1 1 2 2 1 3 3 3 3 12 1 3 3 6

Matrix representation of S3×Dic14 in GL4(𝔽337) generated by

 336 336 0 0 1 0 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 336 336 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 0 1 0 0 0 0 112 94 0 0 243 219
,
 1 0 0 0 0 1 0 0 0 0 145 322 0 0 211 192
G:=sub<GL(4,GF(337))| [336,1,0,0,336,0,0,0,0,0,1,0,0,0,0,1],[1,336,0,0,0,336,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,112,243,0,0,94,219],[1,0,0,0,0,1,0,0,0,0,145,211,0,0,322,192] >;

S3×Dic14 in GAP, Magma, Sage, TeX

S_3\times {\rm Dic}_{14}
% in TeX

G:=Group("S3xDic14");
// GroupNames label

G:=SmallGroup(336,140);
// by ID

G=gap.SmallGroup(336,140);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-3,-7,48,218,50,490,10373]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^2=c^28=1,d^2=c^14,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations

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