Copied to
clipboard

## G = S3×C2×C28order 336 = 24·3·7

### Direct product of C2×C28 and S3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — S3×C2×C28
 Chief series C1 — C3 — C6 — C42 — S3×C14 — S3×C2×C14 — S3×C2×C28
 Lower central C3 — S3×C2×C28
 Upper central C1 — C2×C28

Generators and relations for S3×C2×C28
G = < a,b,c,d | a2=b28=c3=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

Subgroups: 184 in 108 conjugacy classes, 70 normal (22 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C7, C2×C4, C2×C4, C23, Dic3, C12, D6, C2×C6, C14, C14, C14, C22×C4, C21, C4×S3, C2×Dic3, C2×C12, C22×S3, C28, C28, C2×C14, C2×C14, S3×C7, C42, C42, S3×C2×C4, C2×C28, C2×C28, C22×C14, C7×Dic3, C84, S3×C14, C2×C42, C22×C28, S3×C28, Dic3×C14, C2×C84, S3×C2×C14, S3×C2×C28
Quotients: C1, C2, C4, C22, S3, C7, C2×C4, C23, D6, C14, C22×C4, C4×S3, C22×S3, C28, C2×C14, S3×C7, S3×C2×C4, C2×C28, C22×C14, S3×C14, C22×C28, S3×C28, S3×C2×C14, S3×C2×C28

Smallest permutation representation of S3×C2×C28
On 168 points
Generators in S168
(1 77)(2 78)(3 79)(4 80)(5 81)(6 82)(7 83)(8 84)(9 57)(10 58)(11 59)(12 60)(13 61)(14 62)(15 63)(16 64)(17 65)(18 66)(19 67)(20 68)(21 69)(22 70)(23 71)(24 72)(25 73)(26 74)(27 75)(28 76)(29 86)(30 87)(31 88)(32 89)(33 90)(34 91)(35 92)(36 93)(37 94)(38 95)(39 96)(40 97)(41 98)(42 99)(43 100)(44 101)(45 102)(46 103)(47 104)(48 105)(49 106)(50 107)(51 108)(52 109)(53 110)(54 111)(55 112)(56 85)(113 158)(114 159)(115 160)(116 161)(117 162)(118 163)(119 164)(120 165)(121 166)(122 167)(123 168)(124 141)(125 142)(126 143)(127 144)(128 145)(129 146)(130 147)(131 148)(132 149)(133 150)(134 151)(135 152)(136 153)(137 154)(138 155)(139 156)(140 157)
(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 93 154)(2 94 155)(3 95 156)(4 96 157)(5 97 158)(6 98 159)(7 99 160)(8 100 161)(9 101 162)(10 102 163)(11 103 164)(12 104 165)(13 105 166)(14 106 167)(15 107 168)(16 108 141)(17 109 142)(18 110 143)(19 111 144)(20 112 145)(21 85 146)(22 86 147)(23 87 148)(24 88 149)(25 89 150)(26 90 151)(27 91 152)(28 92 153)(29 130 70)(30 131 71)(31 132 72)(32 133 73)(33 134 74)(34 135 75)(35 136 76)(36 137 77)(37 138 78)(38 139 79)(39 140 80)(40 113 81)(41 114 82)(42 115 83)(43 116 84)(44 117 57)(45 118 58)(46 119 59)(47 120 60)(48 121 61)(49 122 62)(50 123 63)(51 124 64)(52 125 65)(53 126 66)(54 127 67)(55 128 68)(56 129 69)
(29 130)(30 131)(31 132)(32 133)(33 134)(34 135)(35 136)(36 137)(37 138)(38 139)(39 140)(40 113)(41 114)(42 115)(43 116)(44 117)(45 118)(46 119)(47 120)(48 121)(49 122)(50 123)(51 124)(52 125)(53 126)(54 127)(55 128)(56 129)(85 146)(86 147)(87 148)(88 149)(89 150)(90 151)(91 152)(92 153)(93 154)(94 155)(95 156)(96 157)(97 158)(98 159)(99 160)(100 161)(101 162)(102 163)(103 164)(104 165)(105 166)(106 167)(107 168)(108 141)(109 142)(110 143)(111 144)(112 145)

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

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

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

168 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3 4A 4B 4C 4D 4E 4F 4G 4H 6A 6B 6C 7A ··· 7F 12A 12B 12C 12D 14A ··· 14R 14S ··· 14AP 21A ··· 21F 28A ··· 28X 28Y ··· 28AV 42A ··· 42R 84A ··· 84X order 1 2 2 2 2 2 2 2 3 4 4 4 4 4 4 4 4 6 6 6 7 ··· 7 12 12 12 12 14 ··· 14 14 ··· 14 21 ··· 21 28 ··· 28 28 ··· 28 42 ··· 42 84 ··· 84 size 1 1 1 1 3 3 3 3 2 1 1 1 1 3 3 3 3 2 2 2 1 ··· 1 2 2 2 2 1 ··· 1 3 ··· 3 2 ··· 2 1 ··· 1 3 ··· 3 2 ··· 2 2 ··· 2

168 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 type + + + + + + + + image C1 C2 C2 C2 C2 C4 C7 C14 C14 C14 C14 C28 S3 D6 D6 C4×S3 S3×C7 S3×C14 S3×C14 S3×C28 kernel S3×C2×C28 S3×C28 Dic3×C14 C2×C84 S3×C2×C14 S3×C14 S3×C2×C4 C4×S3 C2×Dic3 C2×C12 C22×S3 D6 C2×C28 C28 C2×C14 C14 C2×C4 C4 C22 C2 # reps 1 4 1 1 1 8 6 24 6 6 6 48 1 2 1 4 6 12 6 24

Matrix representation of S3×C2×C28 in GL3(𝔽337) generated by

 336 0 0 0 336 0 0 0 336
,
 36 0 0 0 42 0 0 0 42
,
 1 0 0 0 336 336 0 1 0
,
 336 0 0 0 1 0 0 336 336
G:=sub<GL(3,GF(337))| [336,0,0,0,336,0,0,0,336],[36,0,0,0,42,0,0,0,42],[1,0,0,0,336,1,0,336,0],[336,0,0,0,1,336,0,0,336] >;

S3×C2×C28 in GAP, Magma, Sage, TeX

S_3\times C_2\times C_{28}
% in TeX

G:=Group("S3xC2xC28");
// GroupNames label

G:=SmallGroup(336,185);
// by ID

G=gap.SmallGroup(336,185);
# by ID

G:=PCGroup([6,-2,-2,-2,-7,-2,-3,266,8069]);
// Polycyclic

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

׿
×
𝔽