Copied to
clipboard

## G = C3×C4.Dic7order 336 = 24·3·7

### Direct product of C3 and C4.Dic7

Series: Derived Chief Lower central Upper central

 Derived series C1 — C14 — C3×C4.Dic7
 Chief series C1 — C7 — C14 — C28 — C84 — C3×C7⋊C8 — C3×C4.Dic7
 Lower central C7 — C14 — C3×C4.Dic7
 Upper central C1 — C12 — C2×C12

Generators and relations for C3×C4.Dic7
G = < a,b,c,d | a3=b4=1, c14=b2, d2=b2c7, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=c13 >

Smallest permutation representation of C3×C4.Dic7
On 168 points
Generators in S168
(1 71 43)(2 72 44)(3 73 45)(4 74 46)(5 75 47)(6 76 48)(7 77 49)(8 78 50)(9 79 51)(10 80 52)(11 81 53)(12 82 54)(13 83 55)(14 84 56)(15 57 29)(16 58 30)(17 59 31)(18 60 32)(19 61 33)(20 62 34)(21 63 35)(22 64 36)(23 65 37)(24 66 38)(25 67 39)(26 68 40)(27 69 41)(28 70 42)(85 168 133)(86 141 134)(87 142 135)(88 143 136)(89 144 137)(90 145 138)(91 146 139)(92 147 140)(93 148 113)(94 149 114)(95 150 115)(96 151 116)(97 152 117)(98 153 118)(99 154 119)(100 155 120)(101 156 121)(102 157 122)(103 158 123)(104 159 124)(105 160 125)(106 161 126)(107 162 127)(108 163 128)(109 164 129)(110 165 130)(111 166 131)(112 167 132)
(1 22 15 8)(2 23 16 9)(3 24 17 10)(4 25 18 11)(5 26 19 12)(6 27 20 13)(7 28 21 14)(29 50 43 36)(30 51 44 37)(31 52 45 38)(32 53 46 39)(33 54 47 40)(34 55 48 41)(35 56 49 42)(57 78 71 64)(58 79 72 65)(59 80 73 66)(60 81 74 67)(61 82 75 68)(62 83 76 69)(63 84 77 70)(85 92 99 106)(86 93 100 107)(87 94 101 108)(88 95 102 109)(89 96 103 110)(90 97 104 111)(91 98 105 112)(113 120 127 134)(114 121 128 135)(115 122 129 136)(116 123 130 137)(117 124 131 138)(118 125 132 139)(119 126 133 140)(141 148 155 162)(142 149 156 163)(143 150 157 164)(144 151 158 165)(145 152 159 166)(146 153 160 167)(147 154 161 168)
(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 22 100 15 93 8 86)(2 92 23 85 16 106 9 99)(3 105 24 98 17 91 10 112)(4 90 25 111 18 104 11 97)(5 103 26 96 19 89 12 110)(6 88 27 109 20 102 13 95)(7 101 28 94 21 87 14 108)(29 113 50 134 43 127 36 120)(30 126 51 119 44 140 37 133)(31 139 52 132 45 125 38 118)(32 124 53 117 46 138 39 131)(33 137 54 130 47 123 40 116)(34 122 55 115 48 136 41 129)(35 135 56 128 49 121 42 114)(57 148 78 141 71 162 64 155)(58 161 79 154 72 147 65 168)(59 146 80 167 73 160 66 153)(60 159 81 152 74 145 67 166)(61 144 82 165 75 158 68 151)(62 157 83 150 76 143 69 164)(63 142 84 163 77 156 70 149)

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

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

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

102 conjugacy classes

 class 1 2A 2B 3A 3B 4A 4B 4C 6A 6B 6C 6D 7A 7B 7C 8A 8B 8C 8D 12A 12B 12C 12D 12E 12F 14A ··· 14I 21A ··· 21F 24A ··· 24H 28A ··· 28L 42A ··· 42R 84A ··· 84X order 1 2 2 3 3 4 4 4 6 6 6 6 7 7 7 8 8 8 8 12 12 12 12 12 12 14 ··· 14 21 ··· 21 24 ··· 24 28 ··· 28 42 ··· 42 84 ··· 84 size 1 1 2 1 1 1 1 2 1 1 2 2 2 2 2 14 14 14 14 1 1 1 1 2 2 2 ··· 2 2 ··· 2 14 ··· 14 2 ··· 2 2 ··· 2 2 ··· 2

102 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 type + + + + - + - image C1 C2 C2 C3 C4 C4 C6 C6 C12 C12 D7 M4(2) Dic7 D14 Dic7 C3×D7 C3×M4(2) C3×Dic7 C6×D7 C3×Dic7 C4.Dic7 C3×C4.Dic7 kernel C3×C4.Dic7 C3×C7⋊C8 C2×C84 C4.Dic7 C84 C2×C42 C7⋊C8 C2×C28 C28 C2×C14 C2×C12 C21 C12 C12 C2×C6 C2×C4 C7 C4 C4 C22 C3 C1 # reps 1 2 1 2 2 2 4 2 4 4 3 2 3 3 3 6 4 6 6 6 12 24

Matrix representation of C3×C4.Dic7 in GL2(𝔽337) generated by

 208 0 0 208
,
 189 0 213 148
,
 103 0 81 301
,
 183 335 137 154
G:=sub<GL(2,GF(337))| [208,0,0,208],[189,213,0,148],[103,81,0,301],[183,137,335,154] >;

C3×C4.Dic7 in GAP, Magma, Sage, TeX

C_3\times C_4.{\rm Dic}_7
% in TeX

G:=Group("C3xC4.Dic7");
// GroupNames label

G:=SmallGroup(336,64);
// by ID

G=gap.SmallGroup(336,64);
# by ID

G:=PCGroup([6,-2,-2,-3,-2,-2,-7,72,313,69,10373]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^4=1,c^14=b^2,d^2=b^2*c^7,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=c^13>;
// generators/relations

Export

׿
×
𝔽