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G = C3×D56order 336 = 24·3·7

Direct product of C3 and D56

direct product, metacyclic, supersoluble, monomial, 2-hyperelementary

Aliases: C3×D56, C215D8, C569C6, C243D7, C1683C2, D287C6, C6.14D28, C42.24D4, C12.53D14, C84.60C22, C74(C3×D8), C81(C3×D7), C4.9(C6×D7), (C3×D28)⋊7C2, C2.4(C3×D28), C28.32(C2×C6), C14.18(C3×D4), SmallGroup(336,61)

Series: Derived Chief Lower central Upper central

C1C28 — C3×D56
C1C7C14C28C84C3×D28 — C3×D56
C7C14C28 — C3×D56
C1C6C12C24

Generators and relations for C3×D56
 G = < a,b,c | a3=b56=c2=1, ab=ba, ac=ca, cbc=b-1 >

28C2
28C2
14C22
14C22
28C6
28C6
4D7
4D7
7D4
7D4
14C2×C6
14C2×C6
2D14
2D14
4C3×D7
4C3×D7
7D8
7C3×D4
7C3×D4
2C6×D7
2C6×D7
7C3×D8

Smallest permutation representation of C3×D56
On 168 points
Generators in S168
(1 57 164)(2 58 165)(3 59 166)(4 60 167)(5 61 168)(6 62 113)(7 63 114)(8 64 115)(9 65 116)(10 66 117)(11 67 118)(12 68 119)(13 69 120)(14 70 121)(15 71 122)(16 72 123)(17 73 124)(18 74 125)(19 75 126)(20 76 127)(21 77 128)(22 78 129)(23 79 130)(24 80 131)(25 81 132)(26 82 133)(27 83 134)(28 84 135)(29 85 136)(30 86 137)(31 87 138)(32 88 139)(33 89 140)(34 90 141)(35 91 142)(36 92 143)(37 93 144)(38 94 145)(39 95 146)(40 96 147)(41 97 148)(42 98 149)(43 99 150)(44 100 151)(45 101 152)(46 102 153)(47 103 154)(48 104 155)(49 105 156)(50 106 157)(51 107 158)(52 108 159)(53 109 160)(54 110 161)(55 111 162)(56 112 163)
(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 56)(2 55)(3 54)(4 53)(5 52)(6 51)(7 50)(8 49)(9 48)(10 47)(11 46)(12 45)(13 44)(14 43)(15 42)(16 41)(17 40)(18 39)(19 38)(20 37)(21 36)(22 35)(23 34)(24 33)(25 32)(26 31)(27 30)(28 29)(57 112)(58 111)(59 110)(60 109)(61 108)(62 107)(63 106)(64 105)(65 104)(66 103)(67 102)(68 101)(69 100)(70 99)(71 98)(72 97)(73 96)(74 95)(75 94)(76 93)(77 92)(78 91)(79 90)(80 89)(81 88)(82 87)(83 86)(84 85)(113 158)(114 157)(115 156)(116 155)(117 154)(118 153)(119 152)(120 151)(121 150)(122 149)(123 148)(124 147)(125 146)(126 145)(127 144)(128 143)(129 142)(130 141)(131 140)(132 139)(133 138)(134 137)(135 136)(159 168)(160 167)(161 166)(162 165)(163 164)

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

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

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

93 conjugacy classes

class 1 2A2B2C3A3B 4 6A6B6C6D6E6F7A7B7C8A8B12A12B14A14B14C21A···21F24A24B24C24D28A···28F42A···42F56A···56L84A···84L168A···168X
order122233466666677788121214141421···212424242428···2842···4256···5684···84168···168
size112828112112828282822222222222···222222···22···22···22···22···2

93 irreducible representations

dim111111222222222222
type+++++++++
imageC1C2C2C3C6C6D4D7D8C3×D4D14C3×D7C3×D8D28C6×D7D56C3×D28C3×D56
kernelC3×D56C168C3×D28D56C56D28C42C24C21C14C12C8C7C6C4C3C2C1
# reps112224132236466121224

Matrix representation of C3×D56 in GL4(𝔽337) generated by

1000
0100
002080
000208
,
2991800
31917700
0032413
00324324
,
2991800
3133800
00324324
0032413
G:=sub<GL(4,GF(337))| [1,0,0,0,0,1,0,0,0,0,208,0,0,0,0,208],[299,319,0,0,18,177,0,0,0,0,324,324,0,0,13,324],[299,313,0,0,18,38,0,0,0,0,324,324,0,0,324,13] >;

C3×D56 in GAP, Magma, Sage, TeX

C_3\times D_{56}
% in TeX

G:=Group("C3xD56");
// GroupNames label

G:=SmallGroup(336,61);
// by ID

G=gap.SmallGroup(336,61);
# by ID

G:=PCGroup([6,-2,-2,-3,-2,-2,-7,169,223,867,69,10373]);
// Polycyclic

G:=Group<a,b,c|a^3=b^56=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

Export

Subgroup lattice of C3×D56 in TeX

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