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G = C3×C56⋊C2order 336 = 24·3·7

Direct product of C3 and C56⋊C2

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

Aliases: C3×C56⋊C2, C246D7, C1686C2, C5610C6, C218SD16, D28.3C6, C42.23D4, C6.13D28, Dic147C6, C12.52D14, C84.59C22, C82(C3×D7), C4.8(C6×D7), C75(C3×SD16), C2.3(C3×D28), C28.31(C2×C6), (C3×D28).3C2, C14.17(C3×D4), (C3×Dic14)⋊7C2, SmallGroup(336,60)

Series: Derived Chief Lower central Upper central

C1C28 — C3×C56⋊C2
C1C7C14C28C84C3×D28 — C3×C56⋊C2
C7C14C28 — C3×C56⋊C2
C1C6C12C24

Generators and relations for C3×C56⋊C2
 G = < a,b,c | a3=b56=c2=1, ab=ba, ac=ca, cbc=b27 >

28C2
14C22
14C4
28C6
4D7
7Q8
7D4
14C12
14C2×C6
2D14
2Dic7
4C3×D7
7SD16
7C3×Q8
7C3×D4
2C6×D7
2C3×Dic7
7C3×SD16

Smallest permutation representation of C3×C56⋊C2
On 168 points
Generators in S168
(1 59 132)(2 60 133)(3 61 134)(4 62 135)(5 63 136)(6 64 137)(7 65 138)(8 66 139)(9 67 140)(10 68 141)(11 69 142)(12 70 143)(13 71 144)(14 72 145)(15 73 146)(16 74 147)(17 75 148)(18 76 149)(19 77 150)(20 78 151)(21 79 152)(22 80 153)(23 81 154)(24 82 155)(25 83 156)(26 84 157)(27 85 158)(28 86 159)(29 87 160)(30 88 161)(31 89 162)(32 90 163)(33 91 164)(34 92 165)(35 93 166)(36 94 167)(37 95 168)(38 96 113)(39 97 114)(40 98 115)(41 99 116)(42 100 117)(43 101 118)(44 102 119)(45 103 120)(46 104 121)(47 105 122)(48 106 123)(49 107 124)(50 108 125)(51 109 126)(52 110 127)(53 111 128)(54 112 129)(55 57 130)(56 58 131)
(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)
(2 28)(3 55)(4 26)(5 53)(6 24)(7 51)(8 22)(9 49)(10 20)(11 47)(12 18)(13 45)(14 16)(15 43)(17 41)(19 39)(21 37)(23 35)(25 33)(27 31)(30 56)(32 54)(34 52)(36 50)(38 48)(40 46)(42 44)(57 61)(58 88)(60 86)(62 84)(63 111)(64 82)(65 109)(66 80)(67 107)(68 78)(69 105)(70 76)(71 103)(72 74)(73 101)(75 99)(77 97)(79 95)(81 93)(83 91)(85 89)(90 112)(92 110)(94 108)(96 106)(98 104)(100 102)(113 123)(114 150)(115 121)(116 148)(117 119)(118 146)(120 144)(122 142)(124 140)(125 167)(126 138)(127 165)(128 136)(129 163)(130 134)(131 161)(133 159)(135 157)(137 155)(139 153)(141 151)(143 149)(145 147)(152 168)(154 166)(156 164)(158 162)

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

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

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

93 conjugacy classes

class 1 2A2B3A3B4A4B6A6B6C6D7A7B7C8A8B12A12B12C12D14A14B14C21A···21F24A24B24C24D28A···28F42A···42F56A···56L84A···84L168A···168X
order12233446666777881212121214141421···212424242428···2842···4256···5684···84168···168
size112811228112828222222228282222···222222···22···22···22···22···2

93 irreducible representations

dim11111111222222222222
type++++++++
imageC1C2C2C2C3C6C6C6D4D7SD16C3×D4D14C3×D7C3×SD16D28C6×D7C56⋊C2C3×D28C3×C56⋊C2
kernelC3×C56⋊C2C168C3×Dic14C3×D28C56⋊C2C56Dic14D28C42C24C21C14C12C8C7C6C4C3C2C1
# reps11112222132236466121224

Matrix representation of C3×C56⋊C2 in GL2(𝔽337) generated by

2080
0208
,
123177
160159
,
10
143336
G:=sub<GL(2,GF(337))| [208,0,0,208],[123,160,177,159],[1,143,0,336] >;

C3×C56⋊C2 in GAP, Magma, Sage, TeX

C_3\times C_{56}\rtimes C_2
% in TeX

G:=Group("C3xC56:C2");
// GroupNames label

G:=SmallGroup(336,60);
// by ID

G=gap.SmallGroup(336,60);
# by ID

G:=PCGroup([6,-2,-2,-3,-2,-2,-7,169,79,867,69,10373]);
// 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^27>;
// generators/relations

Export

Subgroup lattice of C3×C56⋊C2 in TeX

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