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G = C56⋊S3order 336 = 24·3·7

4th semidirect product of C56 and S3 acting via S3/C3=C2

metacyclic, supersoluble, monomial, 2-hyperelementary

Aliases: C564S3, C245D7, C83D21, C1685C2, D42.1C4, C4.13D42, C28.48D6, C214M4(2), C12.49D14, C84.55C22, Dic21.1C4, C6.6(C4×D7), C21⋊C84C2, C72(C8⋊S3), C14.6(C4×S3), C32(C8⋊D7), C2.3(C4×D21), C42.15(C2×C4), (C4×D21).2C2, SmallGroup(336,91)

Series: Derived Chief Lower central Upper central

C1C42 — C56⋊S3
C1C7C21C42C84C4×D21 — C56⋊S3
C21C42 — C56⋊S3
C1C4C8

Generators and relations for C56⋊S3
 G = < a,b,c | a56=b3=c2=1, ab=ba, cac=a13, cbc=b-1 >

42C2
21C22
21C4
14S3
6D7
21C2×C4
21C8
7Dic3
7D6
3Dic7
3D14
2D21
21M4(2)
7C3⋊C8
7C4×S3
3C4×D7
3C7⋊C8
7C8⋊S3
3C8⋊D7

Smallest permutation representation of C56⋊S3
On 168 points
Generators in S168
(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)
(1 128 100)(2 129 101)(3 130 102)(4 131 103)(5 132 104)(6 133 105)(7 134 106)(8 135 107)(9 136 108)(10 137 109)(11 138 110)(12 139 111)(13 140 112)(14 141 57)(15 142 58)(16 143 59)(17 144 60)(18 145 61)(19 146 62)(20 147 63)(21 148 64)(22 149 65)(23 150 66)(24 151 67)(25 152 68)(26 153 69)(27 154 70)(28 155 71)(29 156 72)(30 157 73)(31 158 74)(32 159 75)(33 160 76)(34 161 77)(35 162 78)(36 163 79)(37 164 80)(38 165 81)(39 166 82)(40 167 83)(41 168 84)(42 113 85)(43 114 86)(44 115 87)(45 116 88)(46 117 89)(47 118 90)(48 119 91)(49 120 92)(50 121 93)(51 122 94)(52 123 95)(53 124 96)(54 125 97)(55 126 98)(56 127 99)
(2 14)(3 27)(4 40)(5 53)(6 10)(7 23)(8 36)(9 49)(11 19)(12 32)(13 45)(16 28)(17 41)(18 54)(20 24)(21 37)(22 50)(25 33)(26 46)(30 42)(31 55)(34 38)(35 51)(39 47)(44 56)(48 52)(57 129)(58 142)(59 155)(60 168)(61 125)(62 138)(63 151)(64 164)(65 121)(66 134)(67 147)(68 160)(69 117)(70 130)(71 143)(72 156)(73 113)(74 126)(75 139)(76 152)(77 165)(78 122)(79 135)(80 148)(81 161)(82 118)(83 131)(84 144)(85 157)(86 114)(87 127)(88 140)(89 153)(90 166)(91 123)(92 136)(93 149)(94 162)(95 119)(96 132)(97 145)(98 158)(99 115)(100 128)(101 141)(102 154)(103 167)(104 124)(105 137)(106 150)(107 163)(108 120)(109 133)(110 146)(111 159)(112 116)

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

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

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

90 conjugacy classes

class 1 2A2B 3 4A4B4C 6 7A7B7C8A8B8C8D12A12B14A14B14C21A···21F24A24B24C24D28A···28F42A···42F56A···56L84A···84L168A···168X
order122344467778888121214141421···212424242428···2842···4256···5684···84168···168
size1142211422222224242222222···222222···22···22···22···22···2

90 irreducible representations

dim1111112222222222222
type++++++++++
imageC1C2C2C2C4C4S3D6D7M4(2)C4×S3D14D21C8⋊S3C4×D7D42C8⋊D7C4×D21C56⋊S3
kernelC56⋊S3C21⋊C8C168C4×D21Dic21D42C56C28C24C21C14C12C8C7C6C4C3C2C1
# reps1111221132236466121224

Matrix representation of C56⋊S3 in GL2(𝔽337) generated by

3713
324292
,
261161
17675
,
10
227336
G:=sub<GL(2,GF(337))| [37,324,13,292],[261,176,161,75],[1,227,0,336] >;

C56⋊S3 in GAP, Magma, Sage, TeX

C_{56}\rtimes S_3
% in TeX

G:=Group("C56:S3");
// GroupNames label

G:=SmallGroup(336,91);
// by ID

G=gap.SmallGroup(336,91);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-3,-7,121,31,50,964,10373]);
// Polycyclic

G:=Group<a,b,c|a^56=b^3=c^2=1,a*b=b*a,c*a*c=a^13,c*b*c=b^-1>;
// generators/relations

Export

Subgroup lattice of C56⋊S3 in TeX

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