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G = S3×C46order 276 = 22·3·23

Direct product of C46 and S3

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

Aliases: S3×C46, C6⋊C46, C1383C2, C694C22, C3⋊(C2×C46), SmallGroup(276,8)

Series: Derived Chief Lower central Upper central

C1C3 — S3×C46
C1C3C69S3×C23 — S3×C46
C3 — S3×C46
C1C46

Generators and relations for S3×C46
 G = < a,b,c | a46=b3=c2=1, ab=ba, ac=ca, cbc=b-1 >

3C2
3C2
3C22
3C46
3C46
3C2×C46

Smallest permutation representation of S3×C46
On 138 points
Generators in S138
(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)
(1 68 125)(2 69 126)(3 70 127)(4 71 128)(5 72 129)(6 73 130)(7 74 131)(8 75 132)(9 76 133)(10 77 134)(11 78 135)(12 79 136)(13 80 137)(14 81 138)(15 82 93)(16 83 94)(17 84 95)(18 85 96)(19 86 97)(20 87 98)(21 88 99)(22 89 100)(23 90 101)(24 91 102)(25 92 103)(26 47 104)(27 48 105)(28 49 106)(29 50 107)(30 51 108)(31 52 109)(32 53 110)(33 54 111)(34 55 112)(35 56 113)(36 57 114)(37 58 115)(38 59 116)(39 60 117)(40 61 118)(41 62 119)(42 63 120)(43 64 121)(44 65 122)(45 66 123)(46 67 124)
(1 24)(2 25)(3 26)(4 27)(5 28)(6 29)(7 30)(8 31)(9 32)(10 33)(11 34)(12 35)(13 36)(14 37)(15 38)(16 39)(17 40)(18 41)(19 42)(20 43)(21 44)(22 45)(23 46)(47 127)(48 128)(49 129)(50 130)(51 131)(52 132)(53 133)(54 134)(55 135)(56 136)(57 137)(58 138)(59 93)(60 94)(61 95)(62 96)(63 97)(64 98)(65 99)(66 100)(67 101)(68 102)(69 103)(70 104)(71 105)(72 106)(73 107)(74 108)(75 109)(76 110)(77 111)(78 112)(79 113)(80 114)(81 115)(82 116)(83 117)(84 118)(85 119)(86 120)(87 121)(88 122)(89 123)(90 124)(91 125)(92 126)

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

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

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

138 conjugacy classes

class 1 2A2B2C 3  6 23A···23V46A···46V46W···46BN69A···69V138A···138V
order12223623···2346···4646···4669···69138···138
size1133221···11···13···32···22···2

138 irreducible representations

dim1111112222
type+++++
imageC1C2C2C23C46C46S3D6S3×C23S3×C46
kernelS3×C46S3×C23C138D6S3C6C46C23C2C1
# reps121224422112222

Matrix representation of S3×C46 in GL2(𝔽47) generated by

430
043
,
012
4346
,
112
046
G:=sub<GL(2,GF(47))| [43,0,0,43],[0,43,12,46],[1,0,12,46] >;

S3×C46 in GAP, Magma, Sage, TeX

S_3\times C_{46}
% in TeX

G:=Group("S3xC46");
// GroupNames label

G:=SmallGroup(276,8);
// by ID

G=gap.SmallGroup(276,8);
# by ID

G:=PCGroup([4,-2,-2,-23,-3,2947]);
// Polycyclic

G:=Group<a,b,c|a^46=b^3=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

Export

Subgroup lattice of S3×C46 in TeX

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