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G = S3xC46order 276 = 22·3·23

Direct product of C46 and S3

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

Aliases: S3xC46, C6:C46, C138:3C2, C69:4C22, C3:(C2xC46), SmallGroup(276,8)

Series: Derived Chief Lower central Upper central

C1C3 — S3xC46
C1C3C69S3xC23 — S3xC46
C3 — S3xC46
C1C46

Generators and relations for S3xC46
 G = < a,b,c | a46=b3=c2=1, ab=ba, ac=ca, cbc=b-1 >

Subgroups: 32 in 20 conjugacy classes, 14 normal (10 characteristic)
Quotients: C1, C2, C22, S3, D6, C23, C46, C2xC46, S3xC23, S3xC46
3C2
3C2
3C22
3C46
3C46
3C2xC46

Smallest permutation representation of S3xC46
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)
(1 130 62)(2 131 63)(3 132 64)(4 133 65)(5 134 66)(6 135 67)(7 136 68)(8 137 69)(9 138 70)(10 93 71)(11 94 72)(12 95 73)(13 96 74)(14 97 75)(15 98 76)(16 99 77)(17 100 78)(18 101 79)(19 102 80)(20 103 81)(21 104 82)(22 105 83)(23 106 84)(24 107 85)(25 108 86)(26 109 87)(27 110 88)(28 111 89)(29 112 90)(30 113 91)(31 114 92)(32 115 47)(33 116 48)(34 117 49)(35 118 50)(36 119 51)(37 120 52)(38 121 53)(39 122 54)(40 123 55)(41 124 56)(42 125 57)(43 126 58)(44 127 59)(45 128 60)(46 129 61)
(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)(47 138)(48 93)(49 94)(50 95)(51 96)(52 97)(53 98)(54 99)(55 100)(56 101)(57 102)(58 103)(59 104)(60 105)(61 106)(62 107)(63 108)(64 109)(65 110)(66 111)(67 112)(68 113)(69 114)(70 115)(71 116)(72 117)(73 118)(74 119)(75 120)(76 121)(77 122)(78 123)(79 124)(80 125)(81 126)(82 127)(83 128)(84 129)(85 130)(86 131)(87 132)(88 133)(89 134)(90 135)(91 136)(92 137)

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

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

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

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+++++
imageC1C2C2C23C46C46S3D6S3xC23S3xC46
kernelS3xC46S3xC23C138D6S3C6C46C23C2C1
# reps121224422112222

Matrix representation of S3xC46 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] >;

S3xC46 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 S3xC46 in TeX

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