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G = S3xDic13order 312 = 23·3·13

Direct product of S3 and Dic13

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

Aliases: S3xDic13, D6.D13, C6.2D26, C26.2D6, Dic39:3C2, C78.2C22, C13:4(C4xS3), C39:5(C2xC4), (S3xC26).C2, (S3xC13):2C4, C2.2(S3xD13), C3:1(C2xDic13), (C3xDic13):1C2, SmallGroup(312,16)

Series: Derived Chief Lower central Upper central

C1C39 — S3xDic13
C1C13C39C78C3xDic13 — S3xDic13
C39 — S3xDic13
C1C2

Generators and relations for S3xDic13
 G = < a,b,c,d | a3=b2=c26=1, d2=c13, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 172 in 32 conjugacy classes, 18 normal (14 characteristic)
Quotients: C1, C2, C4, C22, S3, C2xC4, D6, C4xS3, D13, Dic13, D26, C2xDic13, S3xD13, S3xDic13
3C2
3C2
3C22
13C4
39C4
3C26
3C26
39C2xC4
13C12
13Dic3
3Dic13
3C2xC26
13C4xS3
3C2xDic13

Smallest permutation representation of S3xDic13
On 156 points
Generators in S156
(1 123 76)(2 124 77)(3 125 78)(4 126 53)(5 127 54)(6 128 55)(7 129 56)(8 130 57)(9 105 58)(10 106 59)(11 107 60)(12 108 61)(13 109 62)(14 110 63)(15 111 64)(16 112 65)(17 113 66)(18 114 67)(19 115 68)(20 116 69)(21 117 70)(22 118 71)(23 119 72)(24 120 73)(25 121 74)(26 122 75)(27 102 131)(28 103 132)(29 104 133)(30 79 134)(31 80 135)(32 81 136)(33 82 137)(34 83 138)(35 84 139)(36 85 140)(37 86 141)(38 87 142)(39 88 143)(40 89 144)(41 90 145)(42 91 146)(43 92 147)(44 93 148)(45 94 149)(46 95 150)(47 96 151)(48 97 152)(49 98 153)(50 99 154)(51 100 155)(52 101 156)
(1 14)(2 15)(3 16)(4 17)(5 18)(6 19)(7 20)(8 21)(9 22)(10 23)(11 24)(12 25)(13 26)(27 144)(28 145)(29 146)(30 147)(31 148)(32 149)(33 150)(34 151)(35 152)(36 153)(37 154)(38 155)(39 156)(40 131)(41 132)(42 133)(43 134)(44 135)(45 136)(46 137)(47 138)(48 139)(49 140)(50 141)(51 142)(52 143)(53 113)(54 114)(55 115)(56 116)(57 117)(58 118)(59 119)(60 120)(61 121)(62 122)(63 123)(64 124)(65 125)(66 126)(67 127)(68 128)(69 129)(70 130)(71 105)(72 106)(73 107)(74 108)(75 109)(76 110)(77 111)(78 112)(79 92)(80 93)(81 94)(82 95)(83 96)(84 97)(85 98)(86 99)(87 100)(88 101)(89 102)(90 103)(91 104)
(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)
(1 89 14 102)(2 88 15 101)(3 87 16 100)(4 86 17 99)(5 85 18 98)(6 84 19 97)(7 83 20 96)(8 82 21 95)(9 81 22 94)(10 80 23 93)(11 79 24 92)(12 104 25 91)(13 103 26 90)(27 76 40 63)(28 75 41 62)(29 74 42 61)(30 73 43 60)(31 72 44 59)(32 71 45 58)(33 70 46 57)(34 69 47 56)(35 68 48 55)(36 67 49 54)(37 66 50 53)(38 65 51 78)(39 64 52 77)(105 136 118 149)(106 135 119 148)(107 134 120 147)(108 133 121 146)(109 132 122 145)(110 131 123 144)(111 156 124 143)(112 155 125 142)(113 154 126 141)(114 153 127 140)(115 152 128 139)(116 151 129 138)(117 150 130 137)

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

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

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

48 conjugacy classes

class 1 2A2B2C 3 4A4B4C4D 6 12A12B13A···13F26A···26F26G···26R39A···39F78A···78F
order1222344446121213···1326···2626···2639···3978···78
size1133213133939226262···22···26···64···44···4

48 irreducible representations

dim1111122222244
type+++++++-++-
imageC1C2C2C2C4S3D6C4xS3D13Dic13D26S3xD13S3xDic13
kernelS3xDic13C3xDic13Dic39S3xC26S3xC13Dic13C26C13D6S3C6C2C1
# reps11114112612666

Matrix representation of S3xDic13 in GL4(F157) generated by

1000
0100
000156
001156
,
156000
015600
0001
0010
,
015600
14200
0010
0001
,
1405400
1401700
0010
0001
G:=sub<GL(4,GF(157))| [1,0,0,0,0,1,0,0,0,0,0,1,0,0,156,156],[156,0,0,0,0,156,0,0,0,0,0,1,0,0,1,0],[0,1,0,0,156,42,0,0,0,0,1,0,0,0,0,1],[140,140,0,0,54,17,0,0,0,0,1,0,0,0,0,1] >;

S3xDic13 in GAP, Magma, Sage, TeX

S_3\times {\rm Dic}_{13}
% in TeX

G:=Group("S3xDic13");
// GroupNames label

G:=SmallGroup(312,16);
// by ID

G=gap.SmallGroup(312,16);
# by ID

G:=PCGroup([5,-2,-2,-2,-3,-13,20,168,7204]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^2=c^26=1,d^2=c^13,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations

Export

Subgroup lattice of S3xDic13 in TeX

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