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G = S3×Dic11order 264 = 23·3·11

Direct product of S3 and Dic11

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

Aliases: S3×Dic11, D6.D11, C6.2D22, C22.2D6, Dic333C2, C66.2C22, (S3×C11)⋊C4, C113(C4×S3), C332(C2×C4), (S3×C22).C2, C2.2(S3×D11), C31(C2×Dic11), (C3×Dic11)⋊1C2, SmallGroup(264,6)

Series: Derived Chief Lower central Upper central

C1C33 — S3×Dic11
C1C11C33C66C3×Dic11 — S3×Dic11
C33 — S3×Dic11
C1C2

Generators and relations for S3×Dic11
 G = < a,b,c,d | a3=b2=c22=1, d2=c11, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >

3C2
3C2
3C22
11C4
33C4
3C22
3C22
33C2×C4
11C12
11Dic3
3Dic11
3C2×C22
11C4×S3
3C2×Dic11

Smallest permutation representation of S3×Dic11
On 132 points
Generators in S132
(1 99 116)(2 100 117)(3 101 118)(4 102 119)(5 103 120)(6 104 121)(7 105 122)(8 106 123)(9 107 124)(10 108 125)(11 109 126)(12 110 127)(13 89 128)(14 90 129)(15 91 130)(16 92 131)(17 93 132)(18 94 111)(19 95 112)(20 96 113)(21 97 114)(22 98 115)(23 60 86)(24 61 87)(25 62 88)(26 63 67)(27 64 68)(28 65 69)(29 66 70)(30 45 71)(31 46 72)(32 47 73)(33 48 74)(34 49 75)(35 50 76)(36 51 77)(37 52 78)(38 53 79)(39 54 80)(40 55 81)(41 56 82)(42 57 83)(43 58 84)(44 59 85)
(1 12)(2 13)(3 14)(4 15)(5 16)(6 17)(7 18)(8 19)(9 20)(10 21)(11 22)(23 75)(24 76)(25 77)(26 78)(27 79)(28 80)(29 81)(30 82)(31 83)(32 84)(33 85)(34 86)(35 87)(36 88)(37 67)(38 68)(39 69)(40 70)(41 71)(42 72)(43 73)(44 74)(45 56)(46 57)(47 58)(48 59)(49 60)(50 61)(51 62)(52 63)(53 64)(54 65)(55 66)(89 117)(90 118)(91 119)(92 120)(93 121)(94 122)(95 123)(96 124)(97 125)(98 126)(99 127)(100 128)(101 129)(102 130)(103 131)(104 132)(105 111)(106 112)(107 113)(108 114)(109 115)(110 116)
(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)
(1 63 12 52)(2 62 13 51)(3 61 14 50)(4 60 15 49)(5 59 16 48)(6 58 17 47)(7 57 18 46)(8 56 19 45)(9 55 20 66)(10 54 21 65)(11 53 22 64)(23 130 34 119)(24 129 35 118)(25 128 36 117)(26 127 37 116)(27 126 38 115)(28 125 39 114)(29 124 40 113)(30 123 41 112)(31 122 42 111)(32 121 43 132)(33 120 44 131)(67 110 78 99)(68 109 79 98)(69 108 80 97)(70 107 81 96)(71 106 82 95)(72 105 83 94)(73 104 84 93)(74 103 85 92)(75 102 86 91)(76 101 87 90)(77 100 88 89)

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

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

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

42 conjugacy classes

class 1 2A2B2C 3 4A4B4C4D 6 11A···11E12A12B22A···22E22F···22O33A···33E66A···66E
order122234444611···11121222···2222···2233···3366···66
size113321111333322···222222···26···64···44···4

42 irreducible representations

dim1111122222244
type+++++++-++-
imageC1C2C2C2C4S3D6D11C4×S3Dic11D22S3×D11S3×Dic11
kernelS3×Dic11C3×Dic11Dic33S3×C22S3×C11Dic11C22D6C11S3C6C2C1
# reps11114115210555

Matrix representation of S3×Dic11 in GL4(𝔽397) generated by

039600
139600
0010
0001
,
396100
0100
003960
000396
,
396000
039600
00335396
0033395
,
334000
033400
0045322
00117352
G:=sub<GL(4,GF(397))| [0,1,0,0,396,396,0,0,0,0,1,0,0,0,0,1],[396,0,0,0,1,1,0,0,0,0,396,0,0,0,0,396],[396,0,0,0,0,396,0,0,0,0,335,333,0,0,396,95],[334,0,0,0,0,334,0,0,0,0,45,117,0,0,322,352] >;

S3×Dic11 in GAP, Magma, Sage, TeX

S_3\times {\rm Dic}_{11}
% in TeX

G:=Group("S3xDic11");
// GroupNames label

G:=SmallGroup(264,6);
// by ID

G=gap.SmallGroup(264,6);
# by ID

G:=PCGroup([5,-2,-2,-2,-3,-11,20,168,6004]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^2=c^22=1,d^2=c^11,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 S3×Dic11 in TeX

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