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## G = S3×Dic11order 264 = 23·3·11

### Direct product of S3 and Dic11

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

 Derived series C1 — C33 — S3×Dic11
 Chief series C1 — C11 — C33 — C66 — C3×Dic11 — S3×Dic11
 Lower central C33 — S3×Dic11
 Upper central C1 — C2

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 >

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 2A 2B 2C 3 4A 4B 4C 4D 6 11A ··· 11E 12A 12B 22A ··· 22E 22F ··· 22O 33A ··· 33E 66A ··· 66E order 1 2 2 2 3 4 4 4 4 6 11 ··· 11 12 12 22 ··· 22 22 ··· 22 33 ··· 33 66 ··· 66 size 1 1 3 3 2 11 11 33 33 2 2 ··· 2 22 22 2 ··· 2 6 ··· 6 4 ··· 4 4 ··· 4

42 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 4 4 type + + + + + + + - + + - image C1 C2 C2 C2 C4 S3 D6 D11 C4×S3 Dic11 D22 S3×D11 S3×Dic11 kernel S3×Dic11 C3×Dic11 Dic33 S3×C22 S3×C11 Dic11 C22 D6 C11 S3 C6 C2 C1 # reps 1 1 1 1 4 1 1 5 2 10 5 5 5

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

 0 396 0 0 1 396 0 0 0 0 1 0 0 0 0 1
,
 396 1 0 0 0 1 0 0 0 0 396 0 0 0 0 396
,
 396 0 0 0 0 396 0 0 0 0 335 396 0 0 333 95
,
 334 0 0 0 0 334 0 0 0 0 45 322 0 0 117 352
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

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