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## G = S3×D37order 444 = 22·3·37

### Direct product of S3 and D37

Aliases: S3×D37, C371D6, C31D74, D111⋊C2, C111⋊C22, (S3×C37)⋊C2, (C3×D37)⋊C2, SmallGroup(444,11)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C111 — S3×D37
 Chief series C1 — C37 — C111 — C3×D37 — S3×D37
 Lower central C111 — S3×D37
 Upper central C1

Generators and relations for S3×D37
G = < a,b,c,d | a3=b2=c37=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

3C2
37C2
111C2
111C22
37C6
37S3
3C74
3D37
37D6
3D74

Smallest permutation representation of S3×D37
On 111 points
Generators in S111
(1 58 84)(2 59 85)(3 60 86)(4 61 87)(5 62 88)(6 63 89)(7 64 90)(8 65 91)(9 66 92)(10 67 93)(11 68 94)(12 69 95)(13 70 96)(14 71 97)(15 72 98)(16 73 99)(17 74 100)(18 38 101)(19 39 102)(20 40 103)(21 41 104)(22 42 105)(23 43 106)(24 44 107)(25 45 108)(26 46 109)(27 47 110)(28 48 111)(29 49 75)(30 50 76)(31 51 77)(32 52 78)(33 53 79)(34 54 80)(35 55 81)(36 56 82)(37 57 83)
(38 101)(39 102)(40 103)(41 104)(42 105)(43 106)(44 107)(45 108)(46 109)(47 110)(48 111)(49 75)(50 76)(51 77)(52 78)(53 79)(54 80)(55 81)(56 82)(57 83)(58 84)(59 85)(60 86)(61 87)(62 88)(63 89)(64 90)(65 91)(66 92)(67 93)(68 94)(69 95)(70 96)(71 97)(72 98)(73 99)(74 100)
(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)
(1 37)(2 36)(3 35)(4 34)(5 33)(6 32)(7 31)(8 30)(9 29)(10 28)(11 27)(12 26)(13 25)(14 24)(15 23)(16 22)(17 21)(18 20)(38 40)(41 74)(42 73)(43 72)(44 71)(45 70)(46 69)(47 68)(48 67)(49 66)(50 65)(51 64)(52 63)(53 62)(54 61)(55 60)(56 59)(57 58)(75 92)(76 91)(77 90)(78 89)(79 88)(80 87)(81 86)(82 85)(83 84)(93 111)(94 110)(95 109)(96 108)(97 107)(98 106)(99 105)(100 104)(101 103)

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

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

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

60 conjugacy classes

 class 1 2A 2B 2C 3 6 37A ··· 37R 74A ··· 74R 111A ··· 111R order 1 2 2 2 3 6 37 ··· 37 74 ··· 74 111 ··· 111 size 1 3 37 111 2 74 2 ··· 2 6 ··· 6 4 ··· 4

60 irreducible representations

 dim 1 1 1 1 2 2 2 2 4 type + + + + + + + + + image C1 C2 C2 C2 S3 D6 D37 D74 S3×D37 kernel S3×D37 S3×C37 C3×D37 D111 D37 C37 S3 C3 C1 # reps 1 1 1 1 1 1 18 18 18

Matrix representation of S3×D37 in GL4(𝔽223) generated by

 1 0 0 0 0 1 0 0 0 0 1 32 0 0 202 221
,
 222 0 0 0 0 222 0 0 0 0 222 191 0 0 0 1
,
 11 1 0 0 121 153 0 0 0 0 1 0 0 0 0 1
,
 110 83 0 0 2 113 0 0 0 0 1 0 0 0 0 1
G:=sub<GL(4,GF(223))| [1,0,0,0,0,1,0,0,0,0,1,202,0,0,32,221],[222,0,0,0,0,222,0,0,0,0,222,0,0,0,191,1],[11,121,0,0,1,153,0,0,0,0,1,0,0,0,0,1],[110,2,0,0,83,113,0,0,0,0,1,0,0,0,0,1] >;

S3×D37 in GAP, Magma, Sage, TeX

S_3\times D_{37}
% in TeX

G:=Group("S3xD37");
// GroupNames label

G:=SmallGroup(444,11);
// by ID

G=gap.SmallGroup(444,11);
# by ID

G:=PCGroup([4,-2,-2,-3,-37,54,6915]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^2=c^37=d^2=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations

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