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## G = S3×C51order 306 = 2·32·17

### Direct product of C51 and S3

Aliases: S3×C51, C3⋊C102, C513C6, C321C34, (C3×C51)⋊4C2, SmallGroup(306,6)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — S3×C51
 Chief series C1 — C3 — C51 — C3×C51 — S3×C51
 Lower central C3 — S3×C51
 Upper central C1 — C51

Generators and relations for S3×C51
G = < a,b,c | a51=b3=c2=1, ab=ba, ac=ca, cbc=b-1 >

Smallest permutation representation of S3×C51
On 102 points
Generators in S102
(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)
(1 35 18)(2 36 19)(3 37 20)(4 38 21)(5 39 22)(6 40 23)(7 41 24)(8 42 25)(9 43 26)(10 44 27)(11 45 28)(12 46 29)(13 47 30)(14 48 31)(15 49 32)(16 50 33)(17 51 34)(52 69 86)(53 70 87)(54 71 88)(55 72 89)(56 73 90)(57 74 91)(58 75 92)(59 76 93)(60 77 94)(61 78 95)(62 79 96)(63 80 97)(64 81 98)(65 82 99)(66 83 100)(67 84 101)(68 85 102)
(1 73)(2 74)(3 75)(4 76)(5 77)(6 78)(7 79)(8 80)(9 81)(10 82)(11 83)(12 84)(13 85)(14 86)(15 87)(16 88)(17 89)(18 90)(19 91)(20 92)(21 93)(22 94)(23 95)(24 96)(25 97)(26 98)(27 99)(28 100)(29 101)(30 102)(31 52)(32 53)(33 54)(34 55)(35 56)(36 57)(37 58)(38 59)(39 60)(40 61)(41 62)(42 63)(43 64)(44 65)(45 66)(46 67)(47 68)(48 69)(49 70)(50 71)(51 72)

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

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

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

153 conjugacy classes

 class 1 2 3A 3B 3C 3D 3E 6A 6B 17A ··· 17P 34A ··· 34P 51A ··· 51AF 51AG ··· 51CB 102A ··· 102AF order 1 2 3 3 3 3 3 6 6 17 ··· 17 34 ··· 34 51 ··· 51 51 ··· 51 102 ··· 102 size 1 3 1 1 2 2 2 3 3 1 ··· 1 3 ··· 3 1 ··· 1 2 ··· 2 3 ··· 3

153 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 type + + + image C1 C2 C3 C6 C17 C34 C51 C102 S3 C3×S3 S3×C17 S3×C51 kernel S3×C51 C3×C51 S3×C17 C51 C3×S3 C32 S3 C3 C51 C17 C3 C1 # reps 1 1 2 2 16 16 32 32 1 2 16 32

Matrix representation of S3×C51 in GL2(𝔽103) generated by

 19 0 0 19
,
 46 99 0 56
,
 85 18 45 18
G:=sub<GL(2,GF(103))| [19,0,0,19],[46,0,99,56],[85,45,18,18] >;

S3×C51 in GAP, Magma, Sage, TeX

S_3\times C_{51}
% in TeX

G:=Group("S3xC51");
// GroupNames label

G:=SmallGroup(306,6);
// by ID

G=gap.SmallGroup(306,6);
# by ID

G:=PCGroup([4,-2,-3,-17,-3,3267]);
// Polycyclic

G:=Group<a,b,c|a^51=b^3=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

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